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All Rights Reserved) readonly def +/FullName (CMSY5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 48 /prime put +readonly def +/FontBBox{21 -944 1448 791}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMTT8 +%!PS-AdobeFont-1.1: CMTT8 1.0 +%%CreationDate: 1991 Aug 20 16:46:05 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMTT8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch true def +end readonly def +/FontName /CMTT8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 44 /comma put +dup 46 /period put +dup 48 /zero put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 54 /six put +dup 58 /colon put +dup 64 /at put +dup 71 /G put +dup 97 /a put +dup 99 /c put +dup 100 /d put +dup 101 /e put +dup 103 /g put +dup 108 /l put +dup 109 /m put +dup 110 /n put +dup 111 /o put +dup 112 /p put +dup 114 /r put +dup 115 /s put +dup 117 /u put +dup 122 /z put +readonly def +/FontBBox{-5 -232 545 699}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSAM7) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSAM7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 26 /arrowtailright put +readonly def +/FontBBox{0 -576 1569 1076}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI8 +%!PS-AdobeFont-1.1: CMMI8 1.100 +%%CreationDate: 1996 Jul 23 07:53:54 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +readonly def +/FontBBox{-24 -250 1110 750}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMR8 +%!PS-AdobeFont-1.1: CMR8 1.0 +%%CreationDate: 1991 Aug 20 16:39:40 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +readonly def +/FontBBox{-36 -250 1070 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 +469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 +2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C +68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 +3645B82392D5CAE11A7CB49D7E2E82DCD485CBA1772CE422BB1D7283AD675B65 +48A7EA0069A883EC1DAA3E1F9ECE7586D6CF0A128CD557C7E5D7AA3EA97EBAD3 +9619D1BFCF4A6D64768741EDEA0A5B0EFBBF347CDCBE2E03D756967A16B613DB +0FC45FA2A3312E0C46A5FD0466AB097C58FFEEC40601B8395E52775D0AFCD7DB +8AB317333110531E5C44A4CB4B5ACD571A1A60960B15E450948A5EEA14DD330F +EA209265DB8E1A1FC80DCD3860323FD26C113B041A88C88A21655878680A4466 +FA10403D24BB97152A49B842C180E4D258C9D48F21D057782D90623116830BA3 +9902B3C5F2F2DD01433B0D7099C07DBDE268D0FFED5169BCD03D48B2F058AD62 +D8678C626DC7A3F352152C99BA963EF95F8AD11DB8B0D351210A17E4C2C55AD8 +9EB64172935D3C20A398F3EEEEC31551966A7438EF3FEE422C6D4E05337620D5 +ACC7B52BED984BFAAD36EF9D20748B05D07BE4414A63975125D272FAD83F76E6 +10FFF8363014BE526D580873C5A42B70FA911EC7B86905F13AFE55EB0273F582 +83158793B8CC296B8DE1DCCF1250FD57CB0E035C7EDA3B0092ED940D37A05493 +2EC54E09B984FCA4AB7D2EA182BCF1263AA244B07EC0EA901C077A059F709F30 +4384CB5FA748F2054FAD9A7A43D4EA427918BD414F766531136B60C3477C6632 +BEFE3897B58C19276A301926C2AEF2756B367319772C9B201C49B4D935A8267B +041D6F1783B6AEA4DAC4F5B3507D7032AA640AAB12E343A4E9BDCF419C04A721 +3888B25AF4E293AACED9A6BDC78E61DA1C424C6503CC1885F762B86FECDA8944 +AA3B20AD6019056E01677B9584A3BE08610A05D7F57FFA236EE7DD39FE334E11 +6BD45CC86918CC84B826A8DBFDA9800D1F2EDD676364921987D436D9AF5BB189 +9410AA5596D044CDA2C20B49056536404F6D1800C3E705462F18ADD8D66D50FB +CD33E184956EA73812A752CFB2A98308ACF6E242BA84435314D591A5C97148FC +F3DEDAA3E509C7FA32E2B10371E4962AB8EE4A628DBB985DA5C677846353F89C +3F097478FBB57E4B3B2AF9244574EBF40E8484C1258A34D3624690F98625818E +333E5843BC99F1B87DFCB0B18280C7A1CAE260DDBA08563F9B0E5762EC5290F4 +BB8434F868D419800A636FBB44E9CB60CB68DB47820CC11D8272D0BC99F069CE +6675ADABE03716956866357AB4F275AB34EFBDED7360FBAB671E6C13C56D1365 +B54672A592F34D794DB0CBDA03F346683B18D7A9EBBC909F8393DBDF8A7ECDDC +52F675DF52C847C3D0101AFF927A1EE158CE6E5AB4D9BD505537DE5DAFC0B836 +8441B089BF714E91ACEBCF040C9FF9012DF5EFEAD1AAA9FDA935BCA92ACFAC3D +D6E1F7A2208259F83A54ABA4F5AC4777FF85BA50A7 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndFont +%%BeginFont: CMR5 +%!PS-AdobeFont-1.1: CMR5 1.00B +%%CreationDate: 1992 Feb 19 19:55:02 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.00B) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 49 /one put +dup 50 /two put +dup 51 /three put +readonly def +/FontBBox{-341 -250 1304 965}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI7 +%!PS-AdobeFont-1.1: CMMI7 1.100 +%%CreationDate: 1996 Jul 23 07:53:53 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI7) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +dup 67 /C put +dup 84 /T put +dup 100 /d put +dup 101 /e put +dup 114 /r put +dup 117 /u put +readonly def +/FontBBox{0 -250 1171 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE +D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +dup 66 /B put +dup 68 /D put +dup 82 /R put +dup 105 /i put +dup 106 /j put +dup 109 /m put +dup 110 /n put +dup 114 /r put +dup 120 /x put +dup 121 /y put +readonly def +/FontBBox{11 -250 1241 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 91 /bracketleft put +dup 93 /bracketright put +readonly def +/FontBBox{-20 -250 1193 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 +469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /minus put +dup 17 /equivalence put +dup 18 /reflexsubset put +dup 40 /arrowdblleft put +dup 41 /arrowdblright put +dup 44 /arrowdblboth put +dup 50 /element put +dup 56 /universal put +dup 57 /existential put +dup 58 /logicalnot put +dup 59 /emptyset put +dup 93 /unionmulti put +dup 102 /braceleft put +dup 103 /braceright put +dup 104 /angbracketleft put +dup 105 /angbracketright put +dup 106 /bar put +dup 107 /bardbl put +readonly def +/FontBBox{-30 -958 1146 777}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 3 /asteriskmath put +dup 48 /prime put +dup 106 /bar put +readonly def +/FontBBox{-4 -948 1329 786}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: MSAM10 +%!PS-AdobeFont-1.1: MSAM10 2.1 +%%CreationDate: 1993 Sep 17 09:05:00 +% Math Symbol fonts were designed by the American Mathematical Society. +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSAM10) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSAM10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 26 /arrowtailright put +readonly def +/FontBBox{8 -463 1331 1003}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 +469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 +2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 +4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A +C8812CBCD402444EAEA7A50EC5FD93D6A04C2783B50EA48059E3E7407537CB8D +4C206846EF0764C05289733920E2399E58AD8F137C229F3CE3E34D2D1EAB2D53 +20D44EFAC8EFA4D14A2EFE389D952527F98D0E49BD5BD2C8D58FF9CB9C78D974 +75C2AB5467D73D2B5E277A3FDC35909938A9DF0EB91BD9159D3437BE22EE4544 +3429AC8E2BFBE34AE54D3BA3AD04BDF3F4F43A2B43992DF88678681B3AB32CFD +A23E2C98D1AF00AB206AC95B78BBE6316F7A0AB6BD3236C28C76288B3C25D1EB +E9ABB3576C5EC15A71D26177F5883E9B48293D59015615E2EEAF2E9BA04151ED +5497B9A1C41CBA44BAFF13EA218F5EAC11952EE336AD1DBE6CE92F002EAA3B3D +3BE4C3792F3405763C4BD93EFC3B4FC34193439561841BA989DD8D9F9AEE7A7B +24AEB4654B35023C9720B8F31AA9452E29753FB7915CB29977E725611E37C0B7 +784BCC26FACF8A7A0EB1E54290D27FFE52B2D87FAD080AD15EE1984C37E0EB30 +122C3012D3A16B09C28903D138352AB5462674B6CFB63F1371768D094DDF288C +36FB9B58443F872D61F2CD8CED42FE0EFF3D7E9952A172BB1AFECB60BF79F2B6 +04265FDE4F78BC9FD619AA733CD0412F1D9A7C13B271BF827DCBDC8ABAE24FF0 +74D3C220621D7FF0EFE62D835A221D0A7C139E2E6681FC2BBA58FA3B80D416EC +3854C63BA040A4262B458340DAA18AA6AEA3BBAC61615CB85982B18664D3D3AF +340C65B969071CF2D0CABEB80E04623D0526F862ECA8280EEE236C535F70561A +854181132E677674AD5E14C6636F57541D3C821F0776D2CB9B8526D4B826791A +0B179B386F9E08F78BC145792DCAD1EE054DD45D526E316F2A64C62E0122EF78 +4BA88AF1BC2E215CFF920867091DE1E2D3BB97891F7F159F34BEA32196F2658F +820CAB07716DC582F24F0547CF4DD759925B76727FEDB72F87E3E692AA101778 +27C5B3D24B15A903D12FDA3904F6E956795FDAF1A3A31537A39B07979970FA77 +EE4E5E53E5440661B79D13261EAD82FA978966EE540AB93AAF22CF95B7D67C30 +9E5CD0DF0FCF934E59067EDBA99AA7A0BF295D084766392D3B2E7228FE059782 +62024B67417FDD3EC7D67025E769B3FF0AF4C8DC2102A544E6B583DFAAF633FA +7E19437F88C7BF6280A4E30DCB153A869DA8816B26DBFE1D14EB3677CEBE607A +FB372256 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndFont +%%BeginFont: CMR9 +%!PS-AdobeFont-1.1: CMR9 1.0 +%%CreationDate: 1991 Aug 20 16:39:59 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 35 /numbersign put +dup 38 /ampersand put +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 58 /colon put +dup 61 /equal put +dup 91 /bracketleft put +dup 93 /bracketright put +readonly def +/FontBBox{-39 -250 1036 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def +/FullName (CMMI9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 15 /epsilon1 put +dup 18 /theta put +dup 30 /phi put +dup 58 /period put +dup 59 /comma put +dup 61 /slash put +dup 62 /greater put +dup 65 /A put +dup 66 /B put +dup 67 /C put +dup 68 /D put +dup 69 /E put +dup 72 /H put +dup 76 /L put +dup 84 /T put +dup 97 /a put +dup 98 /b put +dup 99 /c put +dup 100 /d put +dup 101 /e put +dup 103 /g put +dup 105 /i put +dup 106 /j put +dup 108 /l put +dup 110 /n put +dup 111 /o put +dup 114 /r put +dup 115 /s put +dup 116 /t put +dup 117 /u put +dup 120 /x put +dup 121 /y put +readonly def +/FontBBox{-29 -250 1075 750}readonly def +currentdict end +currentfile eexec 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6 +/Parent 4972 0 R +/Kids [1241 0 R 1251 0 R 1278 0 R 1311 0 R 1344 0 R 1376 0 R] +>> endobj +1411 0 obj << +/Type /Pages +/Count 6 +/Parent 4972 0 R +/Kids [1408 0 R 1441 0 R 1472 0 R 1505 0 R 1533 0 R 1563 0 R] +>> endobj +1596 0 obj << +/Type /Pages +/Count 6 +/Parent 4972 0 R +/Kids [1593 0 R 1612 0 R 1617 0 R 1630 0 R 1640 0 R 1647 0 R] +>> endobj +1657 0 obj << +/Type /Pages +/Count 6 +/Parent 4972 0 R +/Kids [1654 0 R 1659 0 R 1663 0 R 1666 0 R 1676 0 R 1685 0 R] +>> endobj +1696 0 obj << +/Type /Pages +/Count 6 +/Parent 4972 0 R +/Kids [1692 0 R 1698 0 R 1703 0 R 1710 0 R 1716 0 R 1724 0 R] +>> endobj +1734 0 obj << +/Type /Pages +/Count 6 +/Parent 4972 0 R +/Kids [1731 0 R 1738 0 R 1745 0 R 1751 0 R 1756 0 R 1764 0 R] +>> endobj +1779 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [1775 0 R 1783 0 R 1793 0 R 1807 0 R 1818 0 R 1831 0 R] +>> endobj +1842 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [1838 0 R 1846 0 R 1853 0 R 1861 0 R 1867 0 R 1876 0 R] +>> endobj +1889 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [1881 0 R 1893 0 R 1910 0 R 1918 0 R 1926 0 R 1931 0 R] +>> endobj +1938 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [1935 0 R 1951 0 R 1958 0 R 1974 0 R 1991 0 R 1995 0 R] +>> endobj +2006 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [2002 0 R 2011 0 R 2019 0 R 2024 0 R 2029 0 R 2036 0 R] +>> endobj +2049 0 obj << +/Type /Pages +/Count 6 +/Parent 4973 0 R +/Kids [2044 0 R 2055 0 R 2066 0 R 2074 0 R 2078 0 R 2083 0 R] +>> endobj +2093 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2088 0 R 2095 0 R 2105 0 R 2114 0 R 2119 0 R 2127 0 R] +>> endobj +2136 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2133 0 R 2138 0 R 2147 0 R 2157 0 R 2162 0 R 2171 0 R] +>> endobj +2185 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2179 0 R 2187 0 R 2193 0 R 2199 0 R 2204 0 R 2211 0 R] +>> endobj +2218 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2215 0 R 2220 0 R 2229 0 R 2241 0 R 2248 0 R 2256 0 R] +>> endobj +2269 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2266 0 R 2272 0 R 2280 0 R 2287 0 R 2292 0 R 2303 0 R] +>> endobj +2311 0 obj << +/Type /Pages +/Count 6 +/Parent 4974 0 R +/Kids [2308 0 R 2313 0 R 2318 0 R 2323 0 R 2329 0 R 2335 0 R] +>> endobj +2344 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2341 0 R 2348 0 R 2357 0 R 2364 0 R 2371 0 R 2378 0 R] +>> endobj +2390 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2386 0 R 2394 0 R 2399 0 R 2405 0 R 2412 0 R 2419 0 R] +>> endobj +2440 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2436 0 R 2443 0 R 2453 0 R 2457 0 R 2462 0 R 2466 0 R] +>> endobj +2473 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2470 0 R 2475 0 R 2479 0 R 2483 0 R 2487 0 R 2491 0 R] +>> endobj +2498 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2495 0 R 2500 0 R 2504 0 R 2508 0 R 2512 0 R 2516 0 R] +>> endobj +2523 0 obj << +/Type /Pages +/Count 6 +/Parent 4975 0 R +/Kids [2520 0 R 2525 0 R 2529 0 R 2533 0 R 2537 0 R 2541 0 R] +>> endobj +2548 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2545 0 R 2550 0 R 2554 0 R 2558 0 R 2562 0 R 2566 0 R] +>> endobj +2574 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2571 0 R 2577 0 R 2582 0 R 2586 0 R 2591 0 R 2595 0 R] +>> endobj +2602 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2599 0 R 2604 0 R 2610 0 R 2614 0 R 2618 0 R 2622 0 R] +>> endobj +2632 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2626 0 R 2634 0 R 2638 0 R 2648 0 R 2652 0 R 2657 0 R] +>> endobj +2666 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2663 0 R 2669 0 R 2673 0 R 2677 0 R 2681 0 R 2685 0 R] +>> endobj +2692 0 obj << +/Type /Pages +/Count 6 +/Parent 4976 0 R +/Kids [2689 0 R 2694 0 R 2699 0 R 2703 0 R 2707 0 R 2711 0 R] +>> endobj +2718 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2715 0 R 2720 0 R 2724 0 R 2728 0 R 2732 0 R 2736 0 R] +>> endobj +2743 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2740 0 R 2745 0 R 2749 0 R 2753 0 R 2757 0 R 2762 0 R] +>> endobj +2769 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2766 0 R 2771 0 R 2775 0 R 2779 0 R 2783 0 R 2787 0 R] +>> endobj +2794 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2791 0 R 2796 0 R 2800 0 R 2804 0 R 2808 0 R 2812 0 R] +>> endobj +2819 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2816 0 R 2821 0 R 2825 0 R 2829 0 R 2833 0 R 2837 0 R] +>> endobj +2844 0 obj << +/Type /Pages +/Count 6 +/Parent 4977 0 R +/Kids [2841 0 R 2846 0 R 2850 0 R 2854 0 R 2858 0 R 2862 0 R] +>> endobj +2869 0 obj << +/Type /Pages +/Count 6 +/Parent 4978 0 R +/Kids [2866 0 R 2871 0 R 2875 0 R 2879 0 R 2883 0 R 2887 0 R] +>> endobj +2894 0 obj << +/Type /Pages +/Count 6 +/Parent 4978 0 R +/Kids [2891 0 R 2896 0 R 2900 0 R 2904 0 R 2908 0 R 2912 0 R] +>> endobj +2919 0 obj << +/Type /Pages +/Count 6 +/Parent 4978 0 R 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obj << +/Type /Pages +/Count 6 +/Parent 4979 0 R +/Kids [3116 0 R 3121 0 R 3125 0 R 3129 0 R 3133 0 R 3137 0 R] +>> endobj +3144 0 obj << +/Type /Pages +/Count 6 +/Parent 4979 0 R +/Kids [3141 0 R 3146 0 R 3150 0 R 3154 0 R 3158 0 R 3162 0 R] +>> endobj +3169 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3166 0 R 3171 0 R 3175 0 R 3179 0 R 3183 0 R 3187 0 R] +>> endobj +3197 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3194 0 R 3199 0 R 3203 0 R 3207 0 R 3211 0 R 3215 0 R] +>> endobj +3222 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3219 0 R 3224 0 R 3228 0 R 3232 0 R 3236 0 R 3240 0 R] +>> endobj +3247 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3244 0 R 3249 0 R 3253 0 R 3257 0 R 3261 0 R 3265 0 R] +>> endobj +3272 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3269 0 R 3274 0 R 3278 0 R 3282 0 R 3287 0 R 3292 0 R] +>> endobj +3465 0 obj << +/Type /Pages +/Count 6 +/Parent 4980 0 R +/Kids [3462 0 R 3612 0 R 3760 0 R 3928 0 R 4102 0 R 4269 0 R] +>> endobj +4427 0 obj << +/Type /Pages +/Count 5 +/Parent 4981 0 R +/Kids [4424 0 R 4593 0 R 4754 0 R 4889 0 R 4893 0 R] +>> endobj +4972 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [1249 0 R 1411 0 R 1596 0 R 1657 0 R 1696 0 R 1734 0 R] +>> endobj +4973 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [1779 0 R 1842 0 R 1889 0 R 1938 0 R 2006 0 R 2049 0 R] +>> endobj +4974 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [2093 0 R 2136 0 R 2185 0 R 2218 0 R 2269 0 R 2311 0 R] +>> endobj +4975 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [2344 0 R 2390 0 R 2440 0 R 2473 0 R 2498 0 R 2523 0 R] +>> endobj +4976 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [2548 0 R 2574 0 R 2602 0 R 2632 0 R 2666 0 R 2692 0 R] +>> endobj +4977 0 obj << +/Type /Pages +/Count 36 +/Parent 4982 0 R +/Kids [2718 0 R 2743 0 R 2769 0 R 2794 0 R 2819 0 R 2844 0 R] +>> endobj +4978 0 obj << +/Type /Pages +/Count 36 +/Parent 4983 0 R +/Kids [2869 0 R 2894 0 R 2919 0 R 2944 0 R 2969 0 R 2994 0 R] +>> endobj +4979 0 obj << +/Type /Pages +/Count 36 +/Parent 4983 0 R +/Kids [3019 0 R 3044 0 R 3069 0 R 3094 0 R 3119 0 R 3144 0 R] +>> endobj +4980 0 obj << +/Type /Pages +/Count 36 +/Parent 4983 0 R +/Kids [3169 0 R 3197 0 R 3222 0 R 3247 0 R 3272 0 R 3465 0 R] +>> endobj +4981 0 obj << +/Type /Pages +/Count 5 +/Parent 4983 0 R +/Kids [4427 0 R] +>> endobj +4982 0 obj << +/Type /Pages +/Count 216 +/Parent 4984 0 R +/Kids [4972 0 R 4973 0 R 4974 0 R 4975 0 R 4976 0 R 4977 0 R] +>> endobj +4983 0 obj << +/Type /Pages +/Count 113 +/Parent 4984 0 R +/Kids [4978 0 R 4979 0 R 4980 0 R 4981 0 R] +>> endobj +4984 0 obj << +/Type /Pages +/Count 329 +/Kids [4982 0 R 4983 0 R] +>> endobj +4985 0 obj << +/Type /Outlines +/First 3 0 R +/Last 1238 0 R +/Count 7 +>> endobj +1238 0 obj << +/Title 1239 0 R +/A 1236 0 R +/Parent 4985 0 R +/Prev 1234 0 R +>> endobj +1234 0 obj << +/Title 1235 0 R +/A 1233 0 R +/Parent 4985 0 R +/Prev 615 0 R +/Next 1238 0 R +>> endobj +1231 0 obj << +/Title 1232 0 R +/A 1229 0 R +/Parent 1211 0 R +/Prev 1227 0 R +>> endobj +1227 0 obj << +/Title 1228 0 R +/A 1225 0 R +/Parent 1211 0 R +/Prev 1215 0 R +/Next 1231 0 R +>> endobj +1223 0 obj << +/Title 1224 0 R +/A 1221 0 R +/Parent 1215 0 R +/Prev 1219 0 R +>> endobj +1219 0 obj << +/Title 1220 0 R +/A 1217 0 R +/Parent 1215 0 R +/Next 1223 0 R +>> endobj +1215 0 obj << +/Title 1216 0 R +/A 1213 0 R +/Parent 1211 0 R +/Next 1227 0 R +/First 1219 0 R +/Last 1223 0 R +/Count -2 +>> endobj +1211 0 obj << +/Title 1212 0 R +/A 1209 0 R +/Parent 615 0 R +/Prev 1127 0 R +/First 1215 0 R +/Last 1231 0 R +/Count -3 +>> endobj +1207 0 obj << +/Title 1208 0 R +/A 1205 0 R +/Parent 1195 0 R +/Prev 1203 0 R +>> endobj +1203 0 obj << +/Title 1204 0 R +/A 1201 0 R +/Parent 1195 0 R +/Prev 1199 0 R +/Next 1207 0 R +>> endobj +1199 0 obj << +/Title 1200 0 R +/A 1197 0 R +/Parent 1195 0 R +/Next 1203 0 R +>> endobj +1195 0 obj << +/Title 1196 0 R +/A 1193 0 R +/Parent 1127 0 R +/Prev 1163 0 R +/First 1199 0 R +/Last 1207 0 R +/Count -3 +>> endobj +1191 0 obj << +/Title 1192 0 R +/A 1189 0 R +/Parent 1163 0 R +/Prev 1187 0 R +>> endobj +1187 0 obj << +/Title 1188 0 R +/A 1185 0 R +/Parent 1163 0 R +/Prev 1183 0 R +/Next 1191 0 R +>> endobj +1183 0 obj << +/Title 1184 0 R +/A 1181 0 R +/Parent 1163 0 R +/Prev 1179 0 R +/Next 1187 0 R +>> endobj +1179 0 obj << +/Title 1180 0 R +/A 1177 0 R +/Parent 1163 0 R +/Prev 1175 0 R +/Next 1183 0 R +>> endobj +1175 0 obj << +/Title 1176 0 R +/A 1173 0 R +/Parent 1163 0 R +/Prev 1171 0 R +/Next 1179 0 R +>> endobj +1171 0 obj << +/Title 1172 0 R +/A 1169 0 R +/Parent 1163 0 R +/Prev 1167 0 R +/Next 1175 0 R +>> endobj +1167 0 obj << +/Title 1168 0 R +/A 1165 0 R +/Parent 1163 0 R +/Next 1171 0 R +>> endobj +1163 0 obj << +/Title 1164 0 R +/A 1161 0 R +/Parent 1127 0 R +/Prev 1143 0 R +/Next 1195 0 R +/First 1167 0 R +/Last 1191 0 R +/Count -7 +>> endobj +1159 0 obj << +/Title 1160 0 R +/A 1157 0 R +/Parent 1143 0 R +/Prev 1155 0 R +>> endobj +1155 0 obj << +/Title 1156 0 R +/A 1153 0 R +/Parent 1143 0 R +/Prev 1151 0 R +/Next 1159 0 R +>> endobj +1151 0 obj << +/Title 1152 0 R +/A 1149 0 R +/Parent 1143 0 R +/Prev 1147 0 R +/Next 1155 0 R +>> endobj +1147 0 obj << +/Title 1148 0 R +/A 1145 0 R +/Parent 1143 0 R +/Next 1151 0 R +>> endobj +1143 0 obj << +/Title 1144 0 R +/A 1141 0 R +/Parent 1127 0 R +/Prev 1135 0 R +/Next 1163 0 R +/First 1147 0 R +/Last 1159 0 R +/Count -4 +>> endobj +1139 0 obj << +/Title 1140 0 R +/A 1137 0 R +/Parent 1135 0 R +>> endobj +1135 0 obj << +/Title 1136 0 R +/A 1133 0 R +/Parent 1127 0 R +/Prev 1131 0 R +/Next 1143 0 R +/First 1139 0 R +/Last 1139 0 R +/Count -1 +>> endobj +1131 0 obj << +/Title 1132 0 R +/A 1129 0 R +/Parent 1127 0 R +/Next 1135 0 R +>> endobj +1127 0 obj << +/Title 1128 0 R +/A 1125 0 R +/Parent 615 0 R +/Prev 1123 0 R +/Next 1211 0 R +/First 1131 0 R +/Last 1195 0 R +/Count -5 +>> endobj +1123 0 obj << +/Title 1124 0 R +/A 1121 0 R +/Parent 615 0 R +/Prev 1119 0 R +/Next 1127 0 R +>> endobj +1119 0 obj << +/Title 1120 0 R +/A 1117 0 R +/Parent 615 0 R +/Prev 1103 0 R +/Next 1123 0 R +>> endobj +1115 0 obj << +/Title 1116 0 R +/A 1113 0 R +/Parent 1103 0 R +/Prev 1111 0 R +>> endobj +1111 0 obj << +/Title 1112 0 R +/A 1109 0 R +/Parent 1103 0 R +/Prev 1107 0 R +/Next 1115 0 R +>> endobj +1107 0 obj << +/Title 1108 0 R +/A 1105 0 R +/Parent 1103 0 R +/Next 1111 0 R +>> endobj +1103 0 obj << +/Title 1104 0 R +/A 1101 0 R +/Parent 615 0 R +/Prev 1099 0 R +/Next 1119 0 R +/First 1107 0 R +/Last 1115 0 R +/Count -3 +>> endobj +1099 0 obj << +/Title 1100 0 R +/A 1097 0 R +/Parent 615 0 R +/Prev 1087 0 R +/Next 1103 0 R +>> endobj +1095 0 obj << +/Title 1096 0 R +/A 1093 0 R +/Parent 1091 0 R +>> endobj +1091 0 obj << +/Title 1092 0 R +/A 1089 0 R +/Parent 1087 0 R +/First 1095 0 R +/Last 1095 0 R +/Count -1 +>> endobj +1087 0 obj << +/Title 1088 0 R +/A 1085 0 R +/Parent 615 0 R +/Prev 1071 0 R +/Next 1099 0 R +/First 1091 0 R +/Last 1091 0 R +/Count -1 +>> endobj +1083 0 obj << +/Title 1084 0 R +/A 1081 0 R +/Parent 1071 0 R +/Prev 1079 0 R +>> endobj +1079 0 obj << +/Title 1080 0 R +/A 1077 0 R +/Parent 1071 0 R +/Prev 1075 0 R +/Next 1083 0 R +>> endobj +1075 0 obj << +/Title 1076 0 R +/A 1073 0 R +/Parent 1071 0 R +/Next 1079 0 R +>> endobj +1071 0 obj << +/Title 1072 0 R +/A 1069 0 R +/Parent 615 0 R +/Prev 1043 0 R +/Next 1087 0 R +/First 1075 0 R +/Last 1083 0 R +/Count -3 +>> endobj +1067 0 obj << +/Title 1068 0 R +/A 1065 0 R +/Parent 1047 0 R +/Prev 1063 0 R +>> endobj +1063 0 obj << +/Title 1064 0 R +/A 1061 0 R +/Parent 1047 0 R +/Prev 1059 0 R +/Next 1067 0 R +>> endobj +1059 0 obj << +/Title 1060 0 R +/A 1057 0 R +/Parent 1047 0 R +/Prev 1055 0 R +/Next 1063 0 R +>> endobj +1055 0 obj << +/Title 1056 0 R +/A 1053 0 R +/Parent 1047 0 R +/Prev 1051 0 R +/Next 1059 0 R +>> endobj +1051 0 obj << +/Title 1052 0 R +/A 1049 0 R +/Parent 1047 0 R +/Next 1055 0 R +>> endobj +1047 0 obj << +/Title 1048 0 R +/A 1045 0 R +/Parent 1043 0 R +/First 1051 0 R +/Last 1067 0 R +/Count -5 +>> endobj +1043 0 obj << +/Title 1044 0 R +/A 1041 0 R +/Parent 615 0 R +/Prev 1039 0 R +/Next 1071 0 R +/First 1047 0 R +/Last 1047 0 R +/Count -1 +>> endobj +1039 0 obj << +/Title 1040 0 R +/A 1037 0 R +/Parent 615 0 R +/Prev 1007 0 R +/Next 1043 0 R +>> endobj +1035 0 obj << +/Title 1036 0 R +/A 1033 0 R +/Parent 1011 0 R +/Prev 1031 0 R +>> endobj +1031 0 obj << +/Title 1032 0 R +/A 1029 0 R +/Parent 1011 0 R +/Prev 1027 0 R +/Next 1035 0 R +>> endobj +1027 0 obj << +/Title 1028 0 R +/A 1025 0 R +/Parent 1011 0 R +/Prev 1023 0 R +/Next 1031 0 R +>> endobj +1023 0 obj << +/Title 1024 0 R +/A 1021 0 R +/Parent 1011 0 R +/Prev 1019 0 R +/Next 1027 0 R +>> endobj +1019 0 obj << +/Title 1020 0 R +/A 1017 0 R +/Parent 1011 0 R +/Prev 1015 0 R +/Next 1023 0 R +>> endobj +1015 0 obj << +/Title 1016 0 R +/A 1013 0 R +/Parent 1011 0 R +/Next 1019 0 R +>> endobj +1011 0 obj << +/Title 1012 0 R +/A 1009 0 R +/Parent 1007 0 R +/First 1015 0 R +/Last 1035 0 R +/Count -6 +>> endobj +1007 0 obj << +/Title 1008 0 R +/A 1005 0 R +/Parent 615 0 R +/Prev 991 0 R +/Next 1039 0 R +/First 1011 0 R +/Last 1011 0 R +/Count -1 +>> endobj +1003 0 obj << +/Title 1004 0 R +/A 1001 0 R +/Parent 995 0 R +/Prev 999 0 R +>> endobj +999 0 obj << +/Title 1000 0 R +/A 997 0 R +/Parent 995 0 R +/Next 1003 0 R +>> endobj +995 0 obj << +/Title 996 0 R +/A 993 0 R +/Parent 991 0 R +/First 999 0 R +/Last 1003 0 R +/Count -2 +>> endobj +991 0 obj << +/Title 992 0 R +/A 989 0 R +/Parent 615 0 R +/Prev 987 0 R +/Next 1007 0 R +/First 995 0 R +/Last 995 0 R +/Count -1 +>> endobj +987 0 obj << +/Title 988 0 R +/A 985 0 R +/Parent 615 0 R +/Prev 967 0 R +/Next 991 0 R +>> endobj +983 0 obj << +/Title 984 0 R +/A 981 0 R +/Parent 971 0 R +/Prev 979 0 R +>> endobj +979 0 obj << +/Title 980 0 R +/A 977 0 R +/Parent 971 0 R +/Prev 975 0 R +/Next 983 0 R +>> endobj +975 0 obj << +/Title 976 0 R +/A 973 0 R +/Parent 971 0 R +/Next 979 0 R +>> endobj +971 0 obj << +/Title 972 0 R +/A 969 0 R +/Parent 967 0 R +/First 975 0 R +/Last 983 0 R +/Count -3 +>> endobj +967 0 obj << +/Title 968 0 R +/A 965 0 R +/Parent 615 0 R +/Prev 943 0 R +/Next 987 0 R +/First 971 0 R +/Last 971 0 R +/Count -1 +>> endobj +963 0 obj << +/Title 964 0 R +/A 961 0 R +/Parent 947 0 R +/Prev 959 0 R +>> endobj +959 0 obj << +/Title 960 0 R +/A 957 0 R +/Parent 947 0 R +/Prev 955 0 R +/Next 963 0 R +>> endobj +955 0 obj << +/Title 956 0 R +/A 953 0 R +/Parent 947 0 R +/Prev 951 0 R +/Next 959 0 R +>> endobj +951 0 obj << +/Title 952 0 R +/A 949 0 R +/Parent 947 0 R +/Next 955 0 R +>> endobj +947 0 obj << +/Title 948 0 R +/A 945 0 R +/Parent 943 0 R +/First 951 0 R +/Last 963 0 R +/Count -4 +>> endobj +943 0 obj << +/Title 944 0 R +/A 941 0 R +/Parent 615 0 R +/Prev 923 0 R +/Next 967 0 R +/First 947 0 R +/Last 947 0 R +/Count -1 +>> endobj +939 0 obj << +/Title 940 0 R +/A 937 0 R +/Parent 923 0 R +/Prev 927 0 R +>> endobj +935 0 obj << +/Title 936 0 R +/A 933 0 R +/Parent 927 0 R +/Prev 931 0 R +>> endobj +931 0 obj << +/Title 932 0 R +/A 929 0 R +/Parent 927 0 R +/Next 935 0 R +>> endobj +927 0 obj << +/Title 928 0 R +/A 925 0 R +/Parent 923 0 R +/Next 939 0 R +/First 931 0 R +/Last 935 0 R +/Count -2 +>> endobj +923 0 obj << +/Title 924 0 R +/A 921 0 R +/Parent 615 0 R +/Prev 903 0 R +/Next 943 0 R +/First 927 0 R +/Last 939 0 R +/Count -2 +>> endobj +919 0 obj << +/Title 920 0 R +/A 917 0 R +/Parent 907 0 R +/Prev 915 0 R +>> endobj +915 0 obj << +/Title 916 0 R +/A 913 0 R +/Parent 907 0 R +/Prev 911 0 R +/Next 919 0 R +>> endobj +911 0 obj << +/Title 912 0 R +/A 909 0 R +/Parent 907 0 R +/Next 915 0 R +>> endobj +907 0 obj << +/Title 908 0 R +/A 905 0 R +/Parent 903 0 R +/First 911 0 R +/Last 919 0 R +/Count -3 +>> endobj +903 0 obj << +/Title 904 0 R +/A 901 0 R +/Parent 615 0 R +/Prev 899 0 R +/Next 923 0 R +/First 907 0 R +/Last 907 0 R +/Count -1 +>> endobj +899 0 obj << +/Title 900 0 R +/A 897 0 R +/Parent 615 0 R +/Prev 895 0 R +/Next 903 0 R +>> endobj +895 0 obj << +/Title 896 0 R +/A 893 0 R +/Parent 615 0 R +/Prev 887 0 R +/Next 899 0 R +>> endobj +891 0 obj << +/Title 892 0 R +/A 889 0 R +/Parent 887 0 R +>> endobj +887 0 obj << +/Title 888 0 R +/A 885 0 R +/Parent 615 0 R +/Prev 879 0 R +/Next 895 0 R +/First 891 0 R +/Last 891 0 R +/Count -1 +>> endobj +883 0 obj << +/Title 884 0 R +/A 881 0 R +/Parent 879 0 R +>> endobj +879 0 obj << +/Title 880 0 R +/A 877 0 R +/Parent 615 0 R +/Prev 867 0 R +/Next 887 0 R +/First 883 0 R +/Last 883 0 R +/Count -1 +>> endobj +875 0 obj << +/Title 876 0 R +/A 873 0 R +/Parent 867 0 R +/Prev 871 0 R +>> endobj +871 0 obj << +/Title 872 0 R +/A 869 0 R +/Parent 867 0 R +/Next 875 0 R +>> endobj +867 0 obj << +/Title 868 0 R +/A 865 0 R +/Parent 615 0 R +/Prev 759 0 R +/Next 879 0 R +/First 871 0 R +/Last 875 0 R +/Count -2 +>> endobj +863 0 obj << +/Title 864 0 R +/A 861 0 R +/Parent 847 0 R +/Prev 851 0 R +>> endobj +859 0 obj << +/Title 860 0 R +/A 857 0 R +/Parent 851 0 R +/Prev 855 0 R +>> endobj +855 0 obj << +/Title 856 0 R +/A 853 0 R +/Parent 851 0 R +/Next 859 0 R +>> endobj +851 0 obj << +/Title 852 0 R +/A 849 0 R +/Parent 847 0 R +/Next 863 0 R +/First 855 0 R +/Last 859 0 R +/Count -2 +>> endobj +847 0 obj << +/Title 848 0 R +/A 845 0 R +/Parent 759 0 R +/Prev 831 0 R +/First 851 0 R +/Last 863 0 R +/Count -2 +>> endobj +843 0 obj << +/Title 844 0 R +/A 841 0 R +/Parent 831 0 R +/Prev 839 0 R +>> endobj +839 0 obj << +/Title 840 0 R +/A 837 0 R +/Parent 831 0 R +/Prev 835 0 R +/Next 843 0 R +>> endobj +835 0 obj << +/Title 836 0 R +/A 833 0 R +/Parent 831 0 R +/Next 839 0 R +>> endobj +831 0 obj << +/Title 832 0 R +/A 829 0 R +/Parent 759 0 R +/Prev 827 0 R +/Next 847 0 R +/First 835 0 R +/Last 843 0 R +/Count -3 +>> endobj +827 0 obj << +/Title 828 0 R +/A 825 0 R +/Parent 759 0 R +/Prev 823 0 R +/Next 831 0 R +>> endobj +823 0 obj << +/Title 824 0 R +/A 821 0 R +/Parent 759 0 R +/Prev 811 0 R +/Next 827 0 R +>> endobj +819 0 obj << +/Title 820 0 R +/A 817 0 R +/Parent 811 0 R +/Prev 815 0 R +>> endobj +815 0 obj << +/Title 816 0 R +/A 813 0 R +/Parent 811 0 R +/Next 819 0 R +>> endobj +811 0 obj << +/Title 812 0 R +/A 809 0 R +/Parent 759 0 R +/Prev 799 0 R +/Next 823 0 R +/First 815 0 R +/Last 819 0 R +/Count -2 +>> endobj +807 0 obj << +/Title 808 0 R +/A 805 0 R +/Parent 799 0 R +/Prev 803 0 R +>> endobj +803 0 obj << +/Title 804 0 R +/A 801 0 R +/Parent 799 0 R +/Next 807 0 R +>> endobj +799 0 obj << +/Title 800 0 R +/A 797 0 R +/Parent 759 0 R +/Prev 779 0 R +/Next 811 0 R +/First 803 0 R +/Last 807 0 R +/Count -2 +>> endobj +795 0 obj << +/Title 796 0 R +/A 793 0 R +/Parent 779 0 R +/Prev 791 0 R +>> endobj +791 0 obj << +/Title 792 0 R +/A 789 0 R +/Parent 779 0 R +/Prev 787 0 R +/Next 795 0 R +>> endobj +787 0 obj << +/Title 788 0 R +/A 785 0 R +/Parent 779 0 R +/Prev 783 0 R +/Next 791 0 R +>> endobj +783 0 obj << +/Title 784 0 R +/A 781 0 R +/Parent 779 0 R +/Next 787 0 R +>> endobj +779 0 obj << +/Title 780 0 R +/A 777 0 R +/Parent 759 0 R +/Prev 771 0 R +/Next 799 0 R +/First 783 0 R +/Last 795 0 R +/Count -4 +>> endobj +775 0 obj << +/Title 776 0 R +/A 773 0 R +/Parent 771 0 R +>> endobj +771 0 obj << +/Title 772 0 R +/A 769 0 R +/Parent 759 0 R +/Prev 767 0 R +/Next 779 0 R +/First 775 0 R +/Last 775 0 R +/Count -1 +>> endobj +767 0 obj << +/Title 768 0 R +/A 765 0 R +/Parent 759 0 R +/Prev 763 0 R +/Next 771 0 R +>> endobj +763 0 obj << +/Title 764 0 R +/A 761 0 R +/Parent 759 0 R +/Next 767 0 R +>> endobj +759 0 obj << +/Title 760 0 R +/A 757 0 R +/Parent 615 0 R +/Prev 747 0 R +/Next 867 0 R +/First 763 0 R +/Last 847 0 R +/Count -10 +>> endobj +755 0 obj << +/Title 756 0 R +/A 753 0 R +/Parent 747 0 R +/Prev 751 0 R +>> endobj +751 0 obj << +/Title 752 0 R +/A 749 0 R +/Parent 747 0 R +/Next 755 0 R +>> endobj +747 0 obj << +/Title 748 0 R +/A 745 0 R +/Parent 615 0 R +/Prev 739 0 R +/Next 759 0 R +/First 751 0 R +/Last 755 0 R +/Count -2 +>> endobj +743 0 obj << +/Title 744 0 R +/A 741 0 R +/Parent 739 0 R +>> endobj +739 0 obj << +/Title 740 0 R +/A 737 0 R +/Parent 615 0 R +/Prev 719 0 R +/Next 747 0 R +/First 743 0 R +/Last 743 0 R +/Count -1 +>> endobj +735 0 obj << +/Title 736 0 R +/A 733 0 R +/Parent 719 0 R +/Prev 731 0 R +>> endobj +731 0 obj << +/Title 732 0 R +/A 729 0 R +/Parent 719 0 R +/Prev 727 0 R +/Next 735 0 R +>> endobj +727 0 obj << +/Title 728 0 R +/A 725 0 R +/Parent 719 0 R +/Prev 723 0 R +/Next 731 0 R +>> endobj +723 0 obj << +/Title 724 0 R +/A 721 0 R +/Parent 719 0 R +/Next 727 0 R +>> endobj +719 0 obj << +/Title 720 0 R +/A 717 0 R +/Parent 615 0 R +/Prev 683 0 R +/Next 739 0 R +/First 723 0 R +/Last 735 0 R +/Count -4 +>> endobj +715 0 obj << +/Title 716 0 R +/A 713 0 R +/Parent 683 0 R +/Prev 711 0 R +>> endobj +711 0 obj << +/Title 712 0 R +/A 709 0 R +/Parent 683 0 R +/Prev 707 0 R +/Next 715 0 R +>> endobj +707 0 obj << +/Title 708 0 R +/A 705 0 R +/Parent 683 0 R +/Prev 703 0 R +/Next 711 0 R +>> endobj +703 0 obj << +/Title 704 0 R +/A 701 0 R +/Parent 683 0 R +/Prev 691 0 R +/Next 707 0 R +>> endobj +699 0 obj << +/Title 700 0 R +/A 697 0 R +/Parent 691 0 R +/Prev 695 0 R +>> endobj +695 0 obj << +/Title 696 0 R +/A 693 0 R +/Parent 691 0 R +/Next 699 0 R +>> endobj +691 0 obj << +/Title 692 0 R +/A 689 0 R +/Parent 683 0 R +/Prev 687 0 R +/Next 703 0 R +/First 695 0 R +/Last 699 0 R +/Count -2 +>> endobj +687 0 obj << +/Title 688 0 R +/A 685 0 R +/Parent 683 0 R +/Next 691 0 R +>> endobj +683 0 obj << +/Title 684 0 R +/A 681 0 R +/Parent 615 0 R +/Prev 679 0 R +/Next 719 0 R +/First 687 0 R +/Last 715 0 R +/Count -6 +>> endobj +679 0 obj << +/Title 680 0 R +/A 677 0 R +/Parent 615 0 R +/Prev 651 0 R +/Next 683 0 R +>> endobj +675 0 obj << +/Title 676 0 R +/A 673 0 R +/Parent 651 0 R +/Prev 671 0 R +>> endobj +671 0 obj << +/Title 672 0 R +/A 669 0 R +/Parent 651 0 R +/Prev 667 0 R +/Next 675 0 R +>> endobj +667 0 obj << +/Title 668 0 R +/A 665 0 R +/Parent 651 0 R +/Prev 663 0 R +/Next 671 0 R +>> endobj +663 0 obj << +/Title 664 0 R +/A 661 0 R +/Parent 651 0 R +/Prev 659 0 R +/Next 667 0 R +>> endobj +659 0 obj << +/Title 660 0 R +/A 657 0 R +/Parent 651 0 R +/Prev 655 0 R +/Next 663 0 R +>> endobj +655 0 obj << +/Title 656 0 R +/A 653 0 R +/Parent 651 0 R +/Next 659 0 R +>> endobj +651 0 obj << +/Title 652 0 R +/A 649 0 R +/Parent 615 0 R +/Prev 619 0 R +/Next 679 0 R +/First 655 0 R +/Last 675 0 R +/Count -6 +>> endobj +647 0 obj << +/Title 648 0 R +/A 645 0 R +/Parent 627 0 R +/Prev 643 0 R +>> endobj +643 0 obj << +/Title 644 0 R +/A 641 0 R +/Parent 627 0 R +/Prev 639 0 R +/Next 647 0 R +>> endobj +639 0 obj << +/Title 640 0 R +/A 637 0 R +/Parent 627 0 R +/Prev 635 0 R +/Next 643 0 R +>> endobj +635 0 obj << +/Title 636 0 R +/A 633 0 R +/Parent 627 0 R +/Prev 631 0 R +/Next 639 0 R +>> endobj +631 0 obj << +/Title 632 0 R +/A 629 0 R +/Parent 627 0 R +/Next 635 0 R +>> endobj +627 0 obj << +/Title 628 0 R +/A 625 0 R +/Parent 619 0 R +/Prev 623 0 R +/First 631 0 R +/Last 647 0 R +/Count -5 +>> endobj +623 0 obj << +/Title 624 0 R +/A 621 0 R +/Parent 619 0 R +/Next 627 0 R +>> endobj +619 0 obj << +/Title 620 0 R +/A 617 0 R +/Parent 615 0 R +/Next 651 0 R +/First 623 0 R +/Last 627 0 R +/Count -2 +>> endobj +615 0 obj << +/Title 616 0 R +/A 613 0 R +/Parent 4985 0 R +/Prev 15 0 R +/Next 1234 0 R +/First 619 0 R +/Last 1211 0 R +/Count -30 +>> endobj +611 0 obj << +/Title 612 0 R +/A 609 0 R +/Parent 599 0 R +/Prev 607 0 R +>> endobj +607 0 obj << +/Title 608 0 R +/A 605 0 R +/Parent 599 0 R +/Prev 603 0 R +/Next 611 0 R +>> endobj +603 0 obj << +/Title 604 0 R +/A 601 0 R +/Parent 599 0 R +/Next 607 0 R +>> endobj +599 0 obj << +/Title 600 0 R +/A 597 0 R +/Parent 15 0 R +/Prev 575 0 R +/First 603 0 R +/Last 611 0 R +/Count -3 +>> endobj +595 0 obj << +/Title 596 0 R +/A 593 0 R +/Parent 575 0 R +/Prev 591 0 R +>> endobj +591 0 obj << +/Title 592 0 R +/A 589 0 R +/Parent 575 0 R +/Prev 587 0 R +/Next 595 0 R +>> endobj +587 0 obj << +/Title 588 0 R +/A 585 0 R +/Parent 575 0 R +/Prev 583 0 R +/Next 591 0 R +>> endobj +583 0 obj << +/Title 584 0 R +/A 581 0 R +/Parent 575 0 R +/Prev 579 0 R +/Next 587 0 R +>> endobj +579 0 obj << +/Title 580 0 R +/A 577 0 R +/Parent 575 0 R +/Next 583 0 R +>> endobj +575 0 obj << +/Title 576 0 R +/A 573 0 R +/Parent 15 0 R +/Prev 547 0 R +/Next 599 0 R +/First 579 0 R +/Last 595 0 R +/Count -5 +>> endobj +571 0 obj << +/Title 572 0 R +/A 569 0 R +/Parent 547 0 R +/Prev 567 0 R +>> endobj +567 0 obj << +/Title 568 0 R +/A 565 0 R +/Parent 547 0 R +/Prev 563 0 R +/Next 571 0 R +>> endobj +563 0 obj << +/Title 564 0 R +/A 561 0 R +/Parent 547 0 R +/Prev 559 0 R +/Next 567 0 R +>> endobj +559 0 obj << +/Title 560 0 R +/A 557 0 R +/Parent 547 0 R +/Prev 555 0 R +/Next 563 0 R +>> endobj +555 0 obj << +/Title 556 0 R +/A 553 0 R +/Parent 547 0 R +/Prev 551 0 R +/Next 559 0 R +>> endobj +551 0 obj << +/Title 552 0 R +/A 549 0 R +/Parent 547 0 R +/Next 555 0 R +>> endobj +547 0 obj << +/Title 548 0 R +/A 545 0 R +/Parent 15 0 R +/Prev 531 0 R +/Next 575 0 R +/First 551 0 R +/Last 571 0 R +/Count -6 +>> endobj +543 0 obj << +/Title 544 0 R +/A 541 0 R +/Parent 531 0 R +/Prev 539 0 R +>> endobj +539 0 obj << +/Title 540 0 R +/A 537 0 R +/Parent 531 0 R +/Prev 535 0 R +/Next 543 0 R +>> endobj +535 0 obj << +/Title 536 0 R +/A 533 0 R +/Parent 531 0 R +/Next 539 0 R +>> endobj +531 0 obj << +/Title 532 0 R +/A 529 0 R +/Parent 15 0 R +/Prev 471 0 R +/Next 547 0 R +/First 535 0 R +/Last 543 0 R +/Count -3 +>> endobj +527 0 obj << +/Title 528 0 R +/A 525 0 R +/Parent 471 0 R +/Prev 523 0 R +>> endobj +523 0 obj << +/Title 524 0 R +/A 521 0 R +/Parent 471 0 R +/Prev 511 0 R +/Next 527 0 R +>> endobj +519 0 obj << +/Title 520 0 R +/A 517 0 R +/Parent 511 0 R +/Prev 515 0 R +>> endobj +515 0 obj << +/Title 516 0 R +/A 513 0 R +/Parent 511 0 R +/Next 519 0 R +>> endobj +511 0 obj << +/Title 512 0 R +/A 509 0 R +/Parent 471 0 R +/Prev 491 0 R +/Next 523 0 R +/First 515 0 R +/Last 519 0 R +/Count -2 +>> endobj +507 0 obj << +/Title 508 0 R +/A 505 0 R +/Parent 491 0 R +/Prev 503 0 R +>> endobj +503 0 obj << +/Title 504 0 R +/A 501 0 R +/Parent 491 0 R +/Prev 499 0 R +/Next 507 0 R +>> endobj +499 0 obj << +/Title 500 0 R +/A 497 0 R +/Parent 491 0 R +/Prev 495 0 R +/Next 503 0 R +>> endobj +495 0 obj << +/Title 496 0 R +/A 493 0 R +/Parent 491 0 R +/Next 499 0 R +>> endobj +491 0 obj << +/Title 492 0 R +/A 489 0 R +/Parent 471 0 R +/Prev 487 0 R +/Next 511 0 R +/First 495 0 R +/Last 507 0 R +/Count -4 +>> endobj +487 0 obj << +/Title 488 0 R +/A 485 0 R +/Parent 471 0 R +/Prev 479 0 R +/Next 491 0 R +>> endobj +483 0 obj << +/Title 484 0 R +/A 481 0 R +/Parent 479 0 R +>> endobj +479 0 obj << +/Title 480 0 R +/A 477 0 R +/Parent 471 0 R +/Prev 475 0 R +/Next 487 0 R +/First 483 0 R +/Last 483 0 R +/Count -1 +>> endobj +475 0 obj << +/Title 476 0 R +/A 473 0 R +/Parent 471 0 R +/Next 479 0 R +>> endobj +471 0 obj << +/Title 472 0 R +/A 469 0 R +/Parent 15 0 R +/Prev 455 0 R +/Next 531 0 R +/First 475 0 R +/Last 527 0 R +/Count -7 +>> endobj +467 0 obj << +/Title 468 0 R +/A 465 0 R +/Parent 455 0 R +/Prev 463 0 R +>> endobj +463 0 obj << +/Title 464 0 R +/A 461 0 R +/Parent 455 0 R +/Prev 459 0 R +/Next 467 0 R +>> endobj +459 0 obj << +/Title 460 0 R +/A 457 0 R +/Parent 455 0 R +/Next 463 0 R +>> endobj +455 0 obj << +/Title 456 0 R +/A 453 0 R +/Parent 15 0 R +/Prev 351 0 R +/Next 471 0 R +/First 459 0 R +/Last 467 0 R +/Count -3 +>> endobj +451 0 obj << +/Title 452 0 R +/A 449 0 R +/Parent 427 0 R +/Prev 447 0 R +>> endobj +447 0 obj << +/Title 448 0 R +/A 445 0 R +/Parent 427 0 R +/Prev 443 0 R +/Next 451 0 R +>> endobj +443 0 obj << +/Title 444 0 R +/A 441 0 R +/Parent 427 0 R +/Prev 439 0 R +/Next 447 0 R +>> endobj +439 0 obj << +/Title 440 0 R +/A 437 0 R +/Parent 427 0 R +/Prev 435 0 R +/Next 443 0 R +>> endobj +435 0 obj << +/Title 436 0 R +/A 433 0 R +/Parent 427 0 R +/Prev 431 0 R +/Next 439 0 R +>> endobj +431 0 obj << +/Title 432 0 R +/A 429 0 R +/Parent 427 0 R +/Next 435 0 R +>> endobj +427 0 obj << +/Title 428 0 R +/A 425 0 R +/Parent 351 0 R +/Prev 395 0 R +/First 431 0 R +/Last 451 0 R +/Count -6 +>> endobj +423 0 obj << +/Title 424 0 R +/A 421 0 R +/Parent 395 0 R +/Prev 419 0 R +>> endobj +419 0 obj << +/Title 420 0 R +/A 417 0 R +/Parent 395 0 R +/Prev 415 0 R +/Next 423 0 R +>> endobj +415 0 obj << +/Title 416 0 R +/A 413 0 R +/Parent 395 0 R +/Prev 411 0 R +/Next 419 0 R +>> endobj +411 0 obj << +/Title 412 0 R +/A 409 0 R +/Parent 395 0 R +/Prev 407 0 R +/Next 415 0 R +>> endobj +407 0 obj << +/Title 408 0 R +/A 405 0 R +/Parent 395 0 R +/Prev 403 0 R +/Next 411 0 R +>> endobj +403 0 obj << +/Title 404 0 R +/A 401 0 R +/Parent 395 0 R +/Prev 399 0 R +/Next 407 0 R +>> endobj +399 0 obj << +/Title 400 0 R +/A 397 0 R +/Parent 395 0 R +/Next 403 0 R +>> endobj +395 0 obj << +/Title 396 0 R +/A 393 0 R +/Parent 351 0 R +/Prev 391 0 R +/Next 427 0 R +/First 399 0 R +/Last 423 0 R +/Count -7 +>> endobj +391 0 obj << +/Title 392 0 R +/A 389 0 R +/Parent 351 0 R +/Prev 355 0 R +/Next 395 0 R +>> endobj +387 0 obj << +/Title 388 0 R +/A 385 0 R +/Parent 355 0 R +/Prev 383 0 R +>> endobj +383 0 obj << +/Title 384 0 R +/A 381 0 R +/Parent 355 0 R +/Prev 379 0 R +/Next 387 0 R +>> endobj +379 0 obj << +/Title 380 0 R +/A 377 0 R +/Parent 355 0 R +/Prev 375 0 R +/Next 383 0 R +>> endobj +375 0 obj << +/Title 376 0 R +/A 373 0 R +/Parent 355 0 R +/Prev 371 0 R +/Next 379 0 R +>> endobj +371 0 obj << +/Title 372 0 R +/A 369 0 R +/Parent 355 0 R +/Prev 367 0 R +/Next 375 0 R +>> endobj +367 0 obj << +/Title 368 0 R +/A 365 0 R +/Parent 355 0 R +/Prev 363 0 R +/Next 371 0 R +>> endobj +363 0 obj << +/Title 364 0 R +/A 361 0 R +/Parent 355 0 R +/Prev 359 0 R +/Next 367 0 R +>> endobj +359 0 obj << +/Title 360 0 R +/A 357 0 R +/Parent 355 0 R +/Next 363 0 R +>> endobj +355 0 obj << +/Title 356 0 R +/A 353 0 R +/Parent 351 0 R +/Next 391 0 R +/First 359 0 R +/Last 387 0 R +/Count -8 +>> endobj +351 0 obj << +/Title 352 0 R +/A 349 0 R +/Parent 15 0 R +/Prev 279 0 R +/Next 455 0 R +/First 355 0 R +/Last 427 0 R +/Count -4 +>> endobj +347 0 obj << +/Title 348 0 R +/A 345 0 R +/Parent 279 0 R +/Prev 343 0 R +>> endobj +343 0 obj << +/Title 344 0 R +/A 341 0 R +/Parent 279 0 R +/Prev 331 0 R +/Next 347 0 R +>> endobj +339 0 obj << +/Title 340 0 R +/A 337 0 R +/Parent 331 0 R +/Prev 335 0 R +>> endobj +335 0 obj << +/Title 336 0 R +/A 333 0 R +/Parent 331 0 R +/Next 339 0 R +>> endobj +331 0 obj << +/Title 332 0 R +/A 329 0 R +/Parent 279 0 R +/Prev 315 0 R +/Next 343 0 R +/First 335 0 R +/Last 339 0 R +/Count -2 +>> endobj +327 0 obj << +/Title 328 0 R +/A 325 0 R +/Parent 315 0 R +/Prev 323 0 R +>> endobj +323 0 obj << +/Title 324 0 R +/A 321 0 R +/Parent 315 0 R +/Prev 319 0 R +/Next 327 0 R +>> endobj +319 0 obj << +/Title 320 0 R +/A 317 0 R +/Parent 315 0 R +/Next 323 0 R +>> endobj +315 0 obj << +/Title 316 0 R +/A 313 0 R +/Parent 279 0 R +/Prev 311 0 R +/Next 331 0 R +/First 319 0 R +/Last 327 0 R +/Count -3 +>> endobj +311 0 obj << +/Title 312 0 R +/A 309 0 R +/Parent 279 0 R +/Prev 291 0 R +/Next 315 0 R +>> endobj +307 0 obj << +/Title 308 0 R +/A 305 0 R +/Parent 291 0 R +/Prev 303 0 R +>> endobj +303 0 obj << +/Title 304 0 R +/A 301 0 R +/Parent 291 0 R +/Prev 299 0 R +/Next 307 0 R +>> endobj +299 0 obj << +/Title 300 0 R +/A 297 0 R +/Parent 291 0 R +/Prev 295 0 R +/Next 303 0 R +>> endobj +295 0 obj << +/Title 296 0 R +/A 293 0 R +/Parent 291 0 R +/Next 299 0 R +>> endobj +291 0 obj << +/Title 292 0 R +/A 289 0 R +/Parent 279 0 R +/Prev 287 0 R +/Next 311 0 R +/First 295 0 R +/Last 307 0 R +/Count -4 +>> endobj +287 0 obj << +/Title 288 0 R +/A 285 0 R +/Parent 279 0 R +/Prev 283 0 R +/Next 291 0 R +>> endobj +283 0 obj << +/Title 284 0 R +/A 281 0 R +/Parent 279 0 R +/Next 287 0 R +>> endobj +279 0 obj << +/Title 280 0 R +/A 277 0 R +/Parent 15 0 R +/Prev 167 0 R +/Next 351 0 R +/First 283 0 R +/Last 347 0 R +/Count -8 +>> endobj +275 0 obj << +/Title 276 0 R +/A 273 0 R +/Parent 167 0 R +/Prev 251 0 R +>> endobj +271 0 obj << +/Title 272 0 R +/A 269 0 R +/Parent 251 0 R +/Prev 267 0 R +>> endobj +267 0 obj << +/Title 268 0 R +/A 265 0 R +/Parent 251 0 R +/Prev 263 0 R +/Next 271 0 R +>> endobj +263 0 obj << +/Title 264 0 R +/A 261 0 R +/Parent 251 0 R +/Prev 259 0 R +/Next 267 0 R +>> endobj +259 0 obj << +/Title 260 0 R +/A 257 0 R +/Parent 251 0 R +/Prev 255 0 R +/Next 263 0 R +>> endobj +255 0 obj << +/Title 256 0 R +/A 253 0 R +/Parent 251 0 R +/Next 259 0 R +>> endobj +251 0 obj << +/Title 252 0 R +/A 249 0 R +/Parent 167 0 R +/Prev 227 0 R +/Next 275 0 R +/First 255 0 R +/Last 271 0 R +/Count -5 +>> endobj +247 0 obj << +/Title 248 0 R +/A 245 0 R +/Parent 239 0 R +/Prev 243 0 R +>> endobj +243 0 obj << +/Title 244 0 R +/A 241 0 R +/Parent 239 0 R +/Next 247 0 R +>> endobj +239 0 obj << +/Title 240 0 R +/A 237 0 R +/Parent 227 0 R +/Prev 235 0 R +/First 243 0 R +/Last 247 0 R +/Count -2 +>> endobj +235 0 obj << +/Title 236 0 R +/A 233 0 R +/Parent 227 0 R +/Prev 231 0 R +/Next 239 0 R +>> endobj +231 0 obj << +/Title 232 0 R +/A 229 0 R +/Parent 227 0 R +/Next 235 0 R +>> endobj +227 0 obj << +/Title 228 0 R +/A 225 0 R +/Parent 167 0 R +/Prev 207 0 R +/Next 251 0 R +/First 231 0 R +/Last 239 0 R +/Count -3 +>> endobj +223 0 obj << +/Title 224 0 R +/A 221 0 R +/Parent 207 0 R +/Prev 219 0 R +>> endobj +219 0 obj << +/Title 220 0 R +/A 217 0 R +/Parent 207 0 R +/Prev 215 0 R +/Next 223 0 R +>> endobj +215 0 obj << +/Title 216 0 R +/A 213 0 R +/Parent 207 0 R +/Prev 211 0 R +/Next 219 0 R +>> endobj +211 0 obj << +/Title 212 0 R +/A 209 0 R +/Parent 207 0 R +/Next 215 0 R +>> endobj +207 0 obj << +/Title 208 0 R +/A 205 0 R +/Parent 167 0 R +/Prev 191 0 R +/Next 227 0 R +/First 211 0 R +/Last 223 0 R +/Count -4 +>> endobj +203 0 obj << +/Title 204 0 R +/A 201 0 R +/Parent 191 0 R +/Prev 199 0 R +>> endobj +199 0 obj << +/Title 200 0 R +/A 197 0 R +/Parent 191 0 R +/Prev 195 0 R +/Next 203 0 R +>> endobj +195 0 obj << +/Title 196 0 R +/A 193 0 R +/Parent 191 0 R +/Next 199 0 R +>> endobj +191 0 obj << +/Title 192 0 R +/A 189 0 R +/Parent 167 0 R +/Prev 171 0 R +/Next 207 0 R +/First 195 0 R +/Last 203 0 R +/Count -3 +>> endobj +187 0 obj << +/Title 188 0 R +/A 185 0 R +/Parent 171 0 R +/Prev 183 0 R +>> endobj +183 0 obj << +/Title 184 0 R +/A 181 0 R +/Parent 171 0 R +/Prev 179 0 R +/Next 187 0 R +>> endobj +179 0 obj << +/Title 180 0 R +/A 177 0 R +/Parent 171 0 R +/Prev 175 0 R +/Next 183 0 R +>> endobj +175 0 obj << +/Title 176 0 R +/A 173 0 R +/Parent 171 0 R +/Next 179 0 R +>> endobj +171 0 obj << +/Title 172 0 R +/A 169 0 R +/Parent 167 0 R +/Next 191 0 R +/First 175 0 R +/Last 187 0 R +/Count -4 +>> endobj +167 0 obj << +/Title 168 0 R +/A 165 0 R +/Parent 15 0 R +/Prev 71 0 R +/Next 279 0 R +/First 171 0 R +/Last 275 0 R +/Count -6 +>> endobj +163 0 obj << +/Title 164 0 R +/A 161 0 R +/Parent 151 0 R +/Prev 159 0 R +>> endobj +159 0 obj << +/Title 160 0 R +/A 157 0 R +/Parent 151 0 R +/Prev 155 0 R +/Next 163 0 R +>> endobj +155 0 obj << +/Title 156 0 R +/A 153 0 R +/Parent 151 0 R +/Next 159 0 R +>> endobj +151 0 obj << +/Title 152 0 R +/A 149 0 R +/Parent 71 0 R +/Prev 147 0 R +/First 155 0 R +/Last 163 0 R +/Count -3 +>> endobj +147 0 obj << +/Title 148 0 R +/A 145 0 R +/Parent 71 0 R +/Prev 131 0 R +/Next 151 0 R +>> endobj +143 0 obj << +/Title 144 0 R +/A 141 0 R +/Parent 131 0 R +/Prev 139 0 R +>> endobj +139 0 obj << +/Title 140 0 R +/A 137 0 R +/Parent 131 0 R +/Prev 135 0 R +/Next 143 0 R +>> endobj +135 0 obj << +/Title 136 0 R +/A 133 0 R +/Parent 131 0 R +/Next 139 0 R +>> endobj +131 0 obj << +/Title 132 0 R +/A 129 0 R +/Parent 71 0 R +/Prev 127 0 R +/Next 147 0 R +/First 135 0 R +/Last 143 0 R +/Count -3 +>> endobj +127 0 obj << +/Title 128 0 R +/A 125 0 R +/Parent 71 0 R +/Prev 123 0 R +/Next 131 0 R +>> endobj +123 0 obj << +/Title 124 0 R +/A 121 0 R +/Parent 71 0 R +/Prev 119 0 R +/Next 127 0 R +>> endobj +119 0 obj << +/Title 120 0 R +/A 117 0 R +/Parent 71 0 R +/Prev 115 0 R +/Next 123 0 R +>> endobj +115 0 obj << +/Title 116 0 R +/A 113 0 R +/Parent 71 0 R +/Prev 111 0 R +/Next 119 0 R +>> endobj +111 0 obj << +/Title 112 0 R +/A 109 0 R +/Parent 71 0 R +/Prev 107 0 R +/Next 115 0 R +>> endobj +107 0 obj << +/Title 108 0 R +/A 105 0 R +/Parent 71 0 R +/Prev 103 0 R +/Next 111 0 R +>> endobj +103 0 obj << +/Title 104 0 R +/A 101 0 R +/Parent 71 0 R +/Prev 99 0 R +/Next 107 0 R +>> endobj +99 0 obj << +/Title 100 0 R +/A 97 0 R +/Parent 71 0 R +/Prev 95 0 R +/Next 103 0 R +>> endobj +95 0 obj << +/Title 96 0 R +/A 93 0 R +/Parent 71 0 R +/Prev 91 0 R +/Next 99 0 R +>> endobj +91 0 obj << +/Title 92 0 R +/A 89 0 R +/Parent 71 0 R +/Prev 87 0 R +/Next 95 0 R +>> endobj +87 0 obj << +/Title 88 0 R +/A 85 0 R +/Parent 71 0 R +/Prev 83 0 R +/Next 91 0 R +>> endobj +83 0 obj << +/Title 84 0 R +/A 81 0 R +/Parent 71 0 R +/Prev 79 0 R +/Next 87 0 R +>> endobj +79 0 obj << +/Title 80 0 R +/A 77 0 R +/Parent 71 0 R +/Prev 75 0 R +/Next 83 0 R +>> endobj +75 0 obj << +/Title 76 0 R +/A 73 0 R +/Parent 71 0 R +/Next 79 0 R +>> endobj +71 0 obj << +/Title 72 0 R +/A 69 0 R +/Parent 15 0 R +/Prev 39 0 R +/Next 167 0 R +/First 75 0 R +/Last 151 0 R +/Count -17 +>> endobj +67 0 obj << +/Title 68 0 R +/A 65 0 R +/Parent 39 0 R +/Prev 63 0 R +>> endobj +63 0 obj << +/Title 64 0 R +/A 61 0 R +/Parent 39 0 R +/Prev 59 0 R +/Next 67 0 R +>> endobj +59 0 obj << +/Title 60 0 R +/A 57 0 R +/Parent 39 0 R +/Prev 55 0 R +/Next 63 0 R +>> endobj +55 0 obj << +/Title 56 0 R +/A 53 0 R +/Parent 39 0 R +/Prev 51 0 R +/Next 59 0 R +>> endobj +51 0 obj << +/Title 52 0 R +/A 49 0 R +/Parent 39 0 R +/Prev 47 0 R +/Next 55 0 R +>> endobj +47 0 obj << +/Title 48 0 R +/A 45 0 R +/Parent 39 0 R +/Prev 43 0 R +/Next 51 0 R +>> endobj +43 0 obj << +/Title 44 0 R +/A 41 0 R +/Parent 39 0 R +/Next 47 0 R +>> endobj +39 0 obj << +/Title 40 0 R +/A 37 0 R +/Parent 15 0 R +/Prev 19 0 R +/Next 71 0 R +/First 43 0 R +/Last 67 0 R +/Count -7 +>> endobj +35 0 obj << +/Title 36 0 R +/A 33 0 R +/Parent 19 0 R +/Prev 31 0 R +>> endobj +31 0 obj << +/Title 32 0 R +/A 29 0 R +/Parent 19 0 R +/Prev 27 0 R +/Next 35 0 R +>> endobj +27 0 obj << +/Title 28 0 R +/A 25 0 R +/Parent 19 0 R +/Prev 23 0 R +/Next 31 0 R +>> endobj +23 0 obj << +/Title 24 0 R +/A 21 0 R +/Parent 19 0 R +/Next 27 0 R +>> endobj +19 0 obj << +/Title 20 0 R +/A 17 0 R +/Parent 15 0 R +/Next 39 0 R +/First 23 0 R +/Last 35 0 R +/Count -4 +>> endobj +15 0 obj << +/Title 16 0 R +/A 13 0 R +/Parent 4985 0 R +/Prev 11 0 R +/Next 615 0 R +/First 19 0 R +/Last 599 0 R +/Count -12 +>> endobj +11 0 obj << +/Title 12 0 R +/A 9 0 R +/Parent 4985 0 R +/Prev 7 0 R +/Next 15 0 R +>> endobj +7 0 obj << +/Title 8 0 R +/A 5 0 R +/Parent 4985 0 R +/Prev 3 0 R +/Next 11 0 R +>> endobj +3 0 obj << +/Title 4 0 R +/A 1 0 R +/Parent 4985 0 R +/Next 7 0 R +>> endobj +4986 0 obj << +/Names [(Doc-Start) 1245 0 R (Hfootnote.1) 1651 0 R (Hfootnote.2) 1849 0 R (Hfootnote.3) 2150 0 R (Item.1) 1622 0 R (Item.10) 1690 0 R] +/Limits [(Doc-Start) (Item.10)] +>> endobj +4987 0 obj << +/Names [(Item.11) 1695 0 R (Item.12) 1884 0 R (Item.13) 1885 0 R (Item.14) 1886 0 R (Item.15) 1887 0 R (Item.16) 1888 0 R] +/Limits [(Item.11) (Item.16)] +>> endobj +4988 0 obj << +/Names [(Item.17) 1896 0 R (Item.18) 1897 0 R (Item.19) 1898 0 R (Item.2) 1623 0 R (Item.20) 1899 0 R (Item.21) 1900 0 R] +/Limits [(Item.17) (Item.21)] +>> endobj +4989 0 obj << +/Names [(Item.22) 1913 0 R (Item.23) 1977 0 R (Item.24) 1978 0 R (Item.25) 1979 0 R (Item.26) 1980 0 R (Item.27) 1981 0 R] +/Limits [(Item.22) (Item.27)] +>> endobj +4990 0 obj << +/Names [(Item.28) 1982 0 R (Item.29) 1983 0 R (Item.3) 1624 0 R (Item.30) 2047 0 R (Item.31) 2048 0 R (Item.32) 2098 0 R] +/Limits [(Item.28) (Item.32)] +>> endobj +4991 0 obj << +/Names [(Item.33) 2099 0 R (Item.34) 2100 0 R (Item.35) 2101 0 R (Item.36) 2165 0 R (Item.37) 2166 0 R (Item.38) 2167 0 R] +/Limits [(Item.33) (Item.38)] +>> endobj +4992 0 obj << +/Names [(Item.39) 2168 0 R (Item.4) 1625 0 R (Item.40) 2174 0 R (Item.41) 2182 0 R (Item.42) 2183 0 R (Item.43) 2184 0 R] +/Limits [(Item.39) (Item.43)] +>> endobj +4993 0 obj << +/Names [(Item.44) 2629 0 R (Item.45) 2630 0 R (Item.46) 2631 0 R (Item.47) 2641 0 R (Item.48) 2642 0 R (Item.49) 2643 0 R] +/Limits [(Item.44) (Item.49)] +>> endobj +4994 0 obj << +/Names [(Item.5) 1626 0 R (Item.50) 2644 0 R (Item.51) 2645 0 R (Item.52) 2646 0 R (Item.53) 3190 0 R (Item.54) 3191 0 R] +/Limits [(Item.5) (Item.54)] +>> endobj +4995 0 obj << +/Names [(Item.55) 3192 0 R (Item.6) 1679 0 R (Item.7) 1680 0 R (Item.8) 1681 0 R (Item.9) 1682 0 R (chapter*.1) 1281 0 R] +/Limits [(Item.55) (chapter*.1)] +>> endobj +4996 0 obj << +/Names [(chapter*.2) 1620 0 R (chapter*.42) 3290 0 R (chapter.1) 18 0 R (chapter.10) 546 0 R (chapter.11) 574 0 R (chapter.12) 598 0 R] +/Limits [(chapter*.2) (chapter.12)] +>> endobj +4997 0 obj << +/Names [(chapter.13) 618 0 R (chapter.14) 650 0 R (chapter.15) 678 0 R (chapter.16) 682 0 R (chapter.17) 718 0 R (chapter.18) 738 0 R] +/Limits [(chapter.13) (chapter.18)] +>> endobj +4998 0 obj << +/Names [(chapter.19) 746 0 R (chapter.2) 38 0 R (chapter.20) 758 0 R (chapter.21) 866 0 R (chapter.22) 878 0 R (chapter.23) 886 0 R] +/Limits [(chapter.19) (chapter.23)] +>> endobj +4999 0 obj << +/Names [(chapter.24) 894 0 R (chapter.25) 898 0 R (chapter.26) 902 0 R (chapter.27) 922 0 R (chapter.28) 942 0 R (chapter.29) 966 0 R] +/Limits [(chapter.24) (chapter.29)] +>> endobj +5000 0 obj << +/Names [(chapter.3) 70 0 R (chapter.30) 986 0 R (chapter.31) 990 0 R (chapter.32) 1006 0 R (chapter.33) 1038 0 R (chapter.34) 1042 0 R] +/Limits [(chapter.3) (chapter.34)] +>> endobj +5001 0 obj << +/Names [(chapter.35) 1070 0 R (chapter.36) 1086 0 R (chapter.37) 1098 0 R (chapter.38) 1102 0 R (chapter.39) 1118 0 R (chapter.4) 166 0 R] +/Limits [(chapter.35) (chapter.4)] +>> endobj +5002 0 obj << +/Names [(chapter.40) 1122 0 R (chapter.41) 1126 0 R (chapter.42) 1210 0 R (chapter.5) 278 0 R (chapter.6) 350 0 R (chapter.7) 454 0 R] +/Limits [(chapter.40) (chapter.7)] +>> endobj +5003 0 obj << +/Names [(chapter.8) 470 0 R (chapter.9) 530 0 R (cite.C) 2360 0 R (cite.C99) 2417 0 R (cite.back78) 1652 0 R (cite.curry&feys:book) 1627 0 R] +/Limits [(chapter.8) (cite.curry&feys:book)] +>> endobj +5004 0 obj << +/Names [(cite.damas-milner82) 1954 0 R (cite.gosling-etal:Java) 2361 0 R (cite.hindley69) 1955 0 R (cite.jones:cclasses) 1961 0 R (cite.liang:JNI) 2382 0 R (cite.lindholm-etal:JVM) 2374 0 R] +/Limits [(cite.damas-milner82) (cite.lindholm-etal:JVM)] +>> endobj +5005 0 obj << +/Names [(cite.penfield:complex-apl) 2321 0 R (cite.unicode) 1708 0 R (cite.wadler:classes) 1962 0 R (contents.0) 6 0 R (figure.11.1) 2667 0 R (figure.2.1) 1748 0 R] +/Limits [(cite.penfield:complex-apl) (figure.2.1)] +>> endobj +5006 0 obj << +/Names [(figure.2.2) 1749 0 R (figure.3.1) 1914 0 R (figure.3.2) 1916 0 R (figure.3.3) 1915 0 R (figure.6.1) 1824 0 R (figure.6.2) 2069 0 R] +/Limits [(figure.2.2) (figure.6.2)] +>> endobj +5007 0 obj << +/Names [(figure.6.3) 2070 0 R (page.0) 1244 0 R (page.1) 1253 0 R (page.10) 1726 0 R (page.100) 2421 0 R (page.101) 2438 0 R] +/Limits [(figure.6.3) (page.101)] +>> endobj +5008 0 obj << +/Names [(page.102) 2445 0 R (page.103) 2455 0 R (page.104) 2459 0 R (page.105) 2464 0 R (page.106) 2468 0 R (page.107) 2472 0 R] +/Limits [(page.102) (page.107)] +>> endobj +5009 0 obj << +/Names [(page.108) 2477 0 R (page.109) 2481 0 R (page.11) 1733 0 R (page.110) 2485 0 R (page.111) 2489 0 R (page.112) 2493 0 R] +/Limits [(page.108) (page.112)] +>> endobj +5010 0 obj << +/Names [(page.113) 2497 0 R (page.114) 2502 0 R (page.115) 2506 0 R (page.116) 2510 0 R (page.117) 2514 0 R (page.118) 2518 0 R] +/Limits [(page.113) (page.118)] +>> endobj +5011 0 obj << +/Names [(page.119) 2522 0 R (page.12) 1740 0 R (page.120) 2527 0 R (page.121) 2531 0 R (page.122) 2535 0 R (page.123) 2539 0 R] +/Limits [(page.119) (page.123)] +>> endobj +5012 0 obj << +/Names [(page.124) 2543 0 R (page.125) 2547 0 R (page.126) 2552 0 R (page.127) 2556 0 R (page.128) 2560 0 R (page.129) 2564 0 R] +/Limits [(page.124) (page.129)] +>> endobj +5013 0 obj << +/Names [(page.13) 1747 0 R (page.130) 2568 0 R (page.131) 2573 0 R (page.132) 2579 0 R (page.133) 2584 0 R (page.134) 2588 0 R] +/Limits [(page.13) (page.134)] +>> endobj +5014 0 obj << +/Names [(page.135) 2593 0 R (page.136) 2597 0 R (page.137) 2601 0 R (page.138) 2606 0 R (page.139) 2612 0 R (page.14) 1753 0 R] +/Limits [(page.135) (page.14)] +>> endobj +5015 0 obj << +/Names [(page.140) 2616 0 R (page.141) 2620 0 R (page.142) 2624 0 R (page.143) 2628 0 R (page.144) 2636 0 R (page.145) 2640 0 R] +/Limits [(page.140) (page.145)] +>> endobj +5016 0 obj << +/Names [(page.146) 2650 0 R (page.147) 2654 0 R (page.148) 2659 0 R (page.149) 2665 0 R (page.15) 1758 0 R (page.150) 2671 0 R] +/Limits [(page.146) (page.150)] +>> endobj +5017 0 obj << +/Names [(page.151) 2675 0 R (page.152) 2679 0 R (page.153) 2683 0 R (page.154) 2687 0 R (page.155) 2691 0 R (page.156) 2696 0 R] +/Limits [(page.151) (page.156)] +>> endobj +5018 0 obj << +/Names [(page.157) 2701 0 R (page.158) 2705 0 R (page.159) 2709 0 R (page.16) 1766 0 R (page.160) 2713 0 R (page.161) 2717 0 R] +/Limits [(page.157) (page.161)] +>> endobj +5019 0 obj << +/Names [(page.162) 2722 0 R (page.163) 2726 0 R (page.164) 2730 0 R (page.165) 2734 0 R (page.166) 2738 0 R (page.167) 2742 0 R] +/Limits [(page.162) (page.167)] +>> endobj +5020 0 obj << +/Names [(page.168) 2747 0 R (page.169) 2751 0 R (page.17) 1777 0 R (page.170) 2755 0 R (page.171) 2759 0 R (page.172) 2764 0 R] +/Limits [(page.168) (page.172)] +>> endobj +5021 0 obj << +/Names [(page.173) 2768 0 R (page.174) 2773 0 R (page.175) 2777 0 R (page.176) 2781 0 R (page.177) 2785 0 R (page.178) 2789 0 R] +/Limits [(page.173) (page.178)] +>> endobj +5022 0 obj << +/Names [(page.179) 2793 0 R (page.18) 1785 0 R (page.180) 2798 0 R (page.181) 2802 0 R (page.182) 2806 0 R (page.183) 2810 0 R] +/Limits [(page.179) (page.183)] +>> endobj +5023 0 obj << +/Names [(page.184) 2814 0 R (page.185) 2818 0 R (page.186) 2823 0 R (page.187) 2827 0 R (page.188) 2831 0 R (page.189) 2835 0 R] +/Limits [(page.184) (page.189)] +>> endobj +5024 0 obj << +/Names [(page.19) 1795 0 R (page.190) 2839 0 R (page.191) 2843 0 R (page.192) 2848 0 R (page.193) 2852 0 R (page.194) 2856 0 R] +/Limits [(page.19) (page.194)] +>> endobj +5025 0 obj << +/Names [(page.195) 2860 0 R (page.196) 2864 0 R (page.197) 2868 0 R (page.198) 2873 0 R (page.199) 2877 0 R (page.2) 1668 0 R] +/Limits [(page.195) (page.2)] +>> endobj +5026 0 obj << +/Names [(page.20) 1809 0 R (page.200) 2881 0 R (page.201) 2885 0 R (page.202) 2889 0 R (page.203) 2893 0 R (page.204) 2898 0 R] +/Limits [(page.20) (page.204)] +>> endobj +5027 0 obj << +/Names [(page.205) 2902 0 R (page.206) 2906 0 R (page.207) 2910 0 R (page.208) 2914 0 R (page.209) 2918 0 R (page.21) 1820 0 R] +/Limits [(page.205) (page.21)] +>> endobj +5028 0 obj << +/Names [(page.210) 2923 0 R (page.211) 2927 0 R (page.212) 2931 0 R (page.213) 2935 0 R (page.214) 2939 0 R (page.215) 2943 0 R] +/Limits [(page.210) (page.215)] +>> endobj +5029 0 obj << +/Names [(page.216) 2948 0 R (page.217) 2952 0 R (page.218) 2956 0 R (page.219) 2960 0 R (page.22) 1833 0 R (page.220) 2964 0 R] +/Limits [(page.216) (page.220)] +>> endobj +5030 0 obj << +/Names [(page.221) 2968 0 R (page.222) 2973 0 R (page.223) 2977 0 R (page.224) 2981 0 R (page.225) 2985 0 R (page.226) 2989 0 R] +/Limits [(page.221) (page.226)] +>> endobj +5031 0 obj << +/Names [(page.227) 2993 0 R (page.228) 2998 0 R (page.229) 3002 0 R (page.23) 1840 0 R (page.230) 3006 0 R (page.231) 3010 0 R] +/Limits [(page.227) (page.231)] +>> endobj +5032 0 obj << +/Names [(page.232) 3014 0 R (page.233) 3018 0 R (page.234) 3023 0 R (page.235) 3027 0 R (page.236) 3031 0 R (page.237) 3035 0 R] +/Limits [(page.232) (page.237)] +>> endobj +5033 0 obj << +/Names [(page.238) 3039 0 R (page.239) 3043 0 R (page.24) 1848 0 R (page.240) 3048 0 R (page.241) 3052 0 R (page.242) 3056 0 R] +/Limits [(page.238) (page.242)] +>> endobj +5034 0 obj << +/Names [(page.243) 3060 0 R (page.244) 3064 0 R (page.245) 3068 0 R (page.246) 3073 0 R (page.247) 3077 0 R (page.248) 3081 0 R] +/Limits [(page.243) (page.248)] +>> endobj +5035 0 obj << +/Names [(page.249) 3085 0 R (page.25) 1855 0 R (page.250) 3089 0 R (page.251) 3093 0 R (page.252) 3098 0 R (page.253) 3102 0 R] +/Limits [(page.249) (page.253)] +>> endobj +5036 0 obj << +/Names [(page.254) 3106 0 R (page.255) 3110 0 R (page.256) 3114 0 R (page.257) 3118 0 R (page.258) 3123 0 R (page.259) 3127 0 R] +/Limits [(page.254) (page.259)] +>> endobj +5037 0 obj << +/Names [(page.26) 1863 0 R (page.260) 3131 0 R (page.261) 3135 0 R (page.262) 3139 0 R (page.263) 3143 0 R (page.264) 3148 0 R] +/Limits [(page.26) (page.264)] +>> endobj +5038 0 obj << +/Names [(page.265) 3152 0 R (page.266) 3156 0 R (page.267) 3160 0 R (page.268) 3164 0 R (page.269) 3168 0 R (page.27) 1869 0 R] +/Limits [(page.265) (page.27)] +>> endobj +5039 0 obj << +/Names [(page.270) 3173 0 R (page.271) 3177 0 R (page.272) 3181 0 R (page.273) 3185 0 R (page.274) 3189 0 R (page.275) 3196 0 R] +/Limits [(page.270) (page.275)] +>> endobj +5040 0 obj << +/Names [(page.276) 3201 0 R (page.277) 3205 0 R (page.278) 3209 0 R (page.279) 3213 0 R (page.28) 1878 0 R (page.280) 3217 0 R] +/Limits [(page.276) (page.280)] +>> endobj +5041 0 obj << +/Names [(page.281) 3221 0 R (page.282) 3226 0 R (page.283) 3230 0 R (page.284) 3234 0 R (page.285) 3238 0 R (page.286) 3242 0 R] +/Limits [(page.281) (page.286)] +>> endobj +5042 0 obj << +/Names [(page.287) 3246 0 R (page.288) 3251 0 R (page.289) 3255 0 R (page.29) 1883 0 R (page.290) 3259 0 R (page.291) 3263 0 R] +/Limits [(page.287) (page.291)] +>> endobj +5043 0 obj << +/Names [(page.292) 3267 0 R (page.293) 3271 0 R (page.294) 3276 0 R (page.295) 3280 0 R (page.296) 3284 0 R (page.297) 3289 0 R] +/Limits [(page.292) (page.297)] +>> endobj +5044 0 obj << +/Names [(page.298) 3294 0 R (page.299) 3464 0 R (page.3) 1678 0 R (page.30) 1895 0 R (page.300) 3614 0 R (page.301) 3762 0 R] +/Limits [(page.298) (page.301)] +>> endobj +5045 0 obj << +/Names [(page.302) 3930 0 R (page.303) 4104 0 R (page.304) 4271 0 R (page.305) 4426 0 R (page.306) 4595 0 R (page.307) 4756 0 R] +/Limits [(page.302) (page.307)] +>> endobj +5046 0 obj << +/Names [(page.308) 4891 0 R (page.309) 4895 0 R (page.31) 1912 0 R (page.32) 1920 0 R (page.33) 1928 0 R (page.34) 1933 0 R] +/Limits [(page.308) (page.34)] +>> endobj +5047 0 obj << +/Names [(page.35) 1937 0 R (page.36) 1953 0 R (page.37) 1960 0 R (page.38) 1976 0 R (page.39) 1993 0 R (page.4) 1687 0 R] +/Limits [(page.35) (page.4)] +>> endobj +5048 0 obj << +/Names [(page.40) 1997 0 R (page.41) 2004 0 R (page.42) 2013 0 R (page.43) 2021 0 R (page.44) 2026 0 R (page.45) 2031 0 R] +/Limits [(page.40) (page.45)] +>> endobj +5049 0 obj << +/Names [(page.46) 2038 0 R (page.47) 2046 0 R (page.48) 2057 0 R (page.49) 2068 0 R (page.5) 1694 0 R (page.50) 2076 0 R] +/Limits [(page.46) (page.50)] +>> endobj +5050 0 obj << +/Names [(page.51) 2080 0 R (page.52) 2085 0 R (page.53) 2090 0 R (page.54) 2097 0 R (page.55) 2107 0 R (page.56) 2116 0 R] +/Limits [(page.51) (page.56)] +>> endobj +5051 0 obj << +/Names [(page.57) 2121 0 R (page.58) 2129 0 R (page.59) 2135 0 R (page.6) 1700 0 R (page.60) 2140 0 R (page.61) 2149 0 R] +/Limits [(page.57) (page.61)] +>> endobj +5052 0 obj << +/Names [(page.62) 2159 0 R (page.63) 2164 0 R (page.64) 2173 0 R (page.65) 2181 0 R (page.66) 2189 0 R (page.67) 2195 0 R] +/Limits [(page.62) (page.67)] +>> endobj +5053 0 obj << +/Names [(page.68) 2201 0 R (page.69) 2206 0 R (page.7) 1705 0 R (page.70) 2213 0 R (page.71) 2217 0 R (page.72) 2222 0 R] +/Limits [(page.68) (page.72)] +>> endobj +5054 0 obj << +/Names [(page.73) 2231 0 R (page.74) 2243 0 R (page.75) 2250 0 R (page.76) 2258 0 R (page.77) 2268 0 R (page.78) 2274 0 R] +/Limits [(page.73) (page.78)] +>> endobj +5055 0 obj << +/Names [(page.79) 2282 0 R (page.8) 1712 0 R (page.80) 2289 0 R (page.81) 2294 0 R (page.82) 2305 0 R (page.83) 2310 0 R] +/Limits [(page.79) (page.83)] +>> endobj +5056 0 obj << +/Names [(page.84) 2315 0 R (page.85) 2320 0 R (page.86) 2325 0 R (page.87) 2331 0 R (page.88) 2337 0 R (page.89) 2343 0 R] +/Limits [(page.84) (page.89)] +>> endobj +5057 0 obj << +/Names [(page.9) 1718 0 R (page.90) 2350 0 R (page.91) 2359 0 R (page.92) 2366 0 R (page.93) 2373 0 R (page.94) 2380 0 R] +/Limits [(page.9) (page.94)] +>> endobj +5058 0 obj << +/Names [(page.95) 2388 0 R (page.96) 2396 0 R (page.97) 2401 0 R (page.98) 2407 0 R (page.99) 2414 0 R (page.i) 1280 0 R] +/Limits [(page.95) (page.i)] +>> endobj +5059 0 obj << +/Names [(page.ii) 1313 0 R (page.iii) 1346 0 R (page.iv) 1378 0 R (page.ix) 1535 0 R (page.v) 1410 0 R (page.vi) 1443 0 R] +/Limits [(page.ii) (page.vi)] +>> endobj +5060 0 obj << +/Names [(page.vii) 1474 0 R (page.viii) 1507 0 R (page.x) 1565 0 R (page.xi) 1595 0 R (page.xii) 1614 0 R (page.xiii) 1619 0 R] +/Limits [(page.vii) (page.xiii)] +>> endobj +5061 0 obj << +/Names [(page.xiv) 1632 0 R (page.xv) 1642 0 R (page.xvi) 1649 0 R (page.xvii) 1656 0 R (page.xviii) 1661 0 R (part.1) 14 0 R] +/Limits [(page.xiv) (part.1)] +>> endobj +5062 0 obj << +/Names [(part.2) 614 0 R (preface.0) 10 0 R (section*.10) 1790 0 R (section*.11) 1796 0 R (section*.12) 1810 0 R (section*.13) 1811 0 R] +/Limits [(part.2) (section*.13)] +>> endobj +5063 0 obj << +/Names [(section*.14) 1821 0 R (section*.15) 1822 0 R (section*.16) 1823 0 R (section*.17) 1834 0 R (section*.18) 1841 0 R (section*.19) 1856 0 R] +/Limits [(section*.14) (section*.19)] +>> endobj +5064 0 obj << +/Names [(section*.20) 1864 0 R (section*.21) 1870 0 R (section*.22) 1871 0 R (section*.23) 1879 0 R (section*.24) 2005 0 R (section*.25) 1865 0 R] +/Limits [(section*.20) (section*.25)] +>> endobj +5065 0 obj << +/Names [(section*.26) 2014 0 R (section*.27) 2091 0 R (section*.28) 2092 0 R (section*.29) 2122 0 R (section*.3) 1621 0 R (section*.30) 2123 0 R] +/Limits [(section*.26) (section*.30)] +>> endobj +5066 0 obj << +/Names [(section*.31) 2130 0 R (section*.32) 2332 0 R (section*.33) 2338 0 R (section*.34) 2339 0 R (section*.35) 2381 0 R (section*.36) 2389 0 R] +/Limits [(section*.31) (section*.36)] +>> endobj +5067 0 obj << +/Names [(section*.37) 2402 0 R (section*.38) 2408 0 R (section*.39) 2409 0 R (section*.4) 1633 0 R (section*.40) 2415 0 R (section*.41) 2416 0 R] +/Limits [(section*.37) (section*.41)] +>> endobj +5068 0 obj << +/Names [(section*.43) 1237 0 R (section*.5) 1634 0 R (section*.6) 1638 0 R (section*.7) 1643 0 R (section*.8) 1778 0 R (section*.9) 1786 0 R] +/Limits [(section*.43) (section*.9)] +>> endobj +5069 0 obj << +/Names [(section.1.1) 22 0 R (section.1.2) 26 0 R (section.1.3) 30 0 R (section.1.4) 34 0 R (section.10.1) 550 0 R (section.10.2) 554 0 R] +/Limits [(section.1.1) (section.10.2)] +>> endobj +5070 0 obj << +/Names [(section.10.3) 558 0 R (section.10.4) 562 0 R (section.10.5) 566 0 R (section.10.6) 570 0 R (section.11.1) 578 0 R (section.11.2) 582 0 R] +/Limits [(section.10.3) (section.11.2)] +>> endobj +5071 0 obj << +/Names [(section.11.3) 586 0 R (section.11.4) 590 0 R (section.11.5) 594 0 R (section.12.1) 602 0 R (section.12.2) 606 0 R (section.12.3) 610 0 R] +/Limits [(section.11.3) (section.12.3)] +>> endobj +5072 0 obj << +/Names [(section.13.1) 622 0 R (section.13.2) 626 0 R (section.14.1) 654 0 R (section.14.2) 658 0 R (section.14.3) 662 0 R (section.14.4) 666 0 R] +/Limits [(section.13.1) (section.14.4)] +>> endobj +5073 0 obj << +/Names [(section.14.5) 670 0 R (section.14.6) 674 0 R (section.16.1) 686 0 R (section.16.2) 690 0 R (section.16.3) 702 0 R (section.16.4) 706 0 R] +/Limits [(section.14.5) (section.16.4)] +>> endobj +5074 0 obj << +/Names [(section.16.5) 710 0 R (section.16.6) 714 0 R (section.17.1) 722 0 R (section.17.2) 726 0 R (section.17.3) 730 0 R (section.17.4) 734 0 R] +/Limits [(section.16.5) (section.17.4)] +>> endobj +5075 0 obj << +/Names [(section.18.1) 742 0 R (section.19.1) 750 0 R (section.19.2) 754 0 R (section.2.1) 42 0 R (section.2.2) 46 0 R (section.2.3) 50 0 R] +/Limits [(section.18.1) (section.2.3)] +>> endobj +5076 0 obj << +/Names [(section.2.4) 54 0 R (section.2.5) 58 0 R (section.2.6) 62 0 R (section.2.7) 66 0 R (section.20.1) 762 0 R (section.20.10) 846 0 R] +/Limits [(section.2.4) (section.20.10)] +>> endobj +5077 0 obj << +/Names [(section.20.2) 766 0 R (section.20.3) 770 0 R (section.20.4) 778 0 R (section.20.5) 798 0 R (section.20.6) 810 0 R (section.20.7) 822 0 R] +/Limits [(section.20.2) (section.20.7)] +>> endobj +5078 0 obj << +/Names [(section.20.8) 826 0 R (section.20.9) 830 0 R (section.21.1) 870 0 R (section.21.2) 874 0 R (section.22.1) 882 0 R (section.23.1) 890 0 R] +/Limits [(section.20.8) (section.23.1)] +>> endobj +5079 0 obj << +/Names [(section.26.1) 906 0 R (section.27.1) 926 0 R (section.27.2) 938 0 R (section.28.1) 946 0 R (section.29.1) 970 0 R (section.3.1) 74 0 R] +/Limits [(section.26.1) (section.3.1)] +>> endobj +5080 0 obj << +/Names [(section.3.10) 110 0 R (section.3.11) 114 0 R (section.3.12) 118 0 R (section.3.13) 122 0 R (section.3.14) 126 0 R (section.3.15) 130 0 R] +/Limits [(section.3.10) (section.3.15)] +>> endobj +5081 0 obj << +/Names [(section.3.16) 146 0 R (section.3.17) 150 0 R (section.3.2) 78 0 R (section.3.3) 82 0 R (section.3.4) 86 0 R (section.3.5) 90 0 R] +/Limits [(section.3.16) (section.3.5)] +>> endobj +5082 0 obj << +/Names [(section.3.6) 94 0 R (section.3.7) 98 0 R (section.3.8) 102 0 R (section.3.9) 106 0 R (section.31.1) 994 0 R (section.32.1) 1010 0 R] +/Limits [(section.3.6) (section.32.1)] +>> endobj +5083 0 obj << +/Names [(section.34.1) 1046 0 R (section.35.1) 1074 0 R (section.35.2) 1078 0 R (section.35.3) 1082 0 R (section.36.1) 1090 0 R (section.38.1) 1106 0 R] +/Limits [(section.34.1) (section.38.1)] +>> endobj +5084 0 obj << +/Names [(section.38.2) 1110 0 R (section.38.3) 1114 0 R (section.4.1) 170 0 R (section.4.2) 190 0 R (section.4.3) 206 0 R (section.4.4) 226 0 R] +/Limits [(section.38.2) (section.4.4)] +>> endobj +5085 0 obj << +/Names [(section.4.5) 250 0 R (section.4.6) 274 0 R (section.41.1) 1130 0 R (section.41.2) 1134 0 R (section.41.3) 1142 0 R (section.41.4) 1162 0 R] +/Limits [(section.4.5) (section.41.4)] +>> endobj +5086 0 obj << +/Names [(section.41.5) 1194 0 R (section.42.1) 1214 0 R (section.42.2) 1226 0 R (section.42.3) 1230 0 R (section.5.1) 282 0 R (section.5.2) 286 0 R] +/Limits [(section.41.5) (section.5.2)] +>> endobj +5087 0 obj << +/Names [(section.5.3) 290 0 R (section.5.4) 310 0 R (section.5.5) 314 0 R (section.5.6) 330 0 R (section.5.7) 342 0 R (section.5.8) 346 0 R] +/Limits [(section.5.3) (section.5.8)] +>> endobj +5088 0 obj << +/Names [(section.6.1) 354 0 R (section.6.2) 390 0 R (section.6.3) 394 0 R (section.6.4) 426 0 R (section.7.1) 458 0 R (section.7.2) 462 0 R] +/Limits [(section.6.1) (section.7.2)] +>> endobj +5089 0 obj << +/Names [(section.7.3) 466 0 R (section.8.1) 474 0 R (section.8.2) 478 0 R (section.8.3) 486 0 R (section.8.4) 490 0 R (section.8.5) 510 0 R] +/Limits [(section.7.3) (section.8.5)] +>> endobj +5090 0 obj << +/Names [(section.8.6) 522 0 R (section.8.7) 526 0 R (section.9.1) 534 0 R (section.9.2) 538 0 R (section.9.3) 542 0 R (subsection.13.2.1) 630 0 R] +/Limits [(section.8.6) (subsection.13.2.1)] +>> endobj +5091 0 obj << +/Names [(subsection.13.2.2) 634 0 R (subsection.13.2.3) 638 0 R (subsection.13.2.4) 642 0 R (subsection.13.2.5) 646 0 R (subsection.16.2.1) 694 0 R (subsection.16.2.2) 698 0 R] +/Limits [(subsection.13.2.2) (subsection.16.2.2)] +>> endobj +5092 0 obj << +/Names [(subsection.20.10.1) 850 0 R (subsection.20.10.2) 862 0 R (subsection.20.3.1) 774 0 R (subsection.20.4.1) 782 0 R (subsection.20.4.2) 786 0 R (subsection.20.4.3) 790 0 R] +/Limits [(subsection.20.10.1) (subsection.20.4.3)] +>> endobj +5093 0 obj << +/Names [(subsection.20.4.4) 794 0 R (subsection.20.5.1) 802 0 R (subsection.20.5.2) 806 0 R (subsection.20.6.1) 814 0 R (subsection.20.6.2) 818 0 R (subsection.20.9.1) 834 0 R] +/Limits [(subsection.20.4.4) (subsection.20.9.1)] +>> endobj +5094 0 obj << +/Names [(subsection.20.9.2) 838 0 R (subsection.20.9.3) 842 0 R (subsection.26.1.1) 910 0 R (subsection.26.1.2) 914 0 R (subsection.26.1.3) 918 0 R (subsection.27.1.1) 930 0 R] +/Limits [(subsection.20.9.2) (subsection.27.1.1)] +>> endobj +5095 0 obj << +/Names [(subsection.27.1.2) 934 0 R (subsection.28.1.1) 950 0 R (subsection.28.1.2) 954 0 R (subsection.28.1.3) 958 0 R (subsection.28.1.4) 962 0 R (subsection.29.1.1) 974 0 R] +/Limits [(subsection.27.1.2) (subsection.29.1.1)] +>> endobj +5096 0 obj << +/Names [(subsection.29.1.2) 978 0 R (subsection.29.1.3) 982 0 R (subsection.3.15.1) 134 0 R (subsection.3.15.2) 138 0 R (subsection.3.15.3) 142 0 R (subsection.3.17.1) 154 0 R] +/Limits [(subsection.29.1.2) (subsection.3.17.1)] +>> endobj +5097 0 obj << +/Names [(subsection.3.17.2) 158 0 R (subsection.3.17.3) 162 0 R (subsection.31.1.1) 998 0 R (subsection.31.1.2) 1002 0 R (subsection.32.1.1) 1014 0 R (subsection.32.1.2) 1018 0 R] +/Limits [(subsection.3.17.2) (subsection.32.1.2)] +>> endobj +5098 0 obj << +/Names [(subsection.32.1.3) 1022 0 R (subsection.32.1.4) 1026 0 R (subsection.32.1.5) 1030 0 R (subsection.32.1.6) 1034 0 R (subsection.34.1.1) 1050 0 R (subsection.34.1.2) 1054 0 R] +/Limits [(subsection.32.1.3) (subsection.34.1.2)] +>> endobj +5099 0 obj << +/Names [(subsection.34.1.3) 1058 0 R (subsection.34.1.4) 1062 0 R (subsection.34.1.5) 1066 0 R (subsection.36.1.1) 1094 0 R (subsection.4.1.1) 174 0 R (subsection.4.1.2) 178 0 R] +/Limits [(subsection.34.1.3) (subsection.4.1.2)] +>> endobj +5100 0 obj << +/Names [(subsection.4.1.3) 182 0 R (subsection.4.1.4) 186 0 R (subsection.4.2.1) 194 0 R (subsection.4.2.2) 198 0 R (subsection.4.2.3) 202 0 R (subsection.4.3.1) 210 0 R] +/Limits [(subsection.4.1.3) (subsection.4.3.1)] +>> endobj +5101 0 obj << +/Names [(subsection.4.3.2) 214 0 R (subsection.4.3.3) 218 0 R (subsection.4.3.4) 222 0 R (subsection.4.4.1) 230 0 R (subsection.4.4.2) 234 0 R (subsection.4.4.3) 238 0 R] +/Limits [(subsection.4.3.2) (subsection.4.4.3)] +>> endobj +5102 0 obj << +/Names [(subsection.4.5.1) 254 0 R (subsection.4.5.2) 258 0 R (subsection.4.5.3) 262 0 R (subsection.4.5.4) 266 0 R (subsection.4.5.5) 270 0 R (subsection.41.2.1) 1138 0 R] +/Limits [(subsection.4.5.1) (subsection.41.2.1)] +>> endobj +5103 0 obj << +/Names [(subsection.41.3.1) 1146 0 R (subsection.41.3.2) 1150 0 R (subsection.41.3.3) 1154 0 R (subsection.41.3.4) 1158 0 R (subsection.41.4.1) 1166 0 R (subsection.41.4.2) 1170 0 R] +/Limits [(subsection.41.3.1) (subsection.41.4.2)] +>> endobj +5104 0 obj << +/Names [(subsection.41.4.3) 1174 0 R (subsection.41.4.4) 1178 0 R (subsection.41.4.5) 1182 0 R (subsection.41.4.6) 1186 0 R (subsection.41.4.7) 1190 0 R (subsection.41.5.1) 1198 0 R] +/Limits [(subsection.41.4.3) (subsection.41.5.1)] +>> endobj +5105 0 obj << +/Names [(subsection.41.5.2) 1202 0 R (subsection.41.5.3) 1206 0 R (subsection.42.1.1) 1218 0 R (subsection.42.1.2) 1222 0 R (subsection.5.3.1) 294 0 R (subsection.5.3.2) 298 0 R] +/Limits [(subsection.41.5.2) (subsection.5.3.2)] +>> endobj +5106 0 obj << +/Names [(subsection.5.3.3) 302 0 R (subsection.5.3.4) 306 0 R (subsection.5.5.1) 318 0 R (subsection.5.5.2) 322 0 R (subsection.5.5.3) 326 0 R (subsection.5.6.1) 334 0 R] +/Limits [(subsection.5.3.3) (subsection.5.6.1)] +>> endobj +5107 0 obj << +/Names [(subsection.5.6.2) 338 0 R (subsection.6.1.1) 358 0 R (subsection.6.1.2) 362 0 R (subsection.6.1.3) 366 0 R (subsection.6.1.4) 370 0 R (subsection.6.1.5) 374 0 R] +/Limits [(subsection.5.6.2) (subsection.6.1.5)] +>> endobj +5108 0 obj << +/Names [(subsection.6.1.6) 378 0 R (subsection.6.1.7) 382 0 R (subsection.6.1.8) 386 0 R (subsection.6.3.1) 398 0 R (subsection.6.3.2) 402 0 R (subsection.6.3.3) 406 0 R] +/Limits [(subsection.6.1.6) (subsection.6.3.3)] +>> endobj +5109 0 obj << +/Names [(subsection.6.3.4) 410 0 R (subsection.6.3.5) 414 0 R (subsection.6.3.6) 418 0 R (subsection.6.3.7) 422 0 R (subsection.6.4.1) 430 0 R (subsection.6.4.2) 434 0 R] +/Limits [(subsection.6.3.4) (subsection.6.4.2)] +>> endobj +5110 0 obj << +/Names [(subsection.6.4.3) 438 0 R (subsection.6.4.4) 442 0 R (subsection.6.4.5) 446 0 R (subsection.6.4.6) 450 0 R (subsection.8.2.1) 482 0 R (subsection.8.4.1) 494 0 R] +/Limits [(subsection.6.4.3) (subsection.8.4.1)] +>> endobj +5111 0 obj << +/Names [(subsection.8.4.2) 498 0 R (subsection.8.4.3) 502 0 R (subsection.8.4.4) 506 0 R (subsection.8.5.1) 514 0 R (subsection.8.5.2) 518 0 R (subsubsection.20.10.1.1) 854 0 R] +/Limits [(subsection.8.4.2) (subsubsection.20.10.1.1)] +>> endobj +5112 0 obj << +/Names [(subsubsection.20.10.1.2) 858 0 R (subsubsection.4.4.3.1) 242 0 R (subsubsection.4.4.3.2) 246 0 R (table.4.1) 1768 0 R (table.6.1) 2306 0 R (table.8.1) 2375 0 R] +/Limits [(subsubsection.20.10.1.2) (table.8.1)] +>> endobj +5113 0 obj << +/Names [(table.8.2) 2439 0 R (table.8.3) 2441 0 R (title.0) 2 0 R] +/Limits [(table.8.2) (title.0)] +>> endobj +5114 0 obj << +/Kids [4986 0 R 4987 0 R 4988 0 R 4989 0 R 4990 0 R 4991 0 R] +/Limits [(Doc-Start) (Item.38)] +>> endobj +5115 0 obj << +/Kids [4992 0 R 4993 0 R 4994 0 R 4995 0 R 4996 0 R 4997 0 R] +/Limits [(Item.39) (chapter.18)] +>> endobj +5116 0 obj << +/Kids [4998 0 R 4999 0 R 5000 0 R 5001 0 R 5002 0 R 5003 0 R] +/Limits [(chapter.19) (cite.curry&feys:book)] +>> endobj +5117 0 obj << +/Kids [5004 0 R 5005 0 R 5006 0 R 5007 0 R 5008 0 R 5009 0 R] +/Limits [(cite.damas-milner82) (page.112)] +>> endobj +5118 0 obj << +/Kids [5010 0 R 5011 0 R 5012 0 R 5013 0 R 5014 0 R 5015 0 R] +/Limits [(page.113) (page.145)] +>> endobj +5119 0 obj << +/Kids [5016 0 R 5017 0 R 5018 0 R 5019 0 R 5020 0 R 5021 0 R] +/Limits [(page.146) (page.178)] +>> endobj +5120 0 obj << +/Kids [5022 0 R 5023 0 R 5024 0 R 5025 0 R 5026 0 R 5027 0 R] +/Limits [(page.179) (page.21)] +>> endobj +5121 0 obj << +/Kids [5028 0 R 5029 0 R 5030 0 R 5031 0 R 5032 0 R 5033 0 R] +/Limits [(page.210) (page.242)] +>> endobj +5122 0 obj << +/Kids [5034 0 R 5035 0 R 5036 0 R 5037 0 R 5038 0 R 5039 0 R] +/Limits [(page.243) (page.275)] +>> endobj +5123 0 obj << +/Kids [5040 0 R 5041 0 R 5042 0 R 5043 0 R 5044 0 R 5045 0 R] +/Limits [(page.276) (page.307)] +>> endobj +5124 0 obj << +/Kids [5046 0 R 5047 0 R 5048 0 R 5049 0 R 5050 0 R 5051 0 R] +/Limits [(page.308) (page.61)] +>> endobj +5125 0 obj << +/Kids [5052 0 R 5053 0 R 5054 0 R 5055 0 R 5056 0 R 5057 0 R] +/Limits [(page.62) (page.94)] +>> endobj +5126 0 obj << +/Kids [5058 0 R 5059 0 R 5060 0 R 5061 0 R 5062 0 R 5063 0 R] +/Limits [(page.95) (section*.19)] +>> endobj +5127 0 obj << +/Kids [5064 0 R 5065 0 R 5066 0 R 5067 0 R 5068 0 R 5069 0 R] +/Limits [(section*.20) 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5255 y(4)p eop +%%Page: 5 5 +5 4 bop 593 492 a Fp(unify\(X,Y\))42 b(:-)750 585 y(nonvar\(Y\),)g +(Y=s\(T\),)750 678 y(unify\(S,T\),)750 771 y(\()79 b(S=zero,)41 +b(X=one;)868 865 y(S=one,)80 b(X=even;)868 958 y(S=even,)41 +b(X=odd;)868 1051 y(S=odd,)80 b(X=even)750 1144 y(\).)1918 +492 y(unify\(zero,Y\))42 b(:-)2075 585 y(Y==0.)1918 678 +y(unify\(X,Y\))g(:-)2075 771 y(\(Y==even;)f(Y==odd;)g(Y==zero;)g +(Y==one\),)2075 865 y(X=Y.)1918 958 y(unify\(X,Y\))h(:-)2075 +1051 y(var\(Y\),)2075 1144 y(X=Y.)846 1299 y Fs(Figure)27 +b(2:)36 b(Abstract)27 b(Uni\014cation)h(for)f(the)h(Extended)g(P)n +(arit)n(y)d(Domain)515 1564 y(in)34 b(Figure)f(2.)55 +b(The)34 b(de\014nition)g(of)g Fk(unify\(X,Y\))c Fs(is)j(presen)n(ted)g +(using)h(four)f(clauses)g(corre-)515 1666 y(sp)r(onding)e(to)h(the)g(p) +r(ossible)g(structures)f(of)h(the)g(second)g(argumen)n(t)e +Fo(Y)51 b Fs(whic)n(h)32 b(could)g(b)r(e:)46 b(a)515 +1768 y(non-v)-5 b(ariable)28 b(of)h(the)h(form)f Fk(s)p +Fs(\()p 1510 1768 25 4 v 30 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Bryant)SH +8 SS +33858 11880 MT +(3)SH +13 /Times-Bold AF +28181 15584 MT +(Abstract)SH +10 /Times-Roman AF +8200 17957 MT +(In this paper) +243 W( we present a new data structure for representing Boolean functions and an associated set of)242 W +7200 19062 MT +(manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to) +300 W( the)301 W +7200 20167 MT +(representations introduced by Lee) +255 W( [1]) +SH( and Akers) +254 W( [2],) +SH( but with further restrictions on the ordering of decision)254 W +7200 21272 MT +(variables in the graph. Although a function requires, in the worst case, a graph of) +25 W( size exponential in the number of)26 W +7200 22377 MT +(arguments, many of the functions encountered in typical) +86 W( applications have a more reasonable representation. Our)85 W +7200 23482 MT +(algorithms have time complexity proportional to the sizes of the graphs being operated on, and hence are) +152 W( quite)153 W +7200 24587 MT +(efficient as long as the graphs do not grow too large.) +232 W( We) +713 W( present experimental results from applying these)231 W +7200 25692 MT +(algorithms to problems in logic design verification that demonstrate the practicality of our approach.)SH +/Times-Italic SF +8200 27899 MT +(Index Terms)SH +/Times-Roman SF +(: Boolean) +250 W( functions, symbolic manipulation, binary decision diagrams, logic design verification)SH +12 /Times-Bold AF +7200 31583 MT +(1. Introduction)SH +10 /Times-Roman AF +8200 32960 MT +(Boolean Algebra forms a cornerstone of computer science and digital) +76 W( system design. Many problems in digital)77 W +7200 34337 MT +(logic design and testing, artificial intelligence, and combinatorics can be expressed as a sequence of) +75 W( operations on)74 W +7200 35714 MT +(Boolean functions. Such applications would) +134 W( benefit from efficient algorithms for representing and manipulating)135 W +7200 37091 MT +(Boolean functions symbolically.) +260 W( Unfortunately,) +768 W( many of the tasks one would like to perform with Boolean)259 W +7200 38468 MT +(functions, such as) +263 W( testing whether there exists any assignment of input variables such that a given Boolean)264 W +7200 39845 MT +(expression evaluates) +21 W( to 1 \050satisfiability\051, or two Boolean expressions denote the same function \050equivalence\051 require)20 W +7200 41222 MT +(solutions to NP-Complete or coNP-Complete problems) +166 W( [3].) +SH( Consequently,) +583 W( all known approaches to performing)167 W +7200 42599 MT +(these operations require, in the worst case, an amount of computer time that grows exponentially with) +12 W( the size of the)11 W +7200 43976 MT +(problem. This) +514 W( makes it difficult to compare the relative efficiencies of different) +132 W( approaches to representing and)133 W +7200 45353 MT +(manipulating Boolean functions. In the worst case, all known approaches perform as poorly) +56 W( as the naive approach)55 W +7200 46730 MT +(of representing functions by) +87 W( their truth tables and defining all of the desired operations in terms of their effect on)88 W +7200 48107 MT +(truth table entries.) +67 W( In) +382 W( practice, by utilizing more clever representations and manipulation algorithms, we can often)66 W +7200 49484 MT +(avoid these exponential computations.)SH +8200 51963 MT +(A variety of methods have been developed for representing and manipulating Boolean functions. Those based on)21 W +7200 53340 MT +(classical representations such) +217 W( as truth tables, Karnaugh maps, or canonical sum-of-products form) +216 W( [4]) +SH( are quite)216 W +10800 50 7200 56307 UL +9 SS +8300 57829 MT +(1)SH +11 SS +8750 58192 MT +(This research was funded at the California Institute of Technology by the Defense) +8 W( Advanced Research)7 W +7200 59388 MT +(Projects Agency ARPA Order Number 3771 and at Carnegie-Mellon) +387 W( University by the Defense)388 W +7200 60584 MT +(Advanced Research Projects Agency) +130 W( ARPA Order Number 3597. A preliminary version of this paper)129 W +7200 61780 MT +(was presented under the title "Symbolic) +420 W( Manipulation of Boolean Functions Using a Graphical)421 W +7200 62976 MT +(Representation" at the 22nd Design Automation Conference, Las Vegas, NV, June 1985.)SH +9 SS +8300 65005 MT +(2)SH +11 /Times-Bold AF +8750 65368 MT +(Update:)SH +/Times-Roman SF +13057 XM +(This paper was originally published in)244 W +/Times-BoldItalic SF +31691 XM +(IEEE Transactions on Computers, C-35-8)244 W +/Times-Roman SF +(, pp.)243 W +7200 66564 MT +(677-691, August, 1986. To create this version, we) +244 W( started with the original electronic form of the)245 W +7200 67760 MT +(submission. All) +511 W( of the figures had to be redrawn, since they were in a now defunct format. We) +118 W( have)117 W +7200 68956 MT +(included footnotes \050starting with ")11 W +/Times-Bold SF +(Update:)SH +/Times-Roman SF +("\051 discussing) +297 W( some of the \050minor\051 errors) +11 W( in the original version)12 W +7200 70152 MT +(and giving updates on some of the open problems.)SH +6 SS +8000 71691 MT +(3)SH +8 SS +8300 72000 MT +(Current address: Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA 15213)SH +ES +%%Page: 2 2 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(2)SH +7200 8094 MT +(impractical---)SH +/Times-Italic SF +(every)SH +/Times-Roman SF +15258 XM +(function of)145 W +/Times-Italic SF +20214 XM +(n)SH +/Times-Roman SF +21109 XM +(arguments has a) +145 W( representation of size 2)146 W +8 /Times-Italic AF +37979 7749 MT +(n)SH +10 /Times-Roman AF +38775 8094 MT +(or more. More practical approaches)146 W +7200 9471 MT +(utilize representations that at least for many functions, are not of exponential) +30 W( size. Example representations include)29 W +7200 10848 MT +(as a reduced sum of) +60 W( products) +61 W( [4],) +SH( \050or equivalently as sets of prime cubes) +61 W( [5]\051) +SH( and factored into unate functions) +61 W( [6].)SH +7200 12225 MT +(These representations suffer from several drawbacks. First,) +82 W( certain common functions still require representations)81 W +7200 13602 MT +(of exponential size. For example, the even and odd) +101 W( parity functions serve as worst case examples in all of these)102 W +7200 14979 MT +(representations. Second,) +712 W( while a certain function may) +231 W( have a reasonable representation, performing a simple)230 W +7200 16356 MT +(operation such as complementation could yield a function with an exponential representation. Finally,) +12 W( none of these)13 W +7200 17733 MT +(representations are)72 W +/Times-Italic SF +15119 XM +(canonical forms)72 W +/Times-Roman SF +(, i.e. a given function may have) +72 W( many different representations. Consequently,)71 W +7200 19110 MT +(testing for equivalence or satisfiability can be quite difficult.)SH +8200 21589 MT +(Due to) +153 W( these characteristics, most programs that process a sequence of operations on Boolean functions have)154 W +7200 22966 MT +(rather erratic behavior. They proceed at a reasonable pace,) +153 W( but then suddenly "blow up", either running out of)152 W +7200 24343 MT +(storage or failing to complete an operation in a reasonable amount of time.)SH +8200 26822 MT +(In this paper we present a new class of algorithms for manipulating Boolean functions represented as directed)120 W +7200 28199 MT +(acyclic graphs.) +5 W( Our) +258 W( representation resembles the binary decision diagram notation introduced by Lee) +4 W( [1]) +SH( and further)4 W +7200 29576 MT +(popularized by Akers) +205 W( [2].) +SH( However,) +660 W( we place further restrictions on the ordering of decision) +205 W( variables in the)206 W +7200 30953 MT +(vertices. These) +484 W( restrictions enable the development of algorithms for manipulating the representations) +117 W( in a more)116 W +7200 32330 MT +(efficient manner.)SH +8200 34809 MT +(Our representation has several) +149 W( advantages over previous approaches to Boolean function manipulation. First,)150 W +7200 36186 MT +(most commonly-encountered functions have a reasonable) +179 W( representation. For example, all symmetric functions)178 W +7200 37563 MT +(\050including even and odd parity\051 are represented by) +26 W( graphs where the number of vertices grows at most as the square)27 W +7200 38940 MT +(of the number of arguments. Second, the performance of a program based on our algorithms when) +113 W( processing a)112 W +7200 40317 MT +(sequence of operations) +13 W( degrades slowly, if at all. That is, the time complexity of any single operation is bounded by)14 W +7200 41694 MT +(the product of the graph sizes for the) +27 W( functions being operated on. For example, complementing a function requires)26 W +7200 43071 MT +(time proportional to the size of the function graph, while combining two functions) +12 W( with a binary operation \050of which)13 W +7200 44448 MT +(intersection, subtraction, and testing for implication are) +171 W( special cases\051 requires at most time proportional to the)170 W +7200 45825 MT +(product of the two graph sizes. Finally, our representation in terms of)4 W +/Times-Italic SF +35498 XM +(reduced)SH +/Times-Roman SF +38973 XM +(graphs is a canonical form,) +4 W( i.e. every)5 W +7200 47202 MT +(function has a unique representation. Hence, testing for equivalence simply involves testing) +16 W( whether the two graphs)15 W +7200 48579 MT +(match exactly, while testing for satisfiability simply involves comparing the graph to that of the constant function)SH +/Times-Bold SF +52919 XM +(0)SH +/Times-Roman SF +(.)SH +8200 51058 MT +(Unfortunately, our approach does have) +91 W( its own set of undesirable characteristics. At the start of processing we)92 W +7200 52435 MT +(must choose some ordering of the system) +84 W( inputs as arguments to all of the functions to be represented. For some)83 W +7200 53812 MT +(functions, the size of the) +189 W( graph representing the function is highly sensitive to this ordering. The problem of)190 W +7200 55189 MT +(computing an ordering that) +112 W( minimizes the size of the graph is itself a coNP-Complete problem. Our experience,)111 W +7200 56566 MT +(however, has) +277 W( been that a human with some understanding of the problem domain can generally choose an)278 W +7200 57943 MT +(appropriate ordering without great) +61 W( difficulty. It seems quite likely that using a small set of heuristics, the program)60 W +7200 59320 MT +(itself could select an adequate ordering most of the time. More) +143 W( seriously, there are some functions that can be)144 W +7200 60697 MT +(represented by Boolean expressions or logic circuits of reasonable size but for) +37 W( all input orderings the representation)36 W +7200 62074 MT +(as a function graph is too large to be practical. For) +2 W( example, we prove in an appendix to this paper that the functions)3 W +7200 63451 MT +(describing the outputs of an integer multiplier have graphs that grow exponentially in the word size regardless) +11 W( of the)10 W +7200 64828 MT +(input ordering.) +105 W( With) +462 W( the exception of integer multiplication, our experience has been that such functions seldom)106 W +7200 66205 MT +(arise in digital logic design applications. For other classes of problems, particularly in combinatorics, our methods)52 W +7200 67582 MT +(seem practical only under restricted conditions.)SH +8200 70061 MT +(A variety of graphical representations of discrete functions have be presented) +52 W( and studied extensively. A survey)53 W +7200 71438 MT +(of the literature on the subject) +58 W( by Moret) +57 W( [7]) +SH( cites over 100 references, but none of these describe a sufficient set of)57 W +ES +%%Page: 3 3 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(3)SH +7200 7886 MT +(algorithms to implement a Boolean function manipulation) +29 W( program. Fortune, Hopcroft, and Schmidt) +30 W( [8]) +SH( studied the)30 W +7200 9263 MT +(properties of graphs obeying similar restrictions to ours, showing that two graphs could be) +146 W( tested for functional)145 W +7200 10640 MT +(equivalence in polynomial time and that some functions require much larger) +122 W( graphs under these restrictions than)123 W +7200 12017 MT +(under milder restrictions. Payne) +270 W( [9]) +SH( describes techniques similar to ours for reducing the size of the graph)269 W +7200 13394 MT +(representing a function. Our algorithms for combining two) +95 W( functions with a binary operation, and for composing)96 W +7200 14771 MT +(two functions are new, however, and these capabilities are central to a symbolic manipulation program.)SH +8200 17250 MT +(The next section of this paper contains a) +169 W( formal presentation of function graphs. We define the graphs, the)168 W +7200 18627 MT +(functions they represent, and a class of "reduced" graphs. Then we prove a key property) +4 W( of reduced function graphs:)5 W +7200 20004 MT +(that they form a canonical representation of Boolean functions. In the following section we depart from this formal)31 W +7200 21381 MT +(presentation to give) +290 W( some examples and to discuss issues regarding to the efficiency of our representation.)291 W +7200 22758 MT +(Following this, we develop a set) +44 W( of algorithms for manipulating Boolean functions using our representation. These)43 W +7200 24135 MT +(algorithms utilize) +211 W( many of the classical techniques for graph algorithms, and we assume the reader has some)212 W +7200 25512 MT +(familiarity with these techniques. We then present some experimental) +143 W( investigations into the practicality of our)142 W +7200 26889 MT +(methods. We) +250 W( conclude by suggesting further refinements of our methods.)SH +11 /Times-Bold AF +7200 30506 MT +(1.1. Notation)SH +10 /Times-Roman AF +8200 31883 MT +(We assume the functions to be represented all have the same)113 W +/Times-Italic SF +33965 XM +(n)SH +/Times-Roman SF +34828 XM +(arguments, written)113 W +/Times-Italic SF +42803 XM +(x)SH +8 /Times-Roman AF +43247 32228 MT +(1)SH +10 SS +43647 31883 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +46341 32228 MT +(n)SH +10 /Times-Roman AF +46741 31883 MT +(. In) +477 W( expressing a)114 W +7200 33260 MT +(system such as a combinational logic network or a Boolean expression) +19 W( as a Boolean function, we must choose some)18 W +7200 34637 MT +(ordering of the inputs or atomic variables, and this ordering must be the same for all functions to be represented.)SH +8200 37116 MT +(The function resulting when some argument)23 W +/Times-Italic SF +26280 XM +(x)SH +8 SS +26724 37461 MT +(i)SH +10 /Times-Roman AF +27219 37116 MT +(of function)23 W +/Times-Italic SF +31931 XM +(f)SH +/Times-Roman SF +32632 XM +(is replaced by a constant)23 W +/Times-Italic SF +42829 XM +(b)SH +/Times-Roman SF +43603 XM +(is called a)24 W +/Times-Italic SF +47924 XM +(restriction)SH +/Times-Roman SF +52365 XM +(of)SH +/Times-Italic SF +53472 XM +(f)SH +/Times-Roman SF +(,)SH +7200 38493 MT +(\050sometimes termed a)SH +/Times-Italic SF +15782 XM +(cofactor)SH +/Times-Roman SF +19365 XM +([10]\051 and is denoted)SH +/Times-Italic SF +27641 XM +(f)SH +/Symbol SF +28169 XM +(|)SH +8 /Times-Italic AF +28369 38838 MT +(x)SH +28724 39183 MT +(i)SH +/Times-Roman SF +28946 38838 MT +(=)SH +/Times-Italic SF +(b)SH +10 /Times-Roman AF +29797 38493 MT +(. That) +250 W( is, for any arguments)SH +/Times-Italic SF +41323 XM +(x)SH +8 /Times-Roman AF +41767 38838 MT +(1)SH +10 SS +42167 38493 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +44861 38838 MT +(n)SH +10 /Times-Roman AF +45261 38493 MT +(,)SH +/Times-Italic SF +9200 40398 MT +(f)SH +/Symbol SF +9728 XM +(|)SH +8 /Times-Italic AF +9928 40743 MT +(x)SH +10283 41088 MT +(i)SH +/Times-Roman SF +10505 40743 MT +(=)SH +/Times-Italic SF +(b)SH +10 /Times-Roman AF +11356 40398 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +12133 40743 MT +(1)SH +10 SS +12533 40398 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +15227 40743 MT +(n)SH +10 /Times-Roman AF +15627 40398 MT +(\051 =)SH +/Times-Italic SF +17024 XM +(f)SH +/Times-Roman SF +17452 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +18229 40743 MT +(1)SH +10 SS +18629 40398 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +21323 40743 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +22384 40398 MT +(,)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(x)SH +8 SS +23828 40743 MT +(i)SH +/Times-Roman SF +(+1)SH +10 SS +24901 40398 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +27595 40743 MT +(n)SH +10 /Times-Roman AF +27995 40398 MT +(\051)SH +7200 41949 MT +(Using this notation, the Shannon expansion [11] of a function around variable)SH +/Times-Italic SF +38613 XM +(x)SH +8 SS +39057 42294 MT +(i)SH +10 /Times-Roman AF +39529 41949 MT +(is given by)SH +/Times-Italic SF +9200 43850 MT +(f)SH +/Times-Roman SF +10164 XM +(=)SH +/Times-Italic SF +11414 XM +(x)SH +8 SS +11858 44195 MT +(i)SH +10 /Symbol AF +12080 43850 MT +(\327)SH +/Times-Italic SF +(f)SH +/Symbol SF +12858 XM +(|)SH +8 /Times-Italic AF +13058 44195 MT +(x)SH +13413 44540 MT +(i)SH +/Times-Roman SF +13635 44195 MT +(=1)SH +10 SS +14736 43850 MT +(+)SH +/Times-Italic SF +15550 XM +(x)SH +/Symbol SF +15550 43469 MT +(-)SH +8 /Times-Italic AF +16099 44195 MT +(i)SH +10 /Symbol AF +16321 43850 MT +(\327)SH +/Times-Italic SF +(f)SH +/Symbol SF +17099 XM +(|)SH +8 /Times-Italic AF +17299 44195 MT +(x)SH +17654 44540 MT +(i)SH +/Times-Roman SF +17876 44195 MT +(=0)SH +10 SS +52834 43850 MT +(\0501\051)SH +7200 45401 MT +(Similarly, the function resulting when some argument)254 W +/Times-Italic SF +30810 XM +(x)SH +8 SS +31254 45746 MT +(i)SH +10 /Times-Roman AF +31980 45401 MT +(of function)254 W +/Times-Italic SF +37154 XM +(f)SH +/Times-Roman SF +38086 XM +(is replaced by function)254 W +/Times-Italic SF +48489 XM +(g)SH +/Times-Roman SF +49493 XM +(is called a)254 W +/Times-Italic SF +7200 46778 MT +(composition)SH +/Times-Roman SF +12339 XM +(of)SH +/Times-Italic SF +13422 XM +(f)SH +/Times-Roman SF +13950 XM +(and)SH +/Times-Italic SF +15644 XM +(g)SH +/Times-Roman SF +(, and is denoted)SH +/Times-Italic SF +22671 XM +(f)SH +/Symbol SF +23199 XM +(|)SH +8 /Times-Italic AF +23399 47123 MT +(x)SH +23754 47468 MT +(i)SH +/Times-Roman SF +23976 47123 MT +(=)SH +/Times-Italic SF +(g)SH +10 /Times-Roman AF +24827 46778 MT +(. That) +250 W( is, for any arguments)SH +/Times-Italic SF +36353 XM +(x)SH +8 /Times-Roman AF +36797 47123 MT +(1)SH +10 SS +37197 46778 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +39891 47123 MT +(n)SH +10 /Times-Roman AF +40291 46778 MT +(,)SH +/Times-Italic SF +9200 48683 MT +(f)SH +/Symbol SF +9728 XM +(|)SH +8 /Times-Italic AF +9928 49028 MT +(x)SH +10283 49373 MT +(i)SH +/Times-Roman SF +10505 49028 MT +(=)SH +/Times-Italic SF +(g)SH +10 /Times-Roman AF +11356 48683 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +12133 49028 MT +(1)SH +10 SS +12533 48683 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +15227 49028 MT +(n)SH +10 /Times-Roman AF +15627 48683 MT +(\051 =)SH +/Times-Italic SF +17024 XM +(f)SH +/Times-Roman SF +17452 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +18229 49028 MT +(1)SH +10 SS +18629 48683 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +21323 49028 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +22384 48683 MT +(,)SH +/Times-Italic SF +(g)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +23911 49028 MT +(1)SH +10 SS +24311 48683 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +27005 49028 MT +(n)SH +10 /Times-Roman AF +27405 48683 MT +(\051,)SH +/Times-Italic SF +(x)SH +8 SS +28432 49028 MT +(i)SH +/Times-Roman SF +(+1)SH +10 SS +29505 48683 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +32199 49028 MT +(n)SH +10 /Times-Roman AF +32599 48683 MT +(\051)SH +8200 51162 MT +(Some functions may not depend on all arguments.) +34 W( The)319 W +/Times-Italic SF +30901 XM +(dependency set)35 W +/Times-Roman SF +37302 XM +(of a function)35 W +/Times-Italic SF +42767 XM +(f)SH +/Times-Roman SF +(, denoted)35 W +/Times-Italic SF +47031 XM +(I)SH +8 SS +47364 51507 MT +(f)SH +10 /Times-Roman AF +47736 51162 MT +(, contains those)35 W +7200 52539 MT +(arguments on which the function depends, i.e.)SH +/Times-Italic SF +9200 54444 MT +(I)SH +8 SS +9533 54789 MT +(f)SH +10 /Times-Roman AF +10591 54444 MT +(= {)436 W +/Times-Italic SF +12571 XM +(i)SH +/Times-Roman SF +13099 XM +(|)SH +/Times-Italic SF +13549 XM +(f)SH +/Symbol SF +14077 XM +(|)SH +8 /Times-Italic AF +14277 54789 MT +(x)SH +14632 55134 MT +(i)SH +/Times-Roman SF +14854 54789 MT +(=0)SH +10 /Symbol AF +16105 54444 MT +(\271)SH +/Times-Italic SF +17054 XM +(f)SH +/Symbol SF +17582 XM +(|)SH +8 /Times-Italic AF +17782 54789 MT +(x)SH +18137 55134 MT +(i)SH +/Times-Roman SF +18359 54789 MT +(=1)SH +10 SS +19210 54444 MT +(})SH +8200 56923 MT +(The function which for all values of the arguments) +49 W( yields 1 \050respectively 0\051 is denoted)48 W +/Times-Bold SF +43672 XM +(1)SH +/Times-Roman SF +44470 XM +(\050respectively)SH +/Times-Bold SF +49933 XM +(0)SH +/Times-Roman SF +(\051. These)346 W +7200 58300 MT +(two Boolean functions have dependency sets equal to the empty set.)SH +8200 60779 MT +(A Boolean) +145 W( function can also be viewed as denoting some subset of Boolean)146 W +/Times-Italic SF +40676 XM +(n)SH +/Times-Roman SF +(-space, namely those argument)146 W +7200 62156 MT +(values for which the function evaluates to 1. The)SH +/Times-Italic SF +27224 XM +(satisfying set)SH +/Times-Roman SF +32669 XM +(of a function)SH +/Times-Italic SF +38029 XM +(f)SH +/Times-Roman SF +(, denoted)SH +/Times-Italic SF +42223 XM +(S)SH +8 SS +42723 62501 MT +(f)SH +10 /Times-Roman AF +43095 62156 MT +(, is defined as:)SH +/Times-Italic SF +11200 63947 MT +(S)SH +8 SS +11700 64292 MT +(f)SH +10 /Times-Roman AF +12758 63947 MT +(= {) +150 436 WX(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +15415 64292 MT +(1)SH +10 SS +15815 63947 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +18509 64292 MT +(n)SH +10 /Times-Roman AF +18909 63947 MT +(\051) +150 MX(|)SH +/Times-Italic SF +19842 XM +(f)SH +/Times-Roman SF +20420 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +21197 64292 MT +(1)SH +10 SS +21597 63947 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +24291 64292 MT +(n)SH +10 /Times-Roman AF +24691 63947 MT +(\051 = 1) +150 0 WX(}.)SH +ES +%%Page: 4 4 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(4)SH +12 /Times-Bold AF +7200 8004 MT +(2. Representation)SH +10 /Times-Roman AF +8200 9381 MT +(In this section we define our graphical representation of a Boolean function and prove that it is a canonical form.)SH +/Times-Bold SF +10200 11625 MT +(Definition 1:)120 W +/Times-Roman SF +16051 XM +(A function graph is) +120 W( a rooted, directed graph with vertex set)119 W +/Times-Italic SF +41247 XM +(V)SH +/Times-Roman SF +42227 XM +(containing two types of)119 W +9200 12730 MT +(vertices. A)616 W +/Times-Italic SF +14581 XM +(nonterminal)SH +/Times-Roman SF +19903 XM +(vertex)SH +/Times-Italic SF +22835 XM +(v)SH +/Times-Roman SF +23712 XM +(has as attributes an argument index)183 W +/Times-Italic SF +39141 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +42717 XM +(\316)SH +/Times-Roman SF +43730 XM +({1, . . . ,)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(}, and two)183 W +9200 13835 MT +(children)SH +/Times-Italic SF +12727 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051,)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +18720 XM +(\316)SH +/Times-Italic SF +19733 XM +(V)SH +/Times-Roman SF +(. A)250 W +/Times-Italic SF +22066 XM +(terminal)SH +/Times-Roman SF +25705 XM +(vertex)SH +/Times-Italic SF +28454 XM +(v)SH +/Times-Roman SF +29148 XM +(has as attribute a value)SH +/Times-Italic SF +38507 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +41933 XM +(\316)SH +/Times-Roman SF +42796 XM +({0,1}.)SH +10200 16042 MT +(Furthermore, for) +370 W( any nonterminal vertex)369 W +/Times-Italic SF +28487 XM +(v)SH +/Times-Roman SF +(, if)369 W +/Times-Italic SF +31030 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 is also nonterminal, then we must have)369 W +/Times-Italic SF +9200 17147 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 <)SH +/Times-Italic SF +13540 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051. Similarly,) +2326 W( if)1038 W +/Times-Italic SF +28913 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 is nonterminal, then we must have)1038 W +/Times-Italic SF +9200 18252 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 <)SH +/Times-Italic SF +13540 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051.)SH +8200 20731 MT +(Due to the ordering restriction in our) +150 W( definition, function graphs form a proper subset of conventional binary)149 W +7200 22108 MT +(decision diagrams. Note that) +228 W( this restriction also implies that a function graph must be acyclic, because the)229 W +7200 23485 MT +(nonterminal vertices along any path must have strictly increasing index values.)SH +8200 25964 MT +(We define the correspondence between function graphs and Boolean functions as follows.)SH +/Times-Bold SF +10200 28208 MT +(Definition 2:)SH +/Times-Roman SF +15811 XM +(A function graph)SH +/Times-Italic SF +22893 XM +(G)SH +/Times-Roman SF +23865 XM +(having root vertex)SH +/Times-Italic SF +31447 XM +(v)SH +/Times-Roman SF +32141 XM +(denotes a function)SH +/Times-Italic SF +39723 XM +(f)SH +8 SS +40001 28553 MT +(v)SH +10 /Times-Roman AF +40606 28208 MT +(defined recursively as:)SH +11200 29547 MT +(1.)SH +12200 XM +(If)SH +/Times-Italic SF +13116 XM +(v)SH +/Times-Roman SF +13810 XM +(is a terminal vertex:)SH +14256 30886 MT +(a.)SH +15200 XM +(If)SH +/Times-Italic SF +16116 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051=1, then)SH +/Times-Italic SF +22928 XM +(f)SH +8 SS +23206 31231 MT +(v)SH +10 /Times-Roman AF +23561 30886 MT +(=)SH +/Times-Bold SF +(1)SH +/Times-Roman SF +14200 32644 MT +(b.)SH +15200 XM +(If)SH +/Times-Italic SF +16116 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051=0, then)SH +/Times-Italic SF +22928 XM +(f)SH +8 SS +23206 32989 MT +(v)SH +10 /Times-Roman AF +23561 32644 MT +(=)SH +/Times-Bold SF +(0)SH +/Times-Roman SF +11200 34402 MT +(2.)SH +12200 XM +(If)SH +/Times-Italic SF +13116 XM +(v)SH +/Times-Roman SF +13810 XM +(is a nonterminal vertex with)SH +/Times-Italic SF +25281 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051=)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(, then)SH +/Times-Italic SF +31871 XM +(f)SH +8 SS +32149 34747 MT +(v)SH +10 /Times-Roman AF +32754 34402 MT +(is the function)SH +/Times-Italic SF +14200 36574 MT +(f)SH +8 SS +14478 36919 MT +(v)SH +10 /Times-Roman AF +14833 36574 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +15610 36919 MT +(1)SH +10 SS +16010 36574 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +18704 36919 MT +(n)SH +10 /Times-Roman AF +19104 36574 MT +(\051 =)436 W +/Times-Italic SF +21373 XM +(x)SH +/Symbol SF +21373 36193 MT +(-)SH +8 /Times-Italic AF +21922 36919 MT +(i)SH +10 /Symbol AF +22144 36574 MT +(\327)SH +/Times-Italic SF +(f)SH +8 SS +22672 36919 MT +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 SS +24715 36574 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +25492 36919 MT +(1)SH +10 SS +25892 36574 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +28586 36919 MT +(n)SH +10 /Times-Roman AF +28986 36574 MT +(\051 +)SH +/Times-Italic SF +30383 XM +(x)SH +8 SS +30827 36919 MT +(i)SH +10 /Symbol AF +31049 36574 MT +(\327)SH +/Times-Italic SF +(f)SH +8 SS +31577 36919 MT +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 SS +33886 36574 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +34663 36919 MT +(1)SH +10 SS +35063 36574 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +37757 36919 MT +(n)SH +10 /Times-Roman AF +38157 36574 MT +(\051.)SH +8200 39053 MT +(In other words, we can view a set of argument values)10 W +/Times-Italic SF +29860 XM +(x)SH +8 /Times-Roman AF +30304 39398 MT +(1)SH +10 SS +30704 39053 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +33398 39398 MT +(n)SH +10 /Times-Roman AF +34058 39053 MT +(as describing a path) +10 W( in the graph starting from the)9 W +7200 40430 MT +(root, where if some vertex)41 W +/Times-Italic SF +18180 XM +(v)SH +/Times-Roman SF +18915 XM +(along the path has)41 W +/Times-Italic SF +26578 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +30918 XM +(i)SH +/Times-Roman SF +(, then the path continues to the low child if)41 W +/Times-Italic SF +48966 XM +(x)SH +8 SS +49410 40775 MT +(i)SH +10 /Times-Roman AF +49882 40430 MT +(= 0) +SH( and to)41 W +7200 41807 MT +(the high child if)43 W +/Times-Italic SF +13983 XM +(x)SH +8 SS +14427 42152 MT +(i)SH +10 /Times-Roman AF +14899 41807 MT +(= 1.) +SH( The) +336 W( value of the function for these arguments equals the) +43 W( value of the terminal vertex at the)42 W +7200 43184 MT +(end of the path. Note that the path defined by) +37 W( a set of argument values is unique. Furthermore, every vertex in the)38 W +7200 44561 MT +(graph is contained in at least one path, i.e. no part of the graph is "unreachable.")SH +8200 47040 MT +(Two function graphs are considered) +94 W( isomorphic if they match in both their structure and their attributes. More)93 W +7200 48417 MT +(precisely:)SH +/Times-Bold SF +10200 50661 MT +(Definition 3:)26 W +/Times-Roman SF +15863 XM +(Function graphs)26 W +/Times-Italic SF +22637 XM +(G)SH +/Times-Roman SF +23635 XM +(and)SH +/Times-Italic SF +25355 XM +(G)SH +/Symbol SF +26227 XM +(\242)SH +/Times-Roman SF +26750 XM +(are)SH +/Times-Italic SF +28247 XM +(isomorphic)SH +/Times-Roman SF +33023 XM +(if there exists a one-to-one function)26 W +/Symbol SF +47677 XM +(s)SH +/Times-Roman SF +48557 XM +(from the)27 W +9200 51766 MT +(vertices of)91 W +/Times-Italic SF +13824 XM +(G)SH +/Times-Roman SF +14886 XM +(onto the vertices of)90 W +/Times-Italic SF +23189 XM +(G)SH +/Symbol SF +24061 XM +(\242)SH +/Times-Roman SF +24648 XM +(such that for any vertex)90 W +/Times-Italic SF +34790 XM +(v)SH +/Times-Roman SF +35574 XM +(if)SH +/Symbol SF +36525 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051=)SH +/Times-Italic SF +(v)SH +/Symbol SF +39396 XM +(\242)SH +/Times-Roman SF +(, then either both)90 W +/Times-Italic SF +47030 XM +(v)SH +/Times-Roman SF +47814 XM +(and)SH +/Times-Italic SF +49598 XM +(v)SH +/Symbol SF +50192 XM +(\242)SH +/Times-Roman SF +50779 XM +(are)SH +9200 52871 MT +(terminal vertices with)586 W +/Times-Italic SF +19929 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +24269 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +27362 XM +(\242)SH +/Times-Roman SF +(\051, or both)586 W +/Times-Italic SF +33311 XM +(v)SH +/Times-Roman SF +34591 XM +(and)SH +/Times-Italic SF +36871 XM +(v)SH +/Symbol SF +37465 XM +(\242)SH +/Times-Roman SF +38548 XM +(are nonterminal) +586 W( vertices with)587 W +/Times-Italic SF +9200 53976 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +13540 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +16633 XM +(\242)SH +/Times-Roman SF +(\051,)SH +/Symbol SF +17713 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 =)SH +/Times-Italic SF +22601 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +24973 XM +(\242)SH +/Times-Roman SF +(\051, and)SH +/Symbol SF +27747 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051=)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +35173 XM +(\242)SH +/Times-Roman SF +(\051.)SH +8200 56455 MT +(Note that since a function graph contains) +10 W( only 1 root and the children of any nonterminal vertex are distinguished,)9 W +7200 57832 MT +(the isomorphic mapping)27 W +/Symbol SF +17253 XM +(s)SH +/Times-Roman SF +18133 XM +(between graphs)27 W +/Times-Italic SF +24685 XM +(G)SH +/Times-Roman SF +25684 XM +(and)SH +/Times-Italic SF +27405 XM +(G)SH +/Symbol SF +28277 XM +(\242)SH +/Times-Roman SF +28801 XM +(is quite constrained: the root in)27 W +/Times-Italic SF +41907 XM +(G)SH +/Times-Roman SF +42907 XM +(must map to the root in)28 W +/Times-Italic SF +52631 XM +(G)SH +/Symbol SF +53503 XM +(\242)SH +/Times-Roman SF +(,)SH +7200 59209 MT +(the root's low child in)9 W +/Times-Italic SF +16328 XM +(G)SH +/Times-Roman SF +17309 XM +(must map to the root's low child) +9 W( in)8 W +/Times-Italic SF +31657 XM +(G)SH +/Symbol SF +32529 XM +(\242)SH +/Times-Roman SF +(, and so on all the way down to the terminal vertices.)8 W +7200 60586 MT +(Hence, testing 2 function graphs for isomorphism is quite simple.)SH +/Times-Bold SF +10200 62830 MT +(Definition 4:)55 W +/Times-Roman SF +15921 XM +(For any vertex)55 W +/Times-Italic SF +22168 XM +(v)SH +/Times-Roman SF +22917 XM +(in a function) +55 W( graph)56 W +/Times-Italic SF +30971 XM +(G)SH +/Times-Roman SF +(, the)56 W +/Times-Italic SF +33777 XM +(subgraph rooted by v)56 W +/Times-Roman SF +42778 XM +(is defined as the graph)56 W +9200 63935 MT +(consisting of)SH +/Times-Italic SF +14589 XM +(v)SH +/Times-Roman SF +15283 XM +(and all of its descendants.)SH +/Times-Bold SF +10200 66179 MT +(Lemma 1:)22 W +/Times-Roman SF +14854 XM +(If)SH +/Times-Italic SF +15792 XM +(G)SH +/Times-Roman SF +16786 XM +(is isomorphic to)22 W +/Times-Italic SF +23547 XM +(G)SH +/Symbol SF +24419 XM +(\242)SH +/Times-Roman SF +24938 XM +(by mapping)22 W +/Symbol SF +29982 XM +(s)SH +/Times-Roman SF +(, then) +22 W( for any vertex)21 W +/Times-Italic SF +39022 XM +(v)SH +/Times-Roman SF +39737 XM +(in)SH +/Times-Italic SF +40786 XM +(G)SH +/Times-Roman SF +(, the subgraph rooted by)21 W +/Times-Italic SF +51556 XM +(v)SH +/Times-Roman SF +9200 67284 MT +(is isomorphic to the subgraph rooted by)SH +/Symbol SF +25338 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051.)SH +8200 69763 MT +(The proof of this) +262 W( lemma is straightforward, since the restriction of)263 W +/Symbol SF +37808 XM +(s)SH +/Times-Roman SF +38924 XM +(to)SH +/Times-Italic SF +40215 XM +(v)SH +/Times-Roman SF +41172 XM +(and its descendants forms the)263 W +7200 71140 MT +(isomorphic mapping.)SH +ES +%%Page: 5 5 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(5)SH +8200 7886 MT +(A function graph) +34 W( can be reduced in size without changing the denoted function by eliminating redundant vertices)33 W +7200 9263 MT +(and duplicate subgraphs. The resulting graph will be our primary data structure for representing a Boolean function.)SH +/Times-Bold SF +10200 11507 MT +(Definition 5:)58 W +/Times-Roman SF +15927 XM +(A function graph)58 W +/Times-Italic SF +23183 XM +(G)SH +/Times-Roman SF +24213 XM +(is)SH +/Times-Italic SF +25188 XM +(reduced)SH +/Times-Roman SF +28717 XM +(if it contains no vertex)58 W +/Times-Italic SF +38256 XM +(v)SH +/Times-Roman SF +39008 XM +(with)SH +/Times-Italic SF +41094 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051=)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051, nor does it)59 W +9200 12612 MT +(contain distinct vertices)SH +/Times-Italic SF +18949 XM +(v)SH +/Times-Roman SF +19643 XM +(and)SH +/Times-Italic SF +21337 XM +(v)SH +/Symbol SF +21931 XM +(\242)SH +/Times-Roman SF +22428 XM +(such that the subgraphs rooted by)SH +/Times-Italic SF +36093 XM +(v)SH +/Times-Roman SF +36787 XM +(and)SH +/Times-Italic SF +38481 XM +(v)SH +/Symbol SF +39075 XM +(\242)SH +/Times-Roman SF +39572 XM +(are isomorphic.)SH +8200 15091 MT +(The following lemma follows directly from the definition of reduced function graphs.)SH +/Times-Bold SF +10200 17335 MT +(Lemma 2:)52 W +/Times-Roman SF +14914 XM +(For every vertex)52 W +/Times-Italic SF +21929 XM +(v)SH +/Times-Roman SF +22675 XM +(in a reduced) +52 W( function graph, the subgraph rooted by)51 W +/Times-Italic SF +44076 XM +(v)SH +/Times-Roman SF +44821 XM +(is itself a reduced)51 W +9200 18440 MT +(function graph.)SH +8200 20919 MT +(The following theorem proves a key property of reduced) +156 W( function graphs, namely that they form a canonical)157 W +7200 22296 MT +(representation for Boolean functions, i.e. every function is represented by a unique reduced) +162 W( function graph. In)161 W +7200 23673 MT +(contrast to) +40 W( other canonical representations of Boolean functions, such as canonical sum-of-products form, however,)41 W +7200 25050 MT +(many "interesting" Boolean functions are represented by function graphs of size polynomial in the number) +205 W( of)204 W +7200 26427 MT +(arguments.)SH +/Times-Bold SF +10200 28671 MT +(Theorem 1:)48 W +/Times-Roman SF +15517 XM +(For any Boolean function)48 W +/Times-Italic SF +26208 XM +(f)SH +/Times-Roman SF +(, there is a unique \050up to isomorphism\051 reduced function graph)49 W +9200 29776 MT +(denoting)SH +/Times-Italic SF +12950 XM +(f)SH +/Times-Roman SF +13478 XM +(and any other function graph denoting)SH +/Times-Italic SF +29031 XM +(f)SH +/Times-Roman SF +29559 XM +(contains more vertices.)SH +/Times-Italic SF +8200 32255 MT +(Proof)SH +/Times-Roman SF +10749 XM +(: The proof of) +21 W( this theorem is conceptually straightforward. However, we must take care not to presuppose)20 W +7200 33632 MT +(anything about the possible representations of a function. The proof proceeds by induction on the size of)SH +/Times-Italic SF +49499 XM +(I)SH +8 SS +49832 33977 MT +(f)SH +10 /Times-Roman AF +50204 33632 MT +(.)SH +8200 36111 MT +(For)SH +/Symbol SF +9904 XM +(|)SH +/Times-Italic SF +(I)SH +8 SS +10437 36456 MT +(f)SH +10 /Symbol AF +10809 36111 MT +(|)SH +/Times-Roman SF +11259 XM +(= 0,)SH +/Times-Italic SF +13138 XM +(f)SH +/Times-Roman SF +13731 XM +(must be one of the two constant functions)65 W +/Times-Bold SF +31194 XM +(0)SH +/Times-Roman SF +32009 XM +(or)SH +/Times-Bold SF +33157 XM +(1)SH +/Times-Roman SF +(. Let)380 W +/Times-Italic SF +36185 XM +(G)SH +/Times-Roman SF +37222 XM +(be a reduced function) +65 W( graph denoting the)66 W +7200 37488 MT +(function)SH +/Times-Bold SF +10799 XM +(0)SH +/Times-Roman SF +(. This) +282 W( graph can contain no terminal vertices having value 1, or else there would be some set of argument)16 W +7200 38865 MT +(values for which the function evaluates to 1, since all vertices in a) +168 W( function graph are reachable by some path)169 W +7200 40242 MT +(corresponding to a set of argument values. Now suppose)10 W +/Times-Italic SF +30435 XM +(G)SH +/Times-Roman SF +31417 XM +(contains at least one nonterminal vertex.) +10 W( Then) +269 W( since the)9 W +7200 41619 MT +(graph is acyclic, there must be a) +99 W( nonterminal vertex)100 W +/Times-Italic SF +29034 XM +(v)SH +/Times-Roman SF +29828 XM +(where both)100 W +/Times-Italic SF +34749 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 and)100 W +/Times-Italic SF +39448 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 are terminal vertices, and it)100 W +7200 42996 MT +(follows that)124 W +/Times-Italic SF +12448 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 =)SH +/Times-Italic SF +18899 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 = 0.) +SH( Either) +498 W( these 2 vertices are distinct, in which case they constitute)124 W +7200 44373 MT +(isomorphic subgraphs,) +53 W( or they are identical, in which case)54 W +/Times-Italic SF +31153 XM +(v)SH +/Times-Roman SF +31901 XM +(has)SH +/Times-Italic SF +33538 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +37157 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051. In) +358 W( either case,)54 W +/Times-Italic SF +46896 XM +(G)SH +/Times-Roman SF +47922 XM +(would not be a)54 W +7200 45750 MT +(reduced function graph. Hence, the) +210 W( only reduced function graph denoting the function)209 W +/Times-Bold SF +44831 XM +(0)SH +/Times-Roman SF +45790 XM +(consists of a single)209 W +7200 47127 MT +(terminal vertex with value 0. Similarly, the only reduced function graph denoting)103 W +/Times-Bold SF +41564 XM +(1)SH +/Times-Roman SF +42418 XM +(consists of a single terminal)104 W +7200 48504 MT +(vertex with value 1.)SH +8200 50983 MT +(Next suppose that the statement) +91 W( of the theorem holds for any function)90 W +/Times-Italic SF +37447 XM +(g)SH +/Times-Roman SF +38287 XM +(having)SH +/Symbol SF +41349 XM +(|)SH +/Times-Italic SF +(I)SH +8 SS +41882 51328 MT +(g)SH +10 /Symbol AF +42432 50983 MT +(|)SH +/Times-Roman SF +42882 XM +(<)SH +/Times-Italic SF +43696 XM +(k)SH +/Times-Roman SF +(, and that)90 W +/Symbol SF +48354 XM +(|)SH +/Times-Italic SF +(I)SH +8 SS +48887 51328 MT +(f)SH +10 /Symbol AF +49259 50983 MT +(|)SH +/Times-Roman SF +49709 XM +(=)SH +/Times-Italic SF +50523 XM +(k)SH +/Times-Roman SF +(, where)90 W +/Times-Italic SF +7200 52360 MT +(k)SH +/Times-Roman SF +7894 XM +(> 0.) +SH( Let)584 W +/Times-Italic SF +12042 XM +(i)SH +/Times-Roman SF +12737 XM +(be the minimum value in)167 W +/Times-Italic SF +23822 XM +(I)SH +8 SS +24155 52705 MT +(f)SH +10 /Times-Roman AF +24527 52360 MT +(, i.e. the least argument on which the function)167 W +/Times-Italic SF +44583 XM +(f)SH +/Times-Roman SF +45278 XM +(depends. Define) +584 W( the)168 W +7200 53737 MT +(functions)SH +/Times-Italic SF +11284 XM +(f)SH +8 /Times-Roman AF +11562 54082 MT +(0)SH +10 SS +12324 53737 MT +(and)SH +/Times-Italic SF +14130 XM +(f)SH +8 /Times-Roman AF +14408 54082 MT +(1)SH +10 SS +15170 53737 MT +(as)SH +/Times-Italic SF +16365 XM +(f)SH +/Symbol SF +16893 XM +(|)SH +8 /Times-Italic AF +17093 54082 MT +(x)SH +17448 54427 MT +(i)SH +/Times-Roman SF +17670 54082 MT +(=0)SH +10 SS +18883 53737 MT +(and)SH +/Times-Italic SF +20689 XM +(f)SH +/Symbol SF +21217 XM +(|)SH +8 /Times-Italic AF +21417 54082 MT +(x)SH +21772 54427 MT +(i)SH +/Times-Roman SF +21994 54082 MT +(=1)SH +10 SS +22845 53737 MT +(, respectively.) +112 W( Both)472 W +/Times-Italic SF +31567 XM +(f)SH +8 /Times-Roman AF +31845 54082 MT +(0)SH +10 SS +32606 53737 MT +(and)SH +/Times-Italic SF +34411 XM +(f)SH +8 /Times-Roman AF +34689 54082 MT +(1)SH +10 SS +35450 53737 MT +(have dependency sets of size less than)111 W +/Times-Italic SF +51751 XM +(k)SH +/Times-Roman SF +52556 XM +(and)SH +7200 55288 MT +(hence are represented by unique reduced function graphs. Let)42 W +/Times-Italic SF +32751 XM +(G)SH +/Times-Roman SF +33765 XM +(and)SH +/Times-Italic SF +35501 XM +(G)SH +/Symbol SF +36373 XM +(\242)SH +/Times-Roman SF +36912 XM +(be reduced function) +42 W( graphs for)43 W +/Times-Italic SF +49649 XM +(f)SH +/Times-Roman SF +(. We) +336 W( will)43 W +7200 56665 MT +(show that these two graphs are isomorphic, consisting of a root vertex with index)14 W +/Times-Italic SF +40142 XM +(i)SH +/Times-Roman SF +40684 XM +(and with low) +14 W( and high subgraphs)13 W +7200 58042 MT +(denoting the functions)426 W +/Times-Italic SF +17672 XM +(f)SH +8 /Times-Roman AF +17950 58387 MT +(0)SH +10 SS +19026 58042 MT +(and)SH +/Times-Italic SF +21146 XM +(f)SH +8 /Times-Roman AF +21424 58387 MT +(1)SH +10 SS +21824 58042 MT +(. Let)1104 W +/Times-Italic SF +25438 XM +(v)SH +/Times-Roman SF +26559 XM +(and)SH +/Times-Italic SF +28680 XM +(v)SH +/Symbol SF +29274 XM +(\242)SH +/Times-Roman SF +30198 XM +(be nonterminal vertices in the two graphs such that)427 W +/Times-Italic SF +7200 59419 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +11540 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +14633 XM +(\242)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +16277 XM +(i)SH +/Times-Roman SF +(. The) +426 W( subgraphs rooted) +88 W( by)87 W +/Times-Italic SF +27996 XM +(v)SH +/Times-Roman SF +28777 XM +(and)SH +/Times-Italic SF +30558 XM +(v)SH +/Symbol SF +31152 XM +(\242)SH +/Times-Roman SF +31736 XM +(both denote)87 W +/Times-Italic SF +36854 XM +(f)SH +/Times-Roman SF +(, since)87 W +/Times-Italic SF +40111 XM +(f)SH +/Times-Roman SF +40876 XM +(is independent of the arguments)87 W +/Times-Italic SF +7200 60796 MT +(x)SH +8 /Times-Roman AF +7644 61141 MT +(1)SH +10 SS +8044 60796 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +10738 61141 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +11799 60796 MT +(. The) +722 W( subgraphs rooted by vertices)236 W +/Times-Italic SF +27726 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 and)237 W +/Times-Italic SF +32699 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +35071 XM +(\242)SH +/Times-Roman SF +(\051 both denote the function)237 W +/Times-Italic SF +47085 XM +(f)SH +8 /Times-Roman AF +47363 61141 MT +(0)SH +10 SS +48250 60796 MT +(and hence by)237 W +7200 62173 MT +(induction must be isomorphic according to some mapping)78 W +/Symbol SF +31323 XM +(s)SH +8 /Times-Roman AF +31926 62518 MT +(0)SH +10 SS +32326 62173 MT +(. Similarly,) +406 W( the subgraphs rooted) +78 W( by vertices)77 W +/Times-Italic SF +51112 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +7200 63550 MT +(and)SH +/Times-Italic SF +8894 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Symbol SF +11599 XM +(\242)SH +/Times-Roman SF +(\051 both denote the function)SH +/Times-Italic SF +22428 XM +(f)SH +8 /Times-Roman AF +22706 63895 MT +(1)SH +10 SS +23356 63550 MT +(and hence must be isomorphic according to some mapping)SH +/Symbol SF +47103 XM +(s)SH +8 /Times-Roman AF +47706 63895 MT +(1)SH +10 SS +48106 63550 MT +(.)SH +8200 66029 MT +(We claim that the subgraphs rooted by)SH +/Times-Italic SF +23892 XM +(v)SH +/Times-Roman SF +24586 XM +(and)SH +/Times-Italic SF +26280 XM +(v)SH +/Symbol SF +26874 XM +(\242)SH +/Times-Roman SF +27371 XM +(must be isomorphic according to the mapping)SH +/Symbol SF +45953 XM +(s)SH +/Times-Roman SF +46806 XM +(defined as)SH +/Times-Italic SF +25400 67820 MT +(v)SH +/Symbol SF +25994 XM +(\242)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +28640 XM +(u)SH +/Times-Roman SF +29390 XM +(=)SH +/Times-Italic SF +30204 XM +(v)SH +/Times-Roman SF +(,)SH +/Symbol SF +20000 68925 MT +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)436 W +/Symbol SF +25400 XM +(s)SH +8 /Times-Roman AF +26003 69270 MT +(0)SH +10 SS +26403 68925 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +28640 XM +(u)SH +/Times-Roman SF +29540 XM +(in subgraph rooted by)SH +/Times-Italic SF +38439 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +25400 70131 MT +(s)SH +8 /Times-Roman AF +26003 70476 MT +(1)SH +10 SS +26403 70131 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051,)SH +/Times-Italic SF +28640 XM +(u)SH +/Times-Roman SF +29540 XM +(in subgraph rooted by)SH +/Times-Italic SF +38439 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +7200 71651 MT +(To prove this, we must show that the function)20 W +/Symbol SF +25990 XM +(s)SH +/Times-Roman SF +26863 XM +(is well-defined, and that it is an isomorphic mapping.) +20 W( Observe) +292 W( that)21 W +ES +%%Page: 6 6 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(6)SH +7200 7886 MT +(if vertex)3 W +/Times-Italic SF +10816 XM +(u)SH +/Times-Roman SF +11569 XM +(is contained in both the subgraph rooted by)3 W +/Times-Italic SF +29147 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 and the subgraph rooted by)3 W +/Times-Italic SF +43107 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051, then the subgraphs)2 W +7200 9263 MT +(rooted by)20 W +/Symbol SF +11295 XM +(s)SH +8 /Times-Roman AF +11898 9608 MT +(0)SH +10 SS +12298 9263 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 and)20 W +/Symbol SF +15448 XM +(s)SH +8 /Times-Roman AF +16051 9608 MT +(1)SH +10 SS +16451 9263 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 must be isomorphic to the one rooted by)20 W +/Times-Italic SF +34435 XM +(u)SH +/Times-Roman SF +35205 XM +(and hence to each other. Since)20 W +/Times-Italic SF +48008 XM +(G)SH +/Symbol SF +48880 XM +(\242)SH +/Times-Roman SF +49397 XM +(contains no)20 W +7200 10640 MT +(isomorphic subgraphs, this can only hold if)7 W +/Symbol SF +24804 XM +(s)SH +8 /Times-Roman AF +25407 10985 MT +(0)SH +10 SS +25807 10640 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Symbol SF +28037 XM +(s)SH +8 /Times-Roman AF +28640 10985 MT +(1)SH +10 SS +29040 10640 MT +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051, and hence there is no conflict in the above definition of)7 W +/Symbol SF +53147 XM +(s)SH +/Times-Roman SF +(.)SH +7200 12017 MT +(By similar reasoning, we can see that)74 W +/Symbol SF +22882 XM +(s)SH +/Times-Roman SF +23809 XM +(must be one-to-one---if there) +74 W( were distinct vertices)75 W +/Times-Italic SF +44909 XM +(u)SH +8 /Times-Roman AF +45409 12362 MT +(1)SH +10 SS +46134 12017 MT +(and)SH +/Times-Italic SF +47903 XM +(u)SH +8 /Times-Roman AF +48403 12362 MT +(2)SH +10 SS +49128 12017 MT +(in)SH +/Times-Italic SF +50231 XM +(G)SH +/Times-Roman SF +51278 XM +(having)SH +/Symbol SF +7200 13394 MT +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +8 /Times-Roman AF +8636 13739 MT +(1)SH +10 SS +9036 13394 MT +(\051) +150 MX(=)SH +/Symbol SF +10233 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +8 /Times-Roman AF +11669 13739 MT +(2)SH +10 SS +12069 13394 MT +(\051, then) +33 W( the subgraphs rooted by these two vertices would be isomorphic to the subgraph rooted by)32 W +/Symbol SF +51831 XM +(s)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +8 /Times-Roman AF +53267 13739 MT +(1)SH +10 SS +53667 13394 MT +(\051)SH +7200 14771 MT +(and hence to each other implying that)57 W +/Times-Italic SF +22902 XM +(G)SH +/Times-Roman SF +23931 XM +(is not reduced. Finally, the properties that)57 W +/Symbol SF +41552 XM +(s)SH +/Times-Roman SF +42462 XM +(is onto and) +57 W( is an isomorphic)58 W +7200 16148 MT +(mapping follows directly from its definition and from the fact that both)SH +/Symbol SF +35920 XM +(s)SH +8 /Times-Roman AF +36523 16493 MT +(0)SH +10 SS +37173 16148 MT +(and)SH +/Symbol SF +38867 XM +(s)SH +8 /Times-Roman AF +39470 16493 MT +(1)SH +10 SS +40120 16148 MT +(obey these properties.)SH +8200 18627 MT +(By similar reasoning, we can see that graph)8 W +/Times-Italic SF +25955 XM +(G)SH +/Times-Roman SF +26935 XM +(contains exactly one vertex with)7 W +/Times-Italic SF +40162 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +44502 XM +(i)SH +/Times-Roman SF +(, because if some other)7 W +7200 20004 MT +(such vertex existed, the subgraphs rooted by it and by)53 W +/Times-Italic SF +29477 XM +(v)SH +/Times-Roman SF +30224 XM +(would be isomorphic. We claim in fact that)53 W +/Times-Italic SF +48533 XM +(v)SH +/Times-Roman SF +49281 XM +(must be the)54 W +7200 21381 MT +(root. Suppose) +540 W( instead that there is some vertex)145 W +/Times-Italic SF +27614 XM +(u)SH +/Times-Roman SF +28509 XM +(with)SH +/Times-Italic SF +30682 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +35078 XM +(j)SH +/Times-Roman SF +35606 XM +(<)SH +/Times-Italic SF +36420 XM +(i)SH +/Times-Roman SF +(, but such that there is no other vertex)145 W +/Times-Italic SF +53333 XM +(w)SH +/Times-Roman SF +7200 22758 MT +(having)SH +/Times-Italic SF +10306 XM +(j)SH +/Times-Roman SF +10834 XM +(<)SH +/Times-Italic SF +11648 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(w)SH +/Times-Roman SF +(\051 <)SH +/Times-Italic SF +16211 XM +(i)SH +/Times-Roman SF +(. The function)135 W +/Times-Italic SF +22782 XM +(f)SH +/Times-Roman SF +23445 XM +(does not depend on the)135 W +/Times-Italic SF +33591 XM +(x)SH +8 SS +34035 23103 MT +(j)SH +10 /Times-Roman AF +34642 22758 MT +(and hence the subgraphs rooted by)135 W +/Times-Italic SF +49560 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 and)135 W +/Times-Italic SF +7200 24135 MT +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 both denote)42 W +/Times-Italic SF +15464 XM +(f)SH +/Times-Roman SF +(, but this implies that)42 W +/Times-Italic SF +24620 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +28295 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +32303 XM +(v)SH +/Times-Roman SF +(, i.e.)42 W +/Times-Italic SF +35095 XM +(G)SH +/Times-Roman SF +36109 XM +(is not) +42 W( reduced. Similarly, vertex)41 W +/Times-Italic SF +49688 XM +(v)SH +/Symbol SF +50282 XM +(\242)SH +/Times-Roman SF +50820 XM +(must be)41 W +7200 25512 MT +(the root of)SH +/Times-Italic SF +11616 XM +(G)SH +/Symbol SF +12488 XM +(\242)SH +/Times-Roman SF +(, and hence the two graphs are isomorphic.)SH +8200 27991 MT +(Finally, we can prove that of all the graphs denoting a) +23 W( particular function, only the reduced graph has a minimum)24 W +7200 29368 MT +(number of vertices. Suppose)149 W +/Times-Italic SF +19832 XM +(G)SH +/Times-Roman SF +20953 XM +(is not) +149 W( a reduced graph. Then we can form a smaller graph denoting the same)148 W +7200 30745 MT +(function as follows. If)94 W +/Times-Italic SF +17002 XM +(G)SH +/Times-Roman SF +18069 XM +(contains a vertex)95 W +/Times-Italic SF +25380 XM +(v)SH +/Times-Roman SF +26169 XM +(with)SH +/Times-Italic SF +28292 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +31911 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051, then eliminate this vertex, and for any vertex)95 W +7200 32122 MT +(having)SH +/Times-Italic SF +10216 XM +(v)SH +/Times-Roman SF +10954 XM +(as child, make)44 W +/Times-Italic SF +17085 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 be the child instead. If)44 W +/Times-Italic SF +29613 XM +(G)SH +/Times-Roman SF +30629 XM +(contains distinct vertices)44 W +/Times-Italic SF +40899 XM +(v)SH +/Times-Roman SF +41637 XM +(and)SH +/Times-Italic SF +43375 XM +(v)SH +/Symbol SF +43969 XM +(\242)SH +/Times-Roman SF +44510 XM +(such that) +44 W( the subgraphs)43 W +7200 33499 MT +(rooted by)69 W +/Times-Italic SF +11393 XM +(v)SH +/Times-Roman SF +12156 XM +(and)SH +/Times-Italic SF +13919 XM +(v)SH +/Symbol SF +14513 XM +(\242)SH +/Times-Roman SF +15079 XM +(are isomorphic, then eliminate vertex)69 W +/Times-Italic SF +30588 XM +(v)SH +/Symbol SF +31182 XM +(\242)SH +/Times-Roman SF +31748 XM +(and for any vertex having)69 W +/Times-Italic SF +42618 XM +(v)SH +/Symbol SF +43212 XM +(\242)SH +/Times-Roman SF +43778 XM +(as a) +69 W( child, make)70 W +/Times-Italic SF +50750 XM +(v)SH +/Times-Roman SF +51514 XM +(be the)70 W +7200 34876 MT +(child instead.)SH +12 /Times-Bold AF +7200 38560 MT +(3. Properties)SH +1 SS +20155 7200 59820 PB +%%BeginDocument: figs/fig1.eps +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 201 202 +%%Title: (F:\\txt\\papers\\tc86\\figs\\fig1.CNV) +%%Creator: Canvas (7.0 : Build 270) +%%CreationDate: (04/08/00 @ 02:42 PM) +%%DocumentProcSets: Deneba_CanvasCustomColors +%%+ CVDict +%%DocumentFonts: +%%DocumentNeededFonts: +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%EndComments +%%BeginProlog +%%BeginResource: procset CVDict +%%Title: Deneba standard graphics +%%Version: 1.02 +%%CreationDate: 10/14/99 +%%Copyright: (C) 1994, 1999 Deneba Systems. All Rights Reserved +userdict /CVDict 200 dict put +CVDict begin +/invalidcolortable? true def +/level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def +level2 not { /xshow { +0 currentpoint 5 -1 roll {pop pop moveto dup 1 add 3 1 roll exch dup 4 1 roll exch get 0 rmoveto currentpoint} exch kshow +pop pop pop pop} def } if +/initialize +{CVDict +{dup xcheck{bind}if pop pop}forall +currentscreen /_sp exch def /_sa exch def /_sf exch def +/DefaultScreen{_sf _sa /_sp load}def +/DefaultScreenProc{ /_sp load}def +/setstrokeadjust where{pop true}{false}ifelse +{/sa /setstrokeadjust load def +/li /lineto load def +/mt /moveto load def +/ct /curveto load def +/lh { currentpoint exch pop lineto}bind def +/lv { currentpoint pop exch lineto}bind def} +{/sa{{currentdict StrokeAdjustDict ne{StrokeAdjustDict begin}if}{ currentdict StrokeAdjustDict eq{end}if}ifelse} bind def +userdict /StrokeAdjustDict 100 dict put +StrokeAdjustDict begin +/_xl +{transform +.25 sub round .25 add exch +.25 sub round .25 add exch +itransform +}bind def +/li{ _xl lineto}bind def +/mt{ _xl moveto}bind def +/ct{ _xl curveto}bind def +/lh { dup _xl pop currentpoint exch pop lineto}bind def +/lv { dup _xl pop currentpoint pop exch lineto}bind def +/rp{_xl/cv_b xdf/cv_r xdf +_xl/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath +}bind def +end}ifelse}def +/terminate +{userdict /StrokeAdjustDict known {currentdict StrokeAdjustDict eq{end}if}if +currentdict CVDict eq{end}if}def +/mxMax 64 def +/mxStk [mxMax {matrix} repeat]def +/mxIdx 0 def +/stm{mxStk mxIdx 0 lt {/mxIdx 0 store}if mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def +/rtm{mxIdx 1 ge {/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}if }def +/scl { 72 div exch 72 div exch scale } def +/scli { 72 exch div exch 72 exch div exch scale } def +/_cvmtx matrix def +/_txtmtx matrix def +/_initmtx matrix def +/xdf{exch def}def +/min{2 copy gt{exch}if pop}def +/max{2 copy lt{exch}if pop}def +/CVvec 256 array def +/T true def +/F false def +/_id 0 def +/_encodindex 0 def +/_encodArr 3 array def +/_fd 0 def +/_textSave null def +/_letterSpace 0 def +/_spaceXtra 0 def +/_PageSave null def +/_penh 1.0 def +/_penv 1.0 def +/save0 null def +/_w 0 def +/_h 0 def +/_ds 0 def +/hct false def +/pw 0 def +/wdif 0 def +/temp 0 def +/currot 0 def +/savehandler null def +/ia 0 def +/ix 0 def +/ixn 1 def +/MultipleDataSources false def +/debug +{dup /$eps exch not def{/savehandler errordict /handleerror get def +errordict +/handleerror{$error begin +newerror{/newerror false def +showpage +72 72 scale +/Helvetica findfont .2 scalefont setfont +.25 9.6 moveto +(Error: )show +errorname +{dup type/stringtype ne +{(converted error name will end up in this string)cvs}if +show}exec +.25 9.4 moveto +(Offending Command: )show +/command load +{dup type/stringtype ne +{(converted command name will end up in this string)cvs}if +show}exec +.25 9.2 moveto +(Stack: )show +ostack +{.25 currentpoint .2 sub exch pop moveto +dup type/stringtype ne{dup type/integertype eq +1 index type/realtype eq or +1 index type/nametype eq or +not {type}if +(converted stack will end up in this string)cvs}if +show}forall +showpage +CVDict /savehandler get exec}if +end}put}if}def +/lc{setlinecap}def +/lj{setlinejoin}def +/lm{setmiterlimit}def +/lf{setflat}def +/cp{closepath}def +/np{newpath}def +/$psz 30 array def +/$pix 0 def +/gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def +/gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def +/li{lineto} def +/lh{currentpoint exch pop lineto}def +/lv{currentpoint pop exch lineto}def +/mt{moveto} def +/_cv_mh{currentpoint exch pop moveto}def +/_cv_mv{currentpoint pop exch moveto}def +/ct{curveto} def +/rt{4 -2 roll moveto +dup 0 exch rlineto +exch 0 rlineto +neg 0 exch rlineto +closepath}def +/_hr +{ +{/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} +{/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} +}def +/_cs null def +/fg{setgray}def +/setcmykcolor where {pop true}{false}ifelse +{/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse +/setcolorspace where {pop true}{false}ifelse +{/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def +}{/f3{ setrgbcolor}def/scs{ pop}def}ifelse +/$cc 5 array def +/findcmykcustomcolor where {pop true}{false}ifelse +not {/findcmykcustomcolor{$cc astore}def}if +/setcustomcolor where {pop true}{false}ifelse +not{/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def}if +/fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def +/setseparationgray where {pop true}{false}ifelse +not{/setseparationgray {systemdict /setgray get exec} def}if +/sg{ setseparationgray}def +/sp{/_penv xdf/_penh xdf}def +/rp +{/cv_b xdf/cv_r xdf +/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath}def +/rr{/yradius xdf +/xradius xdf +/cv_b xdf +/cv_r xdf +/cv_t xdf +/cv_l xdf +cv_l xradius add cv_t moveto +cv_r xradius sub cv_t lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +1 0 1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_r cv_b yradius sub lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 1 -1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l xradius add cv_b lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +-1 0 -1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l cv_t yradius add lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 -1 1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +closepath}def +/ep{_cvmtx currentmatrix pop +translate +scale +1 0 moveto +0 0 1 0 360 arc +closepath +_cvmtx setmatrix}def +/wd{_cvmtx currentmatrix pop +translate +scale +0 0 moveto +0 0 1 5 -2 roll arcn +closepath +_cvmtx setmatrix}def +/as{_cvmtx currentmatrix pop +translate +scale +0 0 1 5 -2 roll arcn +_cvmtx setmatrix}def +/ds +{_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def +/df{fill}def +/de{eofill}def +/_gstk 5 array def +/_gstkc 0 def +/_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def +/_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def +/hs{newpath gsave stm}def +/hc{/_hr load exch get exec newpath}def +/he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store +grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq +not{aload pop moveto}{pop}ifelse dup /_cf get setfont +dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def +/id{ /_id xdf}def +/_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def +% Copyright (C) 1990, 1991 by Adobe Systems Incorporated. +% All rights reserved. +level2 { save /dontloadlevel1 xdf }if +/startnoload{{/noload save def} if}def +/endnoload{{noload restore}if }def +/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def +/ncolors 1 def +/colorimage where{pop true}{false}ifelse{/ncolors 0 +statusdict begin +/processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known +{pop {deviceinfo /Colors get}}if}if}ifelse end def +ncolors 0 ne{/colorimage testsystemdict +/setcolortransfer testsystemdict +/currentcolortransfer testsystemdict +/currentcmykcolor testsystemdict +and and and not{/ncolors 0 def}if}if}if +ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if +ncolors 1 eq dup dup not startnoload +{/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] +cvx bind settransfer _w _h 8 _imatrix _ds image}def}if +not endnoload +ncolors dup 3 eq exch 4 eq or +dup dup not startnoload +{/nullproc{{}}def/concatutil{/exec load +7 -1 roll /exec load}def +/defsubclut +{1 add getinterval def}def +/spconcattransfer +{/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld +currentcolortransfer +[{Aclut ncompute}concatutil] cvx +[{Bclut ncompute}concatutil] cvx +[{Cclut ncompute}concatutil] cvx +[{Dclut ncompute}concatutil] cvx +setcolortransfer}def +/setuprgbcluts +{/bit3x rgbclut length 3 sub def +/bit1x bit3x 3 idiv def/rclut rgbclut def +/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not +endnoload ncolors 3 eq dup dup not startnoload +{/3compute{exch bit3x mul +round cvi get 255 div}def +/doclutimage{%/rgbclut xdf pop +setuprgbcluts/3compute rclut gclut bclut dup +spconcattransfer _w _h 8 _imatrix +[/_ds load /exec load /dup load dup]cvx +nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload +{/stuffclut{cmykindex 3 -1 roll put}def +/ftoint{1 exch sub 255 mul round cvi}def +/4compute { exch bit4x mul round cvi get 255 div } def +/computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def +/cmykclut bit4x 4 add string def +/cclut cmykclut def +/mclut cclut 1 bit4x defsubclut +/yclut cclut 2 bit4x defsubclut +/kclut cclut 3 bit4x defsubclut +/cmykindex 0 def +0 1 bit1x +{dup /cmykindex exch bit1x exch sub 4 mul def +3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div +exch bclut exch get 255 div +setrgbcolor currentcmykcolor +ftoint kclut stuffclut +ftoint yclut stuffclut +ftoint mclut stuffclut +ftoint cclut stuffclut +}for}def +/doclutimage{invalidcolortable?{computecmykclut}if +/4compute cclut mclut yclut kclut spconcattransfer +_w _h 8 _imatrix[/_ds load /exec load/dup load dup dup] cvx +nullproc nullproc nullproc true 4 colorimage}def}if +not endnoload +ncolors 0 eq +dup dup not startnoload +{/colorexpand{mystringexp 0 rgbclut +3 copy +7 -1 roll +{3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall +pop pop pop pop pop}def +/createexpandstr{/mystringexp exch st0 length mul string def}def +/doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix +[/_ds load /exec load /colorexpand load /exec load] cvx +false 3 colorimage}def}if not endnoload +level2 { dontloadlevel1 restore }if +level2 not{ save /dontloadlevel2 xdf }if +/rgbclut 0 def +/doclutimage{/hival 255 def +[/Indexed /DeviceRGB hival rgbclut] setcolorspace +_imagedict dup begin +/Width _w def +/Height _h def +/Decode [0 hival] def +/ImageMatrix _imatrix def +/BitsPerComponent 8 def +/Interpolate false def +end image}def +level2 not{ dontloadlevel2 restore }if +%%EndProcSet +/_imagedict 15 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def +/hascolor true def +/level2 systemdict /languagelevel known +{languagelevel 2 ge}{false}ifelse def +/foureq {4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +pop pop pop pop and and and} def +/setupbands{gsave +/_settransfer systemdict /settransfer get def +/_currenttransfer systemdict /currenttransfer get def +/_image systemdict /image get def +/blank 0 _currenttransfer exec +1 _currenttransfer exec eq def +/negative blank{0 _currenttransfer exec 0.5 lt} +{0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def +/inverted? negative def +/_currentgray systemdict /currentgray get def +hascolor{/$b 0 def} {/$b 5 def} ifelse +/setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub +0 1 0 0 setcmykcolor _currentgray 1 exch sub +0 0 1 0 setcmykcolor _currentgray 1 exch sub +0 0 0 1 setcmykcolor _currentgray 1 exch sub +4 {4 copy} repeat +1 0 0 0 foureq{/$b 1 store}if +0 1 0 0 foureq{/$b 2 store}if +0 0 1 0 foureq{/$b 3 store}if +0 0 0 1 foureq{/$b 4 store}if +0 0 0 0 foureq{/$b 6 store}if}if +blank{/$b 6 store}if +$b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def +/doimage{ +level2{image}{begin +Width Height BitsPerComponent ImageMatrix +Decode length 2 eq +{/DataSource load end image}{ +Decode length 6 eq +{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse +MultipleDataSources 3 end DSIColorImage}{ +Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get +DataSource 2 get DataSource 3 get}{/DataSource load}ifelse +MultipleDataSources 4 end DSIColorImage}if}ifelse +}ifelse +}ifelse}def +/dt{/rgbclut xdf /bwclut xdf}def +/cl{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +doclutimage} def +/cd{$i setmatrix translate scale +level2{_imagedict /ColorSpace get setcolorspace}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +/ImageMatrix _imatrix def +currentdict end +doimage} def +/im{$i setmatrix translate scale +level2{/DeviceGray setcolorspace 0 setgray}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +%/Decode [0 1] def +%/Width _w def +%/Height _h def +/ImageMatrix _imatrix def +currentdict end +doimage }def +/colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ +/$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf +$nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub +{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add +cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup +st0 exch get .3 mul +1 index 1 add st0 exch get .59 mul add +1 index 2 add st0 exch get .11 mul add +1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if +255.0 exch sub cvi +exch 4 div cvi exch +st1 3 1 roll put}for st1}}ifelse +image}def}ifelse +/OPIimage{image}def +/$i matrix def +/bi{scale translate $i currentmatrix pop translate scale setupbands}def +/ei{$i setmatrix}def +/ik{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_w _h false _imatrix 5 -1 roll _ds imagemask}def +/rs{2 copy translate +3 -1 roll neg rotate +neg exch neg exch translate +}def +/cs{_cvmtx astore concat} def +/fd{/_fd xdf}def +/_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def +/rf{/_encodindex xdf +FontDirectory exch known { +findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall +dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def +_encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} +{exch pop}ifelse}forall pop}if}if +currentdict end +definefont pop }{pop pop}ifelse} def +/cvsf{ findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore +makefont setfont}def +/mf{3 -1 roll findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch +_fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def +/wrk1 ( ) def +/wdict 16 dict def +/Work75 75 string def +/Nmk{Work75 cvs dup}def +/Npt{put cvn}def +/dhOdh{Nmk 2 79 Npt}def +/dhBdh{Nmk 2 111 Npt}def +/dhSdh{Nmk 2 83 Npt}def +/sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def +/fts{/ftSize exch def}def +/mkFT{dup/FontType get 0 eq{dup maxlength dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +currentdict end +dup/FDepVector 2 copy get +[exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def +tempFT begin +/FontMatrix [1 0 0 1 0 0] def +/FontType 3 def +dup/Encoding get/Encoding exch def +/proc2 2 index def +dup/ft1 exch def +/ft2 1 array def +/FontBBox [0 0 1 1] def +/BuildChar{CVDict begin +exch dup/ftdt xdf begin +wdict begin +/chr xdf/chrst wrk1 dup 0 chr put def +ft2 0 get null eq{ft1 dup length 2 add dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +/PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def +currentdict end +/q exch definefont +ft2 exch 0 exch put}if +ftdt/proc2 get exec end end end}def end tempFT}ifelse +3 index exch +definefont exch pop}def +/mb{dup dhBdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{boldproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/mo{dup dhOdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{outproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/ms{dup dhSdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{ShadChar} exch +mkFT pop pop pop exch pop +}ifelse }def +/outproc{ft1 setfont +gsave chrst sfWidth grestore +setcharwidth dblsh}def +/boldproc{ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +ft1 /PaintType get 2 ne +{0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} +{0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def +/dblsh{currentcolor currentcolorspace 1.0 setgray +chrst 0 0 moveto show setcolorspace setcolor +ft2 0 get setfont chrst 0 0 moveto show}def +/ShadChar{ +ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +chrst .06 0 moveto show +0 .05 translate dblsh}def +/ts{show}def +/ta{/_letterSpace xdf/_spaceXtra xdf}def +/tf{/_textWidth xdf +/_numSpaces xdf +/_stringLength xdf +dup stringwidth pop +_letterSpace _stringLength 1 sub mul add +_spaceXtra _numSpaces mul add +_textWidth exch sub /wdif xdf +_spaceXtra 0 32 +wdif _stringLength div +_letterSpace add +0 6 -1 roll awidthshow}def +/pb{/_PageSave save def/showpage{}def}def +/pe{_PageSave restore}def +/pm{_pm currentmatrix pop}def +/_pm matrix def +/ps{showpage _pm setmatrix}def +level2{/ef {gsave 6 array astore concat dup execform grestore}def} +{/ef{gsave 6 array astore concat newpath dup begin Matrix concat +BBox dup dup 0 get exch 1 get moveto +dup dup 0 get exch 3 get lineto +dup dup 2 get exch 3 get lineto +dup 2 get exch 1 get lineto closepath +clip newpath PaintProc end grestore}def}ifelse +/bc{save 11 1 roll 10 dict begin /BBox 5 1 roll 4 array astore def +/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def +/ec{pop restore}def +/printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def +/setoverprint where {pop}{/setoverprint{pop}def}ifelse +/sstr { /$sstrs exch def /$sstr exch def /$sstry exch def /$sstrx exch def /Courier findfont $sstrs +scalefont setfont $sstrx $sstry mt stm 1 -1 scale $sstr show rtm } def +/DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc + eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} + {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def +/DSIEndPaintProc{currentdict end}def +/DSIBeginPaintSubProc {save 11 1 roll 10 dict dup /pD exch def begin + /BBox 5 1 roll 4 array astore def /Matrix 7 1 roll 6 array astore def + /FormType 1 def /PaintProc { /pp get exec } def /pp} def +/DSIEndPaintSubProc {bind def end pD} def +/iaa {/ix ix ixn add def} def +/ias2 {/igap exch def /_st0 ia ix igap add get def _st0} def +/ias {/st0 ia ix get def iaa st0} def +/i85s {0 ias2 /ASCII85Decode filter st0 readstring pop iaa} def +/irles {0 ias2 /ASCII85Decode filter /RunLengthDecode filter st0 readstring pop iaa} def +initialize +end +%%EndResource +CVDict begin 0 id 1 fd +_encodArr 0 +[ +39/quotesingle 96/grave +128/bullet/bullet/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/bullet/bullet/bullet +/bullet/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark +/scaron +/guilsinglright/oe/bullet/bullet/Ydieresis +/bullet/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron +/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown +/Agrave/Aacute/Acircumflex/Atilde/Adieresis +/Aring +/AE/Ccedilla/Egrave/Eacute/Egrave/Edieresis/Igrave/Iacute/Icircumflex/Idieresis +/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls +/agrave/aacute/acircumflex/atilde/adieresis +/aring +/ae/ccedilla/egrave/eacute/ecircumflex/edieresis +/igrave +/iacute +/icircumflex +/idieresis +/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis +] put +_encodArr 1 +[ +39/quotesingle 96/grave +128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash +/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation +/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash +/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft +/guillemotright/ellipsis/blank/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge +/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex +/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +] put +end +%%EndProlog +%%IncludeFont: CourierNewPSMT +CVDict begin +/___FN2 /Courier dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { 0 }{-1} ifelse end} {pop -1} ifelse rf +end +0.25 0.25 translate +%%BeginSetup +CVDict begin + +0 id +1 fd +0 202 translate +300 -300 scli +%%EndSetup +%%BeginPageSetup +pm +%%EndPageSetup +/DeviceRGB scs +0.0 0.0 0.0 1. f4 +0.1250 0.1250 sp + 0 lc + 2 lj +T sa +%page rect: 50 290 252 491 +stm +0.0 0.0 0.0 1. f4 +4.1667 4.1667 sp +11.5 lm + 0 lj +np +544.7903 626.732 mt +794.7897 460.0657 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 460.0657 mt +794.7897 626.732 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +794.7897 460.0657 mt +794.7897 626.732 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 460.0657 mt +544.7903 626.732 li +ds +rtm +stm +stm +41.6666 fts +/___FN2 cvsf +0.0 0.0 0.0 1. f4 +F sa +691.6235 493.3985 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +629.1237 493.3985 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +504.124 555.8983 mt +(0) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +816.6232 555.8983 mt +(0) show +T sa +rtm +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 793.3983 mt +794.7897 626.732 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 626.732 mt +794.7897 793.3983 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +794.7897 626.732 mt +794.7897 793.3983 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 626.732 mt +544.7903 793.3983 li +ds +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +691.6235 660.0647 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +629.1237 660.0647 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +504.124 722.5646 mt +(0) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +816.6232 722.5646 mt +(0) show +T sa +rtm +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 501.7323 mt +794.7897 335.066 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 335.066 mt +794.7897 501.7323 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +794.7897 335.066 mt +794.7897 501.7323 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +544.7903 335.066 mt +544.7903 501.7323 li +ds +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +691.6235 368.3988 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +629.1237 368.3988 mt +(1) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +504.124 430.8986 mt +(0) show +T sa +rtm +rtm +stm +stm +0.0 0.0 0.0 1. f4 +F sa +816.6232 430.8986 mt +(0) show +T sa +rtm +rtm +stm +0.0 0.0 0.0 1. f4 +np +541.6653 166.6663 mt +666.665 41.6666 li +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +85.4165 168.7605 mt +168.7496 43.7608 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+np +520.8321 437.499 812.498 520.8321 rp +de +rtm +stm +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 624.9985 479.1656 ep +de +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 624.9985 479.1656 ep +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 666.6651 479.1656 ep +de +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 666.6651 479.1656 ep +ds +rtm +stm +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 708.3317 479.1656 ep +de +0.0 0.0 0.0 1. f4 +np +10.4166 10.4166 708.3317 479.1656 ep +ds +rtm +terminate + +%%EndDocument + PE 10 /Times-Roman AF +23545 61430 MT +(Figure 1.Example Function Graphs)SH +8200 63221 MT +(In this section we explore the efficiency of our representation) +118 W( by means of several examples. Figure 1 shows)117 W +7200 64598 MT +(several examples of) +238 W( reduced function graphs. In this figure, a nonterminal vertex is represented by a circle)239 W +7200 65975 MT +(containing the index with the) +37 W( two children indicated by branches labeled 0 \050low\051 and 1 \050high\051. A terminal vertex is)36 W +7200 67352 MT +(represented by a square containing the value.)SH +ES +%%Page: 7 7 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(7)SH +11 /Times-Bold AF +7200 7937 MT +(3.1. Example Functions)SH +10 /Times-Roman AF +8200 9314 MT +(The function which yields the value of the)74 W +/Times-Italic SF +25956 XM +(i)SH +/Times-Roman SF +26384 XM +(th argument is denoted by a graph with a single nonterminal vertex)74 W +7200 10691 MT +(having index)27 W +/Times-Italic SF +12698 XM +(i)SH +/Times-Roman SF +13253 XM +(and having as low child a terminal vertex with value) +27 W( 0 and as high child a terminal vertex with value)26 W +7200 12068 MT +(1. We) +318 W( present this graph mainly to point out that an input variable can be viewed as a) +34 W( Boolean function, and hence)35 W +7200 13445 MT +(can be operated on by the manipulation algorithms described in this paper.)SH +8200 15924 MT +(The odd parity function of)56 W +/Times-Italic SF +19283 XM +(n)SH +/Times-Roman SF +20088 XM +(variables is denoted by a graph containing 2)55 W +/Times-Italic SF +(n)SH +/Times-Roman SF +(+1 vertices. This compares favorably)55 W +7200 17301 MT +(to its representation in reduced sum-of-products form \050requiring 2)27 W +8 /Times-Italic AF +33800 16956 MT +(n)SH +10 /Times-Roman AF +34477 17301 MT +(terms.\051 This) +304 W( graph) +27 W( resembles the familiar parity)28 W +7200 18678 MT +(ladder contact network first described by Shannon) +38 W( [11].) +SH( In) +324 W( fact, we can adapt his construction of a contact network)37 W +7200 20055 MT +(to implement an arbitrary symmetric function to show that any symmetric function of)63 W +/Times-Italic SF +42488 XM +(n)SH +/Times-Roman SF +43301 XM +(arguments is denoted by a)64 W +7200 21432 MT +(reduced function graph having O) +150 0 WX(\050)SH +/Times-Italic SF +(n)SH +8 /Times-Roman AF +21402 21087 MT +(2)SH +10 SS +21802 21432 MT +(\051 vertices.)SH +8200 23911 MT +(As a) +193 W( third example, the graph denoting the function)192 W +/Times-Italic SF +30815 XM +(x)SH +8 /Times-Roman AF +31259 24256 MT +(1)SH +10 /Symbol AF +31659 23911 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +32353 24256 MT +(2)SH +10 SS +33003 23911 MT +(+)SH +/Times-Italic SF +33817 XM +(x)SH +8 /Times-Roman AF +34261 24256 MT +(4)SH +10 SS +35103 23911 MT +(contains 5 vertices as shown. This example)192 W +7200 25288 MT +(illustrates several key properties of reduced function graphs. First, observe that there is no) +74 W( vertex having index 3,)75 W +7200 26665 MT +(because the) +62 W( function is independent of)61 W +/Times-Italic SF +23175 XM +(x)SH +8 /Times-Roman AF +23619 27010 MT +(3)SH +10 SS +24019 26665 MT +(. More) +372 W( generally, a reduced function graph for a function)61 W +/Times-Italic SF +47989 XM +(f)SH +/Times-Roman SF +48578 XM +(contains only)61 W +7200 28042 MT +(vertices having indices in)17 W +/Times-Italic SF +17711 XM +(I)SH +8 SS +18044 28387 MT +(f)SH +10 /Times-Roman AF +18416 28042 MT +(. There) +285 W( are no inefficiencies caused by considering all of the functions to have the same)18 W +/Times-Italic SF +7200 29419 MT +(n)SH +/Times-Roman SF +7955 XM +(arguments. This) +259 W( would not be the case if we represented functions by their truth tables. Second, observe that even)4 W +7200 30796 MT +(for this simple function, several) +1 W( of the subgraphs are shared by different branches. This sharing yields efficiency not)2 W +7200 32173 MT +(only in the size of the) +45 W( function representation, but also in the performance of our algorithms---once some operation)44 W +7200 33550 MT +(has been performed on a subgraph, the result can be utilized by all places sharing this subgraph.)SH +11 /Times-Bold AF +7200 37167 MT +(3.2. Ordering Dependency)SH +1 SS +26242 7200 64514 PB +%%BeginDocument: figs/fig2.eps +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 311 263 +%%Title: (F:\\txt\\papers\\tc86\\figs\\fig2.CNV) +%%Creator: Canvas (7.0 : Build 270) +%%CreationDate: (04/08/00 @ 03:53 PM) +%%DocumentProcSets: Deneba_CanvasCustomColors +%%+ CVDict +%%DocumentFonts: +%%DocumentNeededFonts: +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%EndComments +%%BeginProlog +%%BeginResource: procset CVDict +%%Title: Deneba standard graphics +%%Version: 1.02 +%%CreationDate: 10/14/99 +%%Copyright: (C) 1994, 1999 Deneba Systems. All Rights Reserved +userdict /CVDict 200 dict put +CVDict begin +/invalidcolortable? true def +/level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def +level2 not { /xshow { +0 currentpoint 5 -1 roll {pop pop moveto dup 1 add 3 1 roll exch dup 4 1 roll exch get 0 rmoveto currentpoint} exch kshow +pop pop pop pop} def } if +/initialize +{CVDict +{dup xcheck{bind}if pop pop}forall +currentscreen /_sp exch def /_sa exch def /_sf exch def +/DefaultScreen{_sf _sa /_sp load}def +/DefaultScreenProc{ /_sp load}def +/setstrokeadjust where{pop true}{false}ifelse +{/sa /setstrokeadjust load def +/li /lineto load def +/mt /moveto load def +/ct /curveto load def +/lh { currentpoint exch pop lineto}bind def +/lv { currentpoint pop exch lineto}bind def} +{/sa{{currentdict StrokeAdjustDict ne{StrokeAdjustDict begin}if}{ currentdict StrokeAdjustDict eq{end}if}ifelse} bind def +userdict /StrokeAdjustDict 100 dict put +StrokeAdjustDict begin +/_xl +{transform +.25 sub round .25 add exch +.25 sub round .25 add exch +itransform +}bind def +/li{ _xl lineto}bind def +/mt{ _xl moveto}bind def +/ct{ _xl curveto}bind def +/lh { dup _xl pop currentpoint exch pop lineto}bind def +/lv { dup _xl pop currentpoint pop exch lineto}bind def +/rp{_xl/cv_b xdf/cv_r xdf +_xl/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath +}bind def +end}ifelse}def +/terminate +{userdict /StrokeAdjustDict known {currentdict StrokeAdjustDict eq{end}if}if +currentdict CVDict eq{end}if}def +/mxMax 64 def +/mxStk [mxMax {matrix} repeat]def +/mxIdx 0 def +/stm{mxStk mxIdx 0 lt {/mxIdx 0 store}if mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def +/rtm{mxIdx 1 ge {/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}if }def +/scl { 72 div exch 72 div exch scale } def +/scli { 72 exch div exch 72 exch div exch scale } def +/_cvmtx matrix def +/_txtmtx matrix def +/_initmtx matrix def +/xdf{exch def}def +/min{2 copy gt{exch}if pop}def +/max{2 copy lt{exch}if pop}def +/CVvec 256 array def +/T true def +/F false def +/_id 0 def +/_encodindex 0 def +/_encodArr 3 array def +/_fd 0 def +/_textSave null def +/_letterSpace 0 def +/_spaceXtra 0 def +/_PageSave null def +/_penh 1.0 def +/_penv 1.0 def +/save0 null def +/_w 0 def +/_h 0 def +/_ds 0 def +/hct false def +/pw 0 def +/wdif 0 def +/temp 0 def +/currot 0 def +/savehandler null def +/ia 0 def +/ix 0 def +/ixn 1 def +/MultipleDataSources false def +/debug +{dup /$eps exch not def{/savehandler errordict /handleerror get def +errordict +/handleerror{$error begin +newerror{/newerror false def +showpage +72 72 scale +/Helvetica findfont .2 scalefont setfont +.25 9.6 moveto +(Error: )show +errorname +{dup type/stringtype ne +{(converted error name will end up in this string)cvs}if +show}exec +.25 9.4 moveto +(Offending Command: )show +/command load +{dup type/stringtype ne +{(converted command name will end up in this string)cvs}if +show}exec +.25 9.2 moveto +(Stack: )show +ostack +{.25 currentpoint .2 sub exch pop moveto +dup type/stringtype ne{dup type/integertype eq +1 index type/realtype eq or +1 index type/nametype eq or +not {type}if +(converted stack will end up in this string)cvs}if +show}forall +showpage +CVDict /savehandler get exec}if +end}put}if}def +/lc{setlinecap}def +/lj{setlinejoin}def +/lm{setmiterlimit}def +/lf{setflat}def +/cp{closepath}def +/np{newpath}def +/$psz 30 array def +/$pix 0 def +/gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def +/gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def +/li{lineto} def +/lh{currentpoint exch pop lineto}def +/lv{currentpoint pop exch lineto}def +/mt{moveto} def +/_cv_mh{currentpoint exch pop moveto}def +/_cv_mv{currentpoint pop exch moveto}def +/ct{curveto} def +/rt{4 -2 roll moveto +dup 0 exch rlineto +exch 0 rlineto +neg 0 exch rlineto +closepath}def +/_hr +{ +{/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} +{/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} +}def +/_cs null def +/fg{setgray}def +/setcmykcolor where {pop true}{false}ifelse +{/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse +/setcolorspace where {pop true}{false}ifelse +{/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def +}{/f3{ setrgbcolor}def/scs{ pop}def}ifelse +/$cc 5 array def +/findcmykcustomcolor where {pop true}{false}ifelse +not {/findcmykcustomcolor{$cc astore}def}if +/setcustomcolor where {pop true}{false}ifelse +not{/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def}if +/fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def +/setseparationgray where {pop true}{false}ifelse +not{/setseparationgray {systemdict /setgray get exec} def}if +/sg{ setseparationgray}def +/sp{/_penv xdf/_penh xdf}def +/rp +{/cv_b xdf/cv_r xdf +/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath}def +/rr{/yradius xdf +/xradius xdf +/cv_b xdf +/cv_r xdf +/cv_t xdf +/cv_l xdf +cv_l xradius add cv_t moveto +cv_r xradius sub cv_t lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +1 0 1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_r cv_b yradius sub lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 1 -1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l xradius add cv_b lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +-1 0 -1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l cv_t yradius add lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 -1 1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +closepath}def +/ep{_cvmtx currentmatrix pop +translate +scale +1 0 moveto +0 0 1 0 360 arc +closepath +_cvmtx setmatrix}def +/wd{_cvmtx currentmatrix pop +translate +scale +0 0 moveto +0 0 1 5 -2 roll arcn +closepath +_cvmtx setmatrix}def +/as{_cvmtx currentmatrix pop +translate +scale +0 0 1 5 -2 roll arcn +_cvmtx setmatrix}def +/ds +{_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def +/df{fill}def +/de{eofill}def +/_gstk 5 array def +/_gstkc 0 def +/_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def +/_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def +/hs{newpath gsave stm}def +/hc{/_hr load exch get exec newpath}def +/he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store +grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq +not{aload pop moveto}{pop}ifelse dup /_cf get setfont +dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def +/id{ /_id xdf}def +/_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def +% Copyright (C) 1990, 1991 by Adobe Systems Incorporated. +% All rights reserved. +level2 { save /dontloadlevel1 xdf }if +/startnoload{{/noload save def} if}def +/endnoload{{noload restore}if }def +/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def +/ncolors 1 def +/colorimage where{pop true}{false}ifelse{/ncolors 0 +statusdict begin +/processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known +{pop {deviceinfo /Colors get}}if}if}ifelse end def +ncolors 0 ne{/colorimage testsystemdict +/setcolortransfer testsystemdict +/currentcolortransfer testsystemdict +/currentcmykcolor testsystemdict +and and and not{/ncolors 0 def}if}if}if +ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if +ncolors 1 eq dup dup not startnoload +{/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] +cvx bind settransfer _w _h 8 _imatrix _ds image}def}if +not endnoload +ncolors dup 3 eq exch 4 eq or +dup dup not startnoload +{/nullproc{{}}def/concatutil{/exec load +7 -1 roll /exec load}def +/defsubclut +{1 add getinterval def}def +/spconcattransfer +{/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld +currentcolortransfer +[{Aclut ncompute}concatutil] cvx +[{Bclut ncompute}concatutil] cvx +[{Cclut ncompute}concatutil] cvx +[{Dclut ncompute}concatutil] cvx +setcolortransfer}def +/setuprgbcluts +{/bit3x rgbclut length 3 sub def +/bit1x bit3x 3 idiv def/rclut rgbclut def +/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not +endnoload ncolors 3 eq dup dup not startnoload +{/3compute{exch bit3x mul +round cvi get 255 div}def +/doclutimage{%/rgbclut xdf pop +setuprgbcluts/3compute rclut gclut bclut dup +spconcattransfer _w _h 8 _imatrix +[/_ds load /exec load /dup load dup]cvx +nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload +{/stuffclut{cmykindex 3 -1 roll put}def +/ftoint{1 exch sub 255 mul round cvi}def +/4compute { exch bit4x mul round cvi get 255 div } def +/computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def +/cmykclut bit4x 4 add string def +/cclut cmykclut def +/mclut cclut 1 bit4x defsubclut +/yclut cclut 2 bit4x defsubclut +/kclut cclut 3 bit4x defsubclut +/cmykindex 0 def +0 1 bit1x +{dup /cmykindex exch bit1x exch sub 4 mul def +3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div +exch bclut exch get 255 div +setrgbcolor currentcmykcolor +ftoint kclut stuffclut +ftoint yclut stuffclut +ftoint mclut stuffclut +ftoint cclut stuffclut +}for}def +/doclutimage{invalidcolortable?{computecmykclut}if +/4compute cclut mclut yclut kclut spconcattransfer +_w _h 8 _imatrix[/_ds load /exec load/dup load dup dup] cvx +nullproc nullproc nullproc true 4 colorimage}def}if +not endnoload +ncolors 0 eq +dup dup not startnoload +{/colorexpand{mystringexp 0 rgbclut +3 copy +7 -1 roll +{3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall +pop pop pop pop pop}def +/createexpandstr{/mystringexp exch st0 length mul string def}def +/doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix +[/_ds load /exec load /colorexpand load /exec load] cvx +false 3 colorimage}def}if not endnoload +level2 { dontloadlevel1 restore }if +level2 not{ save /dontloadlevel2 xdf }if +/rgbclut 0 def +/doclutimage{/hival 255 def +[/Indexed /DeviceRGB hival rgbclut] setcolorspace +_imagedict dup begin +/Width _w def +/Height _h def +/Decode [0 hival] def +/ImageMatrix _imatrix def +/BitsPerComponent 8 def +/Interpolate false def +end image}def +level2 not{ dontloadlevel2 restore }if +%%EndProcSet +/_imagedict 15 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def +/hascolor true def +/level2 systemdict /languagelevel known +{languagelevel 2 ge}{false}ifelse def +/foureq {4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +pop pop pop pop and and and} def +/setupbands{gsave +/_settransfer systemdict /settransfer get def +/_currenttransfer systemdict /currenttransfer get def +/_image systemdict /image get def +/blank 0 _currenttransfer exec +1 _currenttransfer exec eq def +/negative blank{0 _currenttransfer exec 0.5 lt} +{0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def +/inverted? negative def +/_currentgray systemdict /currentgray get def +hascolor{/$b 0 def} {/$b 5 def} ifelse +/setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub +0 1 0 0 setcmykcolor _currentgray 1 exch sub +0 0 1 0 setcmykcolor _currentgray 1 exch sub +0 0 0 1 setcmykcolor _currentgray 1 exch sub +4 {4 copy} repeat +1 0 0 0 foureq{/$b 1 store}if +0 1 0 0 foureq{/$b 2 store}if +0 0 1 0 foureq{/$b 3 store}if +0 0 0 1 foureq{/$b 4 store}if +0 0 0 0 foureq{/$b 6 store}if}if +blank{/$b 6 store}if +$b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def +/doimage{ +level2{image}{begin +Width Height BitsPerComponent ImageMatrix +Decode length 2 eq +{/DataSource load end image}{ +Decode length 6 eq +{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse +MultipleDataSources 3 end DSIColorImage}{ +Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get +DataSource 2 get DataSource 3 get}{/DataSource load}ifelse +MultipleDataSources 4 end DSIColorImage}if}ifelse +}ifelse +}ifelse}def +/dt{/rgbclut xdf /bwclut xdf}def +/cl{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +doclutimage} def +/cd{$i setmatrix translate scale +level2{_imagedict /ColorSpace get setcolorspace}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +/ImageMatrix _imatrix def +currentdict end +doimage} def +/im{$i setmatrix translate scale +level2{/DeviceGray setcolorspace 0 setgray}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +%/Decode [0 1] def +%/Width _w def +%/Height _h def +/ImageMatrix _imatrix def +currentdict end +doimage }def +/colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ +/$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf +$nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub +{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add +cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup +st0 exch get .3 mul +1 index 1 add st0 exch get .59 mul add +1 index 2 add st0 exch get .11 mul add +1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if +255.0 exch sub cvi +exch 4 div cvi exch +st1 3 1 roll put}for st1}}ifelse +image}def}ifelse +/OPIimage{image}def +/$i matrix def +/bi{scale translate $i currentmatrix pop translate scale setupbands}def +/ei{$i setmatrix}def +/ik{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_w _h false _imatrix 5 -1 roll _ds imagemask}def +/rs{2 copy translate +3 -1 roll neg rotate +neg exch neg exch translate +}def +/cs{_cvmtx astore concat} def +/fd{/_fd xdf}def +/_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def +/rf{/_encodindex xdf +FontDirectory exch known { +findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall +dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def +_encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} +{exch pop}ifelse}forall pop}if}if +currentdict end +definefont pop }{pop pop}ifelse} def +/cvsf{ findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore +makefont setfont}def +/mf{3 -1 roll findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch +_fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def +/wrk1 ( ) def +/wdict 16 dict def +/Work75 75 string def +/Nmk{Work75 cvs dup}def +/Npt{put cvn}def +/dhOdh{Nmk 2 79 Npt}def +/dhBdh{Nmk 2 111 Npt}def +/dhSdh{Nmk 2 83 Npt}def +/sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def +/fts{/ftSize exch def}def +/mkFT{dup/FontType get 0 eq{dup maxlength dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +currentdict end +dup/FDepVector 2 copy get +[exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def +tempFT begin +/FontMatrix [1 0 0 1 0 0] def +/FontType 3 def +dup/Encoding get/Encoding exch def +/proc2 2 index def +dup/ft1 exch def +/ft2 1 array def +/FontBBox [0 0 1 1] def +/BuildChar{CVDict begin +exch dup/ftdt xdf begin +wdict begin +/chr xdf/chrst wrk1 dup 0 chr put def +ft2 0 get null eq{ft1 dup length 2 add dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +/PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def +currentdict end +/q exch definefont +ft2 exch 0 exch put}if +ftdt/proc2 get exec end end end}def end tempFT}ifelse +3 index exch +definefont exch pop}def +/mb{dup dhBdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{boldproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/mo{dup dhOdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{outproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/ms{dup dhSdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{ShadChar} exch +mkFT pop pop pop exch pop +}ifelse }def +/outproc{ft1 setfont +gsave chrst sfWidth grestore +setcharwidth dblsh}def +/boldproc{ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +ft1 /PaintType get 2 ne +{0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} +{0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def +/dblsh{currentcolor currentcolorspace 1.0 setgray +chrst 0 0 moveto show setcolorspace setcolor +ft2 0 get setfont chrst 0 0 moveto show}def +/ShadChar{ +ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +chrst .06 0 moveto show +0 .05 translate dblsh}def +/ts{show}def +/ta{/_letterSpace xdf/_spaceXtra xdf}def +/tf{/_textWidth xdf +/_numSpaces xdf +/_stringLength xdf +dup stringwidth pop +_letterSpace _stringLength 1 sub mul add +_spaceXtra _numSpaces mul add +_textWidth exch sub /wdif xdf +_spaceXtra 0 32 +wdif _stringLength div +_letterSpace add +0 6 -1 roll awidthshow}def +/pb{/_PageSave save def/showpage{}def}def +/pe{_PageSave restore}def +/pm{_pm currentmatrix pop}def +/_pm matrix def +/ps{showpage _pm setmatrix}def +level2{/ef {gsave 6 array astore concat dup execform grestore}def} +{/ef{gsave 6 array astore concat newpath dup begin Matrix concat +BBox dup dup 0 get exch 1 get moveto +dup dup 0 get exch 3 get lineto +dup dup 2 get exch 3 get lineto +dup 2 get exch 1 get lineto closepath +clip newpath PaintProc end grestore}def}ifelse +/bc{save 11 1 roll 10 dict begin /BBox 5 1 roll 4 array astore def +/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def +/ec{pop restore}def +/printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def +/setoverprint where {pop}{/setoverprint{pop}def}ifelse +/sstr { /$sstrs exch def /$sstr exch def /$sstry exch def /$sstrx exch def /Courier findfont $sstrs +scalefont setfont $sstrx $sstry mt stm 1 -1 scale $sstr show rtm } def +/DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc + eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} + {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def +/DSIEndPaintProc{currentdict end}def +/DSIBeginPaintSubProc {save 11 1 roll 10 dict dup /pD exch def begin + /BBox 5 1 roll 4 array astore def /Matrix 7 1 roll 6 array astore def + /FormType 1 def /PaintProc { /pp get exec } def /pp} def +/DSIEndPaintSubProc {bind def end pD} def +/iaa {/ix ix ixn add def} def +/ias2 {/igap exch def /_st0 ia ix igap add get def _st0} def +/ias {/st0 ia ix get def iaa st0} def +/i85s {0 ias2 /ASCII85Decode filter st0 readstring pop iaa} def +/irles {0 ias2 /ASCII85Decode filter /RunLengthDecode filter st0 readstring pop iaa} def +initialize +end +%%EndResource +CVDict begin 0 id 1 fd +_encodArr 0 +[ +39/quotesingle 96/grave +128/bullet/bullet/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/bullet/bullet/bullet +/bullet/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark +/scaron +/guilsinglright/oe/bullet/bullet/Ydieresis +/bullet/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron +/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown +/Agrave/Aacute/Acircumflex/Atilde/Adieresis +/Aring +/AE/Ccedilla/Egrave/Eacute/Egrave/Edieresis/Igrave/Iacute/Icircumflex/Idieresis +/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls +/agrave/aacute/acircumflex/atilde/adieresis +/aring +/ae/ccedilla/egrave/eacute/ecircumflex/edieresis +/igrave +/iacute +/icircumflex +/idieresis +/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis +] put +_encodArr 1 +[ +39/quotesingle 96/grave +128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash +/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation +/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash +/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft +/guillemotright/ellipsis/blank/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge +/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex +/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +] put +end +%%EndProlog +%%IncludeFont: CourierNewPSMT +%%IncludeFont: TimesNewRomanPSMT +CVDict begin +/___FN2 /Courier dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { 0 }{-1} ifelse end} {pop -1} ifelse rf +/___FN3 /Times-Roman dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { 0 }{-1} ifelse end} {pop -1} ifelse rf +end +0.25 0.25 translate +%%BeginSetup +CVDict begin + +0 id +1 fd +0 263 translate +300 -300 scli +%%EndSetup +%%BeginPageSetup +pm +%%EndPageSetup +/DeviceRGB scs +0.0 0.0 0.0 1. f4 +0.1250 0.1250 sp + 0 lc + 2 lj +T sa +%page rect: 35 270 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Right = high) 25 6 749.98 tf +T sa +rtm +rtm +terminate + +%%EndDocument + PE 10 /Times-Roman AF +19769 66124 MT +(Figure 2.Example of Argument Ordering Dependency)SH +8200 67915 MT +(Figure 2 shows an extreme case of how the ordering of the arguments can affect the size) +45 W( of the graph denoting a)46 W +7200 69292 MT +(function. The) +394 W( functions)72 W +/Times-Italic SF +17348 XM +(x)SH +8 /Times-Roman AF +17792 69637 MT +(1)SH +10 /Symbol AF +18192 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +18886 69637 MT +(2)SH +10 SS +19536 69292 MT +(+)SH +/Times-Italic SF +20350 XM +(x)SH +8 /Times-Roman AF +20794 69637 MT +(3)SH +10 /Symbol AF +21194 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +21888 69637 MT +(4)SH +10 SS +22538 69292 MT +(+)SH +/Times-Italic SF +23352 XM +(x)SH +8 /Times-Roman AF +23796 69637 MT +(5)SH +10 /Symbol AF +24196 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +24890 69637 MT +(6)SH +10 SS +25612 69292 MT +(and)SH +/Times-Italic SF +27378 XM +(x)SH +8 /Times-Roman AF +27822 69637 MT +(1)SH +10 /Symbol AF +28222 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +28916 69637 MT +(4)SH +10 SS +29566 69292 MT +(+)SH +/Times-Italic SF +30380 XM +(x)SH +8 /Times-Roman AF +30824 69637 MT +(2)SH +10 /Symbol AF +31224 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +31918 69637 MT +(5)SH +10 SS +32568 69292 MT +(+)SH +/Times-Italic SF +33382 XM +(x)SH +8 /Times-Roman AF +33826 69637 MT +(3)SH +10 /Symbol AF +34226 69292 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +34920 69637 MT +(6)SH +10 SS +35642 69292 MT +(differ from each other only) +72 W( by a permutation)71 W +7200 70669 MT +(of their arguments, yet one is denoted by a function graph with 8 vertices while the other requires) +155 W( 16 vertices.)156 W +ES +%%Page: 8 8 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(8)SH +7200 7886 MT +(Generalizing this to) +66 W( functions of 2)65 W +/Times-Italic SF +(n)SH +/Times-Roman SF +22035 XM +(arguments, the function)65 W +/Times-Italic SF +31951 XM +(x)SH +8 /Times-Roman AF +32395 8231 MT +(1)SH +10 /Symbol AF +32795 7886 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +33489 8231 MT +(2)SH +10 SS +34139 7886 MT +(+)SH +/Symbol SF +35203 XM +(\327 \327 \327)SH +/Times-Roman SF +36953 XM +(+)SH +/Times-Italic SF +37767 XM +(x)SH +8 /Times-Roman AF +38211 8231 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Symbol AF +39850 7886 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +40544 8231 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +41659 7886 MT +(is denoted by a graph of 2)65 W +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +7200 9300 MT +(vertices, while the function)144 W +/Times-Italic SF +18913 XM +(x)SH +8 /Times-Roman AF +19357 9645 MT +(1)SH +10 /Symbol AF +19757 9300 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +20451 9645 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +21952 9300 MT +(+)SH +/Symbol SF +23016 XM +(\327 \327 \327)SH +/Times-Roman SF +24766 XM +(+)SH +/Times-Italic SF +25580 XM +(x)SH +8 SS +26024 9645 MT +(n)SH +10 /Symbol AF +26424 9300 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +27118 9645 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +28312 9300 MT +(requires 2)144 W +8 /Times-Italic AF +32427 8955 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +34072 9300 MT +(vertices. Consequently,) +539 W( a poor initial choice of)145 W +7200 10677 MT +(input ordering can have very undesirable effects.)SH +8200 13156 MT +(Upon closer examination of) +36 W( these two graphs, we can gain a better intuition of how this problem arises. Imagine)35 W +7200 14533 MT +(a bit-serial processor that computes a Boolean function by examining the arguments)153 W +/Times-Italic SF +42920 XM +(x)SH +8 /Times-Roman AF +43364 14878 MT +(1)SH +10 SS +43764 14533 MT +(,)SH +/Times-Italic SF +44417 XM +(x)SH +8 /Times-Roman AF +44861 14878 MT +(2)SH +10 SS +45261 14533 MT +(, and so) +153 W( on in order,)154 W +7200 15910 MT +(producing output 0 or 1 after the last) +63 W( bit has been read. Such a processor requires internal storage to store enough)62 W +7200 17287 MT +(information about the arguments it has already seen to correctly deduce the value) +32 W( of the function from the values of)33 W +7200 18664 MT +(the remaining arguments. Some functions require little intermediate information. For example, to compute) +150 W( the)149 W +7200 20041 MT +(parity function) +81 W( a bit-serial processor need only store the parity of the arguments it has already seen. Similarly, to)82 W +7200 21418 MT +(compute the function)87 W +/Times-Italic SF +16210 XM +(x)SH +8 /Times-Roman AF +16654 21763 MT +(1)SH +10 /Symbol AF +17054 21418 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +17748 21763 MT +(2)SH +10 SS +18398 21418 MT +(+)SH +/Symbol SF +19462 XM +(\327 \327 \327)SH +/Times-Roman SF +21212 XM +(+)SH +/Times-Italic SF +22026 XM +(x)SH +8 /Times-Roman AF +22470 21763 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Symbol AF +24109 21418 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +24803 21763 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +25603 21418 MT +(, the processor need only store whether any of the preceding pairs) +87 W( of)86 W +7200 22795 MT +(arguments were both 1, and perhaps the value of the previous argument. On the other) +11 W( hand, to compute the function)12 W +/Times-Italic SF +7200 24172 MT +(x)SH +8 /Times-Roman AF +7644 24517 MT +(1)SH +10 /Symbol AF +8044 24172 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +8738 24517 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +10239 24172 MT +(+)SH +/Symbol SF +11303 XM +(\327 \327 \327)SH +/Times-Roman SF +13053 XM +(+)SH +/Times-Italic SF +13867 XM +(x)SH +8 SS +14311 24517 MT +(n)SH +10 /Symbol AF +14711 24172 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +15405 24517 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +16205 24172 MT +(, we would need to store the first)93 W +/Times-Italic SF +30308 XM +(n)SH +/Times-Roman SF +31151 XM +(arguments to correctly deduce the value of the) +93 W( function)92 W +7200 25549 MT +(from the remaining arguments. A function graph can be thought of as such a processor, with) +118 W( the set of vertices)119 W +7200 26926 MT +(having index)84 W +/Times-Italic SF +12812 XM +(i)SH +/Times-Roman SF +13424 XM +(describing the processing of) +84 W( argument)83 W +/Times-Italic SF +29367 XM +(x)SH +8 SS +29811 27271 MT +(i)SH +10 /Times-Roman AF +30033 26926 MT +(. Rather) +416 W( than storing intermediate information as bits in a)83 W +7200 28303 MT +(memory, however, this information is encoded in the set of possible branch destinations. That is,) +97 W( if the bit-serial)98 W +7200 29680 MT +(processor requires)43 W +/Times-Italic SF +14839 XM +(b)SH +/Times-Roman SF +15632 XM +(bits to encode information about the first)43 W +/Times-Italic SF +32514 XM +(i)SH +/Times-Roman SF +33084 XM +(arguments, then in any graph for this function there)42 W +7200 31057 MT +(must be at least 2)110 W +8 /Times-Italic AF +14584 30712 MT +(b)SH +10 /Times-Roman AF +15344 31057 MT +(vertices that are) +110 W( either terminal or are nonterminal with index greater than)111 W +/Times-Italic SF +46500 XM +(i)SH +/Times-Roman SF +47139 XM +(having incoming)111 W +7200 32434 MT +(branches from vertices with index less than or equal to)29 W +/Times-Italic SF +29597 XM +(i)SH +/Times-Roman SF +(. For) +308 W( example, the function)29 W +/Times-Italic SF +41381 XM +(x)SH +8 /Times-Roman AF +41825 32779 MT +(1)SH +10 /Symbol AF +42225 32434 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +42919 32779 MT +(4)SH +10 SS +43569 32434 MT +(+)SH +/Times-Italic SF +44383 XM +(x)SH +8 /Times-Roman AF +44827 32779 MT +(2)SH +10 /Symbol AF +45227 32434 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +45921 32779 MT +(5)SH +10 SS +46571 32434 MT +(+)SH +/Times-Italic SF +47385 XM +(x)SH +8 /Times-Roman AF +47829 32779 MT +(3)SH +10 /Symbol AF +48229 32434 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +48923 32779 MT +(6)SH +10 SS +49601 32434 MT +(requires 2)28 W +8 SS +53600 32089 MT +(3)SH +10 SS +7200 33811 MT +(branches between vertices) +128 W( with index less than or equal to 3 to vertices which are either terminal or have index)129 W +7200 35188 MT +(greater than 3. In fact, the first 3 levels of this) +119 W( graph must form a complete binary tree to obtain this degree of)118 W +7200 36565 MT +(branching. In) +466 W( the generalization of this function, the first)108 W +/Times-Italic SF +31443 XM +(n)SH +/Times-Roman SF +32302 XM +(levels of the graph form a complete binary tree, and)109 W +7200 37942 MT +(hence the number of vertices grows exponentially with the number of arguments.)SH +8200 40421 MT +(To view this from a different perspective, consider the family of functions:)SH +/Times-Italic SF +9200 42212 MT +(f)SH +8 SS +9478 42557 MT +(b)SH +/Times-Roman SF +9878 42902 MT +(1)SH +10278 42557 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +12478 42902 MT +(n)SH +10 /Times-Roman AF +12878 42212 MT +(\050)SH +/Times-Italic SF +(x)SH +8 SS +13655 42557 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +14906 42212 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +17600 42557 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +18400 42212 MT +(\051 =)436 W +/Times-Italic SF +20669 XM +(b)SH +8 /Times-Roman AF +21169 42557 MT +(1)SH +10 /Symbol AF +21569 42212 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +22263 42557 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +23764 42212 MT +(+)SH +/Symbol SF +24828 XM +(\327 \327 \327)SH +/Times-Roman SF +26578 XM +(+)SH +/Times-Italic SF +27392 XM +(b)SH +8 SS +27892 42557 MT +(n)SH +10 /Symbol AF +28292 42212 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +28986 42557 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +29786 42212 MT +(.)SH +7200 43971 MT +(For all 2)26 W +8 /Times-Italic AF +10641 43626 MT +(n)SH +10 /Times-Roman AF +11317 43971 MT +(possible combinations of the values)26 W +/Times-Italic SF +25974 XM +(b)SH +8 /Times-Roman AF +26474 44316 MT +(1)SH +10 SS +26874 43971 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +29624 44316 MT +(n)SH +10 /Times-Roman AF +30024 43971 MT +(, each of these functions is distinct, and hence they must) +26 W( be)25 W +7200 45348 MT +(represented by distinct subgraphs in the graph of the function)SH +/Times-Italic SF +31974 XM +(x)SH +8 /Times-Roman AF +32418 45693 MT +(1)SH +10 /Symbol AF +32818 45348 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +33512 45693 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +35013 45348 MT +(+)SH +/Symbol SF +36077 XM +(\327 \327 \327)SH +/Times-Roman SF +37827 XM +(+)SH +/Times-Italic SF +38641 XM +(x)SH +8 SS +39085 45693 MT +(n)SH +10 /Symbol AF +39485 45348 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +40179 45693 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +40979 45348 MT +(.)SH +8200 47827 MT +(To use our algorithms on anything other than small problems \050e.g. functions of 16 variables or more\051, a user must)19 W +7200 49204 MT +(have an intuition about why certain functions have large function graphs, and how the choice of input ordering may)31 W +7200 50581 MT +(affect this size. In Section 5 we will present examples of how the structure of the problem) +18 W( to be solved can often be)19 W +7200 51958 MT +(exploited to obtain a suitable input ordering.)SH +11 /Times-Bold AF +7200 55575 MT +(3.3. Inherently Complex Functions)SH +10 /Times-Roman AF +8200 56952 MT +(Some functions cannot) +281 W( be represented efficiently with our representation regardless of the input ordering.)280 W +7200 58329 MT +(Unfortunately, the functions representing the output bits of an) +19 W( integer multiplier fall within this class. The appendix)20 W +7200 59706 MT +(contains a proof that for any) +187 W( ordering of the inputs)186 W +/Times-Italic SF +29506 XM +(a)SH +8 /Times-Roman AF +30006 60051 MT +(1)SH +10 SS +30406 59706 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +33156 60051 MT +(n)SH +10 /Times-Roman AF +33992 59706 MT +(and)SH +/Times-Italic SF +35872 XM +(b)SH +8 /Times-Roman AF +36372 60051 MT +(1)SH +10 SS +36772 59706 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +39522 60051 MT +(n)SH +10 /Times-Roman AF +39922 59706 MT +(, at least one of the 2)186 W +/Times-Italic SF +(n)SH +/Times-Roman SF +50278 XM +(functions)SH +7200 61120 MT +(representing the integer product)23 W +/Times-Italic SF +20289 XM +(a)SH +/Symbol SF +20939 XM +(\327)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +21962 XM +(requires a graph containing at least 2)23 W +8 /Times-Italic AF +36819 60775 MT +(n)SH +/Times-Roman SF +(/8)SH +10 SS +38114 61120 MT +(vertices. While) +296 W( this) +23 W( lower bound is not)24 W +7200 62497 MT +(very large for word sizes encountered in) +270 W( practice \050e.g. it equals 256 for)269 W +/Times-Italic SF +39278 XM +(n)SH +/Times-Roman SF +(=64\051, it indicates the exponential)269 W +7200 63874 MT +(complexity of these functions. Furthermore, we suspect the true bound is far worse.)SH +8200 66353 MT +(Empirically, we have found that for word sizes)140 W +/Times-Italic SF +28344 XM +(n)SH +/Times-Roman SF +29234 XM +(less than or equal to 8, the) +140 W( output functions of a multiplier)141 W +7200 67730 MT +(require no more) +100 W( than 5000 vertices for a variety of different input orderings. However, for)99 W +/Times-Italic SF +45332 XM +(n)SH +/Times-Roman SF +46132 XM +(> 10,) +50 W( some outputs)99 W +7200 69107 MT +(require graphs with more than 100,000 vertices and hence become impractical.)SH +8200 71586 MT +(Given the wide variety of techniques used in implementing) +70 W( multipliers \050e.g.) +71 W( [12]\051,) +SH( a canonical form for Boolean)71 W +ES +%%Page: 9 9 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30350 4286 MT +(9)SH +7200 7886 MT +(functions \050along with a set of manipulation algorithms\051 that could efficiently represent multiplication) +89 W( would be of)88 W +7200 9263 MT +(great interest for circuit verification. Unfortunately, these functions seem especially intractable.)SH +12 /Times-Bold AF +7200 12947 MT +(4. Operations)SH +10 /Times-Roman AF +8200 14324 MT +(We view a symbolic manipulation) +15 W( program as executing a sequence of commands that build up representations of)16 W +7200 15701 MT +(functions and determine various properties about them. For example, suppose) +367 W( we wish to construct the)366 W +7200 17078 MT +(representation of) +64 W( the function computed by a combinational logic gate network. Starting from graphs representing)65 W +7200 18455 MT +(the input) +37 W( variables, we proceed through the network, constructing the function computed at the output of each logic)36 W +7200 19832 MT +(gate by applying the gate operator) +36 W( to the functions at the gate inputs. In this process, we can take advantage of any)37 W +7200 21209 MT +(repeated structure by first) +145 W( constructing the functions representing the individual subcircuits \050in terms of a set of)144 W +7200 22586 MT +(auxiliary variables\051 and then composing the subcircuit functions to obtain the complete network functions.) +198 W( A)648 W +7200 23963 MT +(similar procedure is followed to construct the representation of the function denoted) +80 W( by some Boolean expression.)79 W +7200 25340 MT +(At this point we can test various properties of the function, such as whether it equals)241 W +/Times-Bold SF +45080 XM +(0)SH +/Times-Roman SF +46072 XM +(\050satisfiability\051 or)242 W +/Times-Bold SF +53500 XM +(1)SH +/Times-Roman SF +7200 26717 MT +(\050tautology\051, or whether it equals the function denoted by some other expression \050equivalence\051. We can) +39 W( also ask for)38 W +7200 28094 MT +(information about) +184 W( the function's satisfying set, such as to list some member, to list all members, to test some)185 W +7200 29471 MT +(element for membership, etc.)SH +/Times-Bold SF +14400 31246 MT +(Procedure)SH +31039 XM +(Result)SH +41400 XM +(Time Complexity)SH +/Times-Italic SF +14400 32623 MT +(Reduce)SH +23400 XM +(G)SH +/Times-Roman SF +24372 XM +(reduced to canonical form)SH +41400 XM +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +43677 XM +(|\327)SH +/Times-Roman SF +(log)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +46477 XM +(|)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 34000 MT +(Apply)SH +23400 XM +(f)SH +8 /Times-Roman AF +23678 34345 MT +(1)SH +10 SS +24328 34000 MT +()SH +/Times-Italic SF +26706 XM +(f)SH +8 /Times-Roman AF +26984 34345 MT +(2)SH +10 SS +41400 34000 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +43527 34345 MT +(1)SH +10 /Symbol AF +43927 34000 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +45299 34345 MT +(2)SH +10 /Symbol AF +45699 34000 MT +(|)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 35377 MT +(Restrict)SH +23400 XM +(f)SH +/Symbol SF +23928 XM +(|)SH +8 /Times-Italic AF +24128 35722 MT +(x)SH +24483 36067 MT +(i)SH +/Times-Roman SF +24705 35722 MT +(=)SH +/Times-Italic SF +(b)SH +10 /Times-Roman AF +41400 35377 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +43677 XM +(|\327)SH +/Times-Roman SF +(log)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +46477 XM +(|)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 37136 MT +(Compose)SH +23400 XM +(f)SH +8 /Times-Roman AF +23678 37481 MT +(1)SH +10 /Symbol AF +24228 37136 MT +(|)SH +8 /Times-Italic AF +24428 37481 MT +(x)SH +24783 37826 MT +(i)SH +/Times-Roman SF +25005 37481 MT +(=)SH +/Times-Italic SF +(f)SH +/Times-Roman SF +25678 37826 MT +(2)SH +10 SS +41400 37136 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +43527 37481 MT +(1)SH +10 /Symbol AF +43927 37136 MT +(|)SH +8 /Times-Roman AF +44127 36791 MT +(2)SH +10 /Symbol AF +44527 37136 MT +(\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +45699 37481 MT +(2)SH +10 /Symbol AF +46099 37136 MT +(|)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 38687 MT +(Satisfy-one)SH +/Times-Roman SF +23400 XM +(some element of)SH +/Times-Italic SF +30260 XM +(S)SH +8 SS +30760 39032 MT +(f)SH +10 /Times-Roman AF +41400 38687 MT +(O) +150 MX(\050)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 40064 MT +(Satisfy-all)SH +23400 XM +(S)SH +8 SS +23900 40409 MT +(f)SH +10 /Times-Roman AF +41400 40064 MT +(O) +150 MX(\050)SH +/Times-Italic SF +(n)SH +/Symbol SF +(\327|)SH +/Times-Italic SF +(S)SH +8 SS +44055 40409 MT +(f)SH +10 /Symbol AF +44427 40064 MT +(|)SH +/Times-Roman SF +(\051)SH +/Times-Italic SF +14400 41441 MT +(Satisfy-count)SH +/Symbol SF +23400 XM +(|)SH +/Times-Italic SF +(S)SH +8 SS +24100 41786 MT +(f)SH +10 /Symbol AF +24472 41441 MT +(|)SH +/Times-Roman SF +41400 XM +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +43677 XM +(|)SH +/Times-Roman SF +(\051)SH +22920 43051 MT +(Table 1.Summary of Basic Operations)SH +8200 45530 MT +(In this section we will) +223 W( present algorithms to perform basic operations on Boolean functions represented as)222 W +7200 46907 MT +(function graphs as summarized in Table 1.)31 W +8 SS +24411 46562 MT +(4)SH +10 SS +25372 46907 MT +(These few basic operations can be combined to perform a wide variety)30 W +7200 48284 MT +(of operations on Boolean functions. In the table, the) +173 W( function)174 W +/Times-Italic SF +33963 XM +(f)SH +/Times-Roman SF +34815 XM +(is represented by a reduced function graph)174 W +/Times-Italic SF +53278 XM +(G)SH +/Times-Roman SF +7200 49661 MT +(containing)SH +/Symbol SF +11720 XM +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +12792 XM +(|)SH +/Times-Roman SF +13290 XM +(vertices, and similarly for the functions)48 W +/Times-Italic SF +29548 XM +(f)SH +8 /Times-Roman AF +29826 50006 MT +(1)SH +10 SS +30524 49661 MT +(and)SH +/Times-Italic SF +32266 XM +(f)SH +8 /Times-Roman AF +32544 50006 MT +(2)SH +10 SS +32944 49661 MT +(. Our) +346 W( algorithms utilize techniques commonly) +48 W( used)47 W +7200 51038 MT +(in graph algorithms such as ordered) +71 W( traversal, table look-up and vertex encoding. As the table shows, most of the)72 W +7200 52415 MT +(algorithms have time complexity proportional) +98 W( to the size of the graphs being manipulated. Hence, as long as the)97 W +7200 53792 MT +(functions of interest can be represented by reasonably small graphs, our algorithms are quite efficient.)SH +11 /Times-Bold AF +7200 57409 MT +(4.1. Data Structures)SH +10 /Times-Roman AF +8200 58786 MT +(We will express our algorithms in a pseudo-Pascal notation. Each vertex in a) +51 W( function graph is represented by a)52 W +7200 60163 MT +(record declared as follows:)SH +10800 50 7200 65587 UL +6 SS +8000 66926 MT +(4)SH +8 /Times-Bold AF +8300 67235 MT +(Update:)SH +/Times-Roman SF +11605 XM +(Some of the time complexity entries in this table are not strictly correct. An extra factor) +75 W( of \050log)76 W +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Times-Roman SF +44272 67580 MT +(1)SH +/Symbol SF +44822 67235 MT +(|)SH +/Times-Roman SF +(+log)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Times-Roman SF +47193 67580 MT +(2)SH +/Symbol SF +47593 67235 MT +(|)SH +/Times-Roman SF +(\051 should have been)76 W +7200 68304 MT +(included in the time complexity of)20 W +/Times-Italic SF +18558 XM +(Apply)SH +/Times-Roman SF +20644 XM +(and)SH +/Times-Italic SF +22019 XM +(Compose)SH +/Times-Roman SF +25217 XM +(to account for the complexity of reducing) +20 W( the resulting graph. In later research, Wegener)19 W +7200 69228 MT +(and Sieling showed how to perform BDD) +34 W( reduction in linear time \050)35 W +/Times-Italic SF +(Information Processing Letters 48)35 W +/Times-Roman SF +(, pp. 139-144, 1993.\051 Consequently, all of)35 W +7200 70152 MT +(the log factors can be dropped from the table. In practice, most BDD implementations use hash tables rather than) +7 W( the sorting method described in)6 W +7200 71076 MT +(this paper. Assuming retrieving an element from a hash table takes constant time \050a reasonable) +98 W( assumption for a good hash function\051, these)99 W +7200 72000 MT +(implementations also perform BDD reduction in linear time.)SH +ES +%%Page: 10 10 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(10)SH +9200 7886 MT +(type vertex = record)SH +11718 8991 MT +(low, high: vertex;)SH +11718 10096 MT +(index: 1..n+1;)SH +11718 11201 MT +(val: \0500,1,X\051;)SH +11718 12306 MT +(id: integer;)SH +11718 13411 MT +(mark: boolean;)SH +9200 14516 MT +(end;)SH +7200 16036 MT +(Both nonterminal and terminal vertices are represented by the) +12 W( same type of record, but the field values for a vertex)11 W +/Times-Italic SF +53556 XM +(v)SH +/Times-Roman SF +7200 17413 MT +(depend on the vertex type as given in the following table.)SH +/Times-Bold SF +20516 19007 MT +(Field)SH +26800 XM +(Terminal)SH +40478 XM +(Nonterminal)SH +/Times-Roman SF +20850 20112 MT +(low)SH +28022 XM +(null)SH +/Times-Italic SF +41922 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +20711 21217 MT +(high)SH +28022 XM +(null)SH +/Times-Italic SF +41756 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +20489 22322 MT +(index)SH +/Times-Italic SF +28018 XM +(n)SH +/Times-Roman SF +(+1)SH +/Times-Italic SF +41562 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +20989 23427 MT +(val)SH +/Times-Italic SF +27162 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +42839 XM +(X)SH +8200 25906 MT +(The id and mark fields contain auxiliary information used by the algorithms. The id) +151 W( field contains a integer)152 W +7200 27283 MT +(identifier which is unique to that vertex in the graph. It does not) +60 W( matter how the identifiers are ordered among the)59 W +7200 28660 MT +(vertices, only that they range from 1 up to the number of vertices and that they all be different.) +70 W( The) +391 W( mark field is)71 W +7200 30037 MT +(used to mark which vertices have) +96 W( been visited during a traversal of the graph. The procedure)95 W +/Times-Italic SF +46366 XM +(Traverse)SH +/Times-Roman SF +50266 XM +(shown in)95 W +7200 31414 MT +(Figure 3 illustrates) +81 W( a general method used by many of our algorithms for traversing a graph and performing some)82 W +7200 32791 MT +(operation on the vertices. This procedure is called at the top level with the) +97 W( root vertex as argument and with the)96 W +7200 34168 MT +(mark fields of the vertices being either all true or all false. It) +30 W( then systematically visits every vertex in the graph by)31 W +7200 35545 MT +(recursively visiting the subgraphs rooted by the two children. As it) +49 W( visits a vertex, it complements the value of the)48 W +7200 36922 MT +(mark field, so that it can later determine whether a child has already been visited by comparing the two marks. As) +14 W( a)15 W +7200 38299 MT +(vertex is visited, we could perform some operation such as to increment a counter and then set the id field to the)94 W +7200 39676 MT +(value of the counter \050thereby assigning a unique identifier to each vertex.\051 Each vertex is visited exactly once,) +47 W( and)48 W +7200 41053 MT +(assuming the operation at each vertex requires constant time, the complexity of the algorithm) +236 W( is O) +150 235 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +51510 XM +(|)SH +/Times-Roman SF +(\051, i.e.)235 W +7200 42430 MT +(proportional to the number of vertices in) +93 W( the graph. Upon termination, the vertices again all have the same mark)94 W +7200 43807 MT +(value.)SH +9200 45598 MT +(procedure)SH +/Times-Italic SF +13448 XM +(Traverse)SH +/Times-Roman SF +(\050v:vertex\051;)SH +9200 46703 MT +(begin)SH +11718 47808 MT +(v.mark := not v.mark;)SH +11718 48913 MT +(... do something to v ...)SH +11718 50018 MT +(if v.index)SH +/Symbol SF +15951 XM +(\243)SH +/Times-Roman SF +16900 XM +(n)SH +11718 51123 MT +(then begin {v nonterminal})SH +14236 52228 MT +(if v.mark)SH +/Symbol SF +18302 XM +(\271)SH +/Times-Roman SF +19251 XM +(v.low.mark then)SH +/Times-Italic SF +26028 XM +(Traverse)SH +/Times-Roman SF +(\050v.low\051;)SH +14236 53333 MT +(if v.mark)SH +/Symbol SF +18302 XM +(\271)SH +/Times-Roman SF +19251 XM +(v.high.mark then)SH +/Times-Italic SF +26306 XM +(Traverse)SH +/Times-Roman SF +(\050v.high\051;)SH +11718 54438 MT +(end;)SH +9200 55543 MT +(end;)SH +21310 57153 MT +(Figure 3.Implementation of Ordered Traversal)SH +11 /Times-Bold AF +7200 60770 MT +(4.2. Reduction)SH +10 /Times-Roman AF +8200 62147 MT +(The reduction algorithm transforms) +51 W( an arbitrary function graph into a reduced graph denoting the same function.)50 W +7200 63524 MT +(It closely follows an algorithm presented in Example 3.2 of Aho,) +16 W( Hopcroft, and Ullman) +17 W( [13]) +SH( for testing whether two)17 W +7200 64901 MT +(trees are isomorphic.) +39 W( Proceeding) +326 W( from the terminal vertices up to the root, a unique integer identifier is assigned to)38 W +7200 66278 MT +(each unique subgraph root. That is, for each vertex)46 W +/Times-Italic SF +28488 XM +(v)SH +/Times-Roman SF +29228 XM +(it assigns) +46 W( a label)47 W +/Times-Italic SF +36248 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 such that for any two vertices)47 W +/Times-Italic SF +50768 XM +(u)SH +/Times-Roman SF +51565 XM +(and)SH +/Times-Italic SF +53306 XM +(v)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +7200 67655 MT +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +10208 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 if and only if)33 W +/Times-Italic SF +17955 XM +(f)SH +8 SS +18233 68000 MT +(u)SH +10 /Times-Roman AF +18633 67655 MT +(=)SH +/Times-Italic SF +(f)SH +8 SS +19475 68000 MT +(v)SH +10 /Times-Roman AF +20113 67655 MT +(\050in the) +33 W( terminology of Definition 2.\051 Given this labeling, the algorithm constructs a)32 W +7200 69032 MT +(graph with one vertex for each unique label.)SH +8200 71511 MT +(By working from the terminal vertices up to the root, a procedure) +7 W( can label the vertices by the following inductive)8 W +ES +%%Page: 11 11 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(11)SH +7200 7886 MT +(method. First,) +378 W( two terminal vertices should have the same label if and only if they have the same) +64 W( value attributes.)63 W +7200 9263 MT +(Now assume all terminal) +71 W( vertices and all nonterminal vertices with index greater than)72 W +/Times-Italic SF +42599 XM +(i)SH +/Times-Roman SF +43199 XM +(have been labeled. As we)72 W +7200 10640 MT +(proceed with the labeling of vertices with index)41 W +/Times-Italic SF +26858 XM +(i)SH +/Times-Roman SF +(, a vertex)41 W +/Times-Italic SF +31202 XM +(v)SH +/Times-Roman SF +31937 XM +(should have)40 W +/Times-Italic SF +37072 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 equal to that of some vertex that has)40 W +7200 12017 MT +(already been labeled if and only if one of two conditions is satisfied. First, if)36 W +/Times-Italic SF +38829 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 =)SH +/Times-Italic SF +43892 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051, then vertex)37 W +/Times-Italic SF +53556 XM +(v)SH +/Times-Roman SF +7200 13394 MT +(is redundant, and we should set)9 W +/Times-Italic SF +20058 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +23010 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051. Second,) +268 W( if there is some labeled vertex)8 W +/Times-Italic SF +43552 XM +(u)SH +/Times-Roman SF +44310 XM +(with)SH +/Times-Italic SF +46346 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +50742 XM +(i)SH +/Times-Roman SF +51278 XM +(having)SH +/Times-Italic SF +7200 14771 MT +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 =)SH +/Times-Italic SF +12263 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051\051, and)84 W +/Times-Italic SF +18680 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 =)SH +/Times-Italic SF +24076 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051\051, then the reduced subgraphs) +84 W( rooted by these two vertices will)85 W +7200 16148 MT +(be isomorphic, and we should set)SH +/Times-Italic SF +20782 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +23734 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051.)SH +8200 18627 MT +(A sketch of the code is shown in Figure 4. First, the vertices) +89 W( are collected into lists according to their indices.)88 W +7200 20004 MT +(This can be done by a procedure similar to)45 W +/Times-Italic SF +24907 XM +(Traverse)SH +/Times-Roman SF +(, where) +45 W( as a vertex is visited, it is added to the appropriate list.)46 W +7200 21381 MT +(Then we process these lists working from the one containing the terminal vertices up to) +31 W( the one containing the root.)30 W +7200 22758 MT +(For each vertex on a list we create a key of the form \050)13 W +/Times-Italic SF +(value)SH +/Times-Roman SF +(\051 for a terminal vertex or of the form \050)13 W +/Times-Italic SF +(lowid)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +48975 XM +(highid)SH +/Times-Roman SF +(\051 for a)13 W +7200 24135 MT +(nonterminal vertex, where)74 W +/Times-Italic SF +18197 XM +(lowid)SH +/Times-Roman SF +20670 XM +(=)SH +/Times-Italic SF +21484 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051 and)74 W +/Times-Italic SF +27575 XM +(highid)SH +/Times-Roman SF +30381 XM +(=)SH +/Times-Italic SF +31195 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051. If) +398 W( a vertex has)74 W +/Times-Italic SF +42662 XM +(lowid)SH +/Times-Roman SF +45135 XM +(=)SH +/Times-Italic SF +45949 XM +(highid)SH +/Times-Roman SF +(, then we can)73 W +7200 25512 MT +(immediately set)185 W +/Times-Italic SF +14181 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +17133 XM +(lowid)SH +/Times-Roman SF +(. The) +620 W( remaining vertices are sorted according to) +185 W( their keys. Aho, Hopcroft, and)186 W +7200 26889 MT +(Ullman describe) +19 W( a linear-time lexicographic sorting method for this based on bucket sorting. We then work through)18 W +7200 28266 MT +(this sorted list, assigning a given label to all vertices having the same key. We also select one vertex record for each)9 W +7200 29643 MT +(unique label and store a pointer to this vertex) +21 W( in an array indexed by the label. These selected vertices will form the)20 W +7200 31020 MT +(final reduced graph. Hence, we can obtain the reduced version of a subgraph with root)110 W +/Times-Italic SF +44006 XM +(v)SH +/Times-Roman SF +44810 XM +(by accessing) +110 W( the array)111 W +7200 32397 MT +(element with index)83 W +/Times-Italic SF +15364 XM +(id)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 We use this method to modify a vertex record so that its two children are vertices in the)82 W +7200 33774 MT +(reduced graph and to return the root of) +75 W( the final reduced graph when the procedure is exited. Note that the labels)76 W +7200 35151 MT +(assigned to the vertices by this) +115 W( routine can serve as unique identifiers for later routines. Assuming a linear-time)114 W +7200 36528 MT +(sorting routine, the processing at each level requires time proportional to the number of vertices at that level. Each)44 W +7200 37905 MT +(level is processed once, and hence the overall complexity of the algorithm is linear in the number of vertices.)SH +9200 39696 MT +(function)SH +/Times-Italic SF +12783 XM +(Reduce)SH +/Times-Roman SF +(\050v: vertex\051: vertex;)SH +11718 40801 MT +(var subgraph: array[1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +(|)SH +/Times-Roman SF +(] of vertex;)SH +11718 41906 MT +(var vlist: array[1..n+1] of list;)SH +9200 43011 MT +(begin)SH +11718 44116 MT +(Put each vertex u on list vlist[u.index])SH +11718 45221 MT +(nextid := 0;)SH +11718 46326 MT +(for i := n+1 downto 1 do)SH +11718 47431 MT +(begin)SH +14236 48536 MT +(Q := empty set;)SH +14236 49641 MT +(for each u in vlist[i] do)SH +16754 50746 MT +(if u.index = n+1)SH +19272 51851 MT +(then add to Q where key = \050u.value\051 {terminal vertex})SH +19272 52956 MT +(else if u.low.id = u.high.id)SH +21790 54061 MT +(then u.id := u.low.id {redundant vertex})SH +21790 55166 MT +(else add to Q where key = \050u.low.id, u.high.id\051;)SH +14236 56271 MT +(Sort elements of Q by keys;)SH +14236 57376 MT +(oldkey := \050)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1;)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1\051; {unmatchable) +1500 W( key})SH +14236 58481 MT +(for each in Q removed in order do)SH +16754 59586 MT +(if key = oldkey)SH +19272 60691 MT +(then u.id:= nextid;) +SH( {matches) +500 W( existing vertex})SH +19272 61796 MT +(else begin {unique vertex})SH +21790 62901 MT +(nextid := nextid + 1; u.id := nextid; subgraph[nextid] := u;)SH +21790 64006 MT +(u.low := subgraph[u.low.id]; u.high := subgraph[u.high.id];)SH +21790 65111 MT +(oldkey := key;)SH +19272 66216 MT +(end;)SH +11718 67321 MT +(end;)SH +11718 68426 MT +(return\050subgraph[v.id]\051;)SH +9200 69531 MT +(end;)SH +23490 71141 MT +(Figure 4.Implementation of)SH +/Times-Italic SF +34767 XM +(Reduce)SH +ES +%%Page: 12 12 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(12)SH +1 SS +15067 7200 22267 PB +%%BeginDocument: figs/fig5.eps +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 341 151 +%%Title: (C:\\Shared Files\\Documents\\BDD Paper\\fig5.CNV) +%%Creator: Canvas (7.0.1) +%%CreationDate: (05/09/00 @ 03:51 PM) +%%DocumentProcSets: Deneba_CanvasCustomColors +%%+ CVDict +%%DocumentFonts: +%%DocumentNeededFonts: CourierNewPSMT + +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%EndComments +%%BeginProlog +%%BeginResource: procset CVDict +%%Title: Deneba standard graphics +%%Version: 1.02 +%%CreationDate: 10/14/99 +%%Copyright: (C) 1994, 1999 Deneba Systems. All Rights Reserved +userdict /CVDict 200 dict put +CVDict begin +/invalidcolortable? true def +/level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def +level2 not { /xshow { +0 currentpoint 5 -1 roll {pop pop moveto dup 1 add 3 1 roll exch dup 4 1 roll exch get 0 rmoveto currentpoint} exch kshow +pop pop pop pop} def } if +/initialize +{CVDict +{dup xcheck{bind}if pop pop}forall +currentscreen /_sp exch def /_sa exch def /_sf exch def +/DefaultScreen{_sf _sa /_sp load}def +/DefaultScreenProc{ /_sp load}def +/setstrokeadjust where{pop true}{false}ifelse +{/sa /setstrokeadjust load def +/li /lineto load def +/mt /moveto load def +/ct /curveto load def +/lh { currentpoint exch pop lineto}bind def +/lv { currentpoint pop exch lineto}bind def} +{/sa{{currentdict StrokeAdjustDict ne{StrokeAdjustDict begin}if}{ currentdict StrokeAdjustDict eq{end}if}ifelse} bind def +userdict /StrokeAdjustDict 100 dict put +StrokeAdjustDict begin +/_xl +{transform +.25 sub round .25 add exch +.25 sub round .25 add exch +itransform +}bind def +/li{ _xl lineto}bind def +/mt{ _xl moveto}bind def +/ct{ _xl curveto}bind def +/lh { dup _xl pop currentpoint exch pop lineto}bind def +/lv { dup _xl pop currentpoint pop exch lineto}bind def +/rp{_xl/cv_b xdf/cv_r xdf +_xl/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath +}bind def +end}ifelse}def +/terminate +{userdict /StrokeAdjustDict known {currentdict StrokeAdjustDict eq{end}if}if +currentdict CVDict eq{end}if}def +/mxMax 64 def +/mxStk [mxMax {matrix} repeat]def +/mxIdx 0 def +/stm{mxStk mxIdx 0 lt {/mxIdx 0 store}if mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def +/rtm{mxIdx 1 ge {/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}if }def +/scl { 72 div exch 72 div exch scale } def +/scli { 72 exch div exch 72 exch div exch scale } def +/_cvmtx matrix def +/_txtmtx matrix def +/_initmtx matrix def +/xdf{exch def}def +/min{2 copy gt{exch}if pop}def +/max{2 copy lt{exch}if pop}def +/CVvec 256 array def +/T true def +/F false def +/_id 0 def +/_encodindex 0 def +/_encodArr 3 array def +/_fd 0 def +/_textSave null def +/_letterSpace 0 def +/_spaceXtra 0 def +/_PageSave null def +/_penh 1.0 def +/_penv 1.0 def +/save0 null def +/_w 0 def +/_h 0 def +/_ds 0 def +/hct false def +/pw 0 def +/wdif 0 def +/temp 0 def +/currot 0 def +/savehandler null def +/ia 0 def +/ix 0 def +/ixn 1 def +/MultipleDataSources false def +/debug +{dup /$eps exch not def{/savehandler errordict /handleerror get def +errordict +/handleerror{$error begin +newerror{/newerror false def +showpage +72 72 scale +/Helvetica findfont .2 scalefont setfont +.25 9.6 moveto +(Error: )show +errorname +{dup type/stringtype ne +{(converted error name will end up in this string)cvs}if +show}exec +.25 9.4 moveto +(Offending Command: )show +/command load +{dup type/stringtype ne +{(converted command name will end up in this string)cvs}if +show}exec +.25 9.2 moveto +(Stack: )show +ostack +{.25 currentpoint .2 sub exch pop moveto +dup type/stringtype ne{dup type/integertype eq +1 index type/realtype eq or +1 index type/nametype eq or +not {type}if +(converted stack will end up in this string)cvs}if +show}forall +showpage +CVDict /savehandler get exec}if +end}put}if}def +/lc{setlinecap}def +/lj{setlinejoin}def +/lm{setmiterlimit}def +/lf{setflat}def +/cp{closepath}def +/np{newpath}def +/$psz 30 array def +/$pix 0 def +/gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def +/gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def +/li{lineto} def +/lh{currentpoint exch pop lineto}def +/lv{currentpoint pop exch lineto}def +/mt{moveto} def +/_cv_mh{currentpoint exch pop moveto}def +/_cv_mv{currentpoint pop exch moveto}def +/ct{curveto} def +/rt{4 -2 roll moveto +dup 0 exch rlineto +exch 0 rlineto +neg 0 exch rlineto +closepath}def +/_hr +{ +{/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} +{/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} +}def +/_cs null def +/fg{setgray}def +/setcmykcolor where {pop true}{false}ifelse +{/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse +/setcolorspace where {pop true}{false}ifelse +{/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def +}{/f3{ setrgbcolor}def/scs{ pop}def}ifelse +/$cc 5 array def +/findcmykcustomcolor where {pop true}{false}ifelse +not {/findcmykcustomcolor{$cc astore}def}if +/setcustomcolor where {pop true}{false}ifelse +not{/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def}if +/fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def +/setseparationgray where {pop true}{false}ifelse +not{/setseparationgray {systemdict /setgray get exec} def}if +/sg{ setseparationgray}def +/sp{/_penv xdf/_penh xdf}def +/rp +{/cv_b xdf/cv_r xdf +/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath}def +/rr{/yradius xdf +/xradius xdf +/cv_b xdf +/cv_r xdf +/cv_t xdf +/cv_l xdf +cv_l xradius add cv_t moveto +cv_r xradius sub cv_t lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +1 0 1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_r cv_b yradius sub lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 1 -1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l xradius add cv_b lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +-1 0 -1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l cv_t yradius add lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 -1 1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +closepath}def +/ep{_cvmtx currentmatrix pop +translate +scale +1 0 moveto +0 0 1 0 360 arc +closepath +_cvmtx setmatrix}def +/wd{_cvmtx currentmatrix pop +translate +scale +0 0 moveto +0 0 1 5 -2 roll arcn +closepath +_cvmtx setmatrix}def +/as{_cvmtx currentmatrix pop +translate +scale +0 0 1 5 -2 roll arcn +_cvmtx setmatrix}def +/ds +{_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def +/df{fill}def +/de{eofill}def +/_gstk 5 array def +/_gstkc 0 def +/_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def +/_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def +/hs{newpath gsave stm}def +/hc{/_hr load exch get exec newpath}def +/he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store +grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq +not{aload pop moveto}{pop}ifelse dup /_cf get setfont +dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def +/id{ /_id xdf}def +/_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def +% Copyright (C) 1990, 1991 by Adobe Systems Incorporated. +% All rights reserved. +level2 { save /dontloadlevel1 xdf }if +/startnoload{{/noload save def} if}def +/endnoload{{noload restore}if }def +/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def +/ncolors 1 def +/colorimage where{pop true}{false}ifelse{/ncolors 0 +statusdict begin +/processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known +{pop {deviceinfo /Colors get}}if}if}ifelse end def +ncolors 0 ne{/colorimage testsystemdict +/setcolortransfer testsystemdict +/currentcolortransfer testsystemdict +/currentcmykcolor testsystemdict +and and and not{/ncolors 0 def}if}if}if +ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if +ncolors 1 eq dup dup not startnoload +{/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] +cvx bind settransfer _w _h 8 _imatrix _ds image}def}if +not endnoload +ncolors dup 3 eq exch 4 eq or +dup dup not startnoload +{/nullproc{{}}def/concatutil{/exec load +7 -1 roll /exec load}def +/defsubclut +{1 add getinterval def}def +/spconcattransfer +{/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld +currentcolortransfer +[{Aclut ncompute}concatutil] cvx +[{Bclut ncompute}concatutil] cvx +[{Cclut ncompute}concatutil] cvx +[{Dclut ncompute}concatutil] cvx +setcolortransfer}def +/setuprgbcluts +{/bit3x rgbclut length 3 sub def +/bit1x bit3x 3 idiv def/rclut rgbclut def +/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not +endnoload ncolors 3 eq dup dup not startnoload +{/3compute{exch bit3x mul +round cvi get 255 div}def +/doclutimage{%/rgbclut xdf pop +setuprgbcluts/3compute rclut gclut bclut dup +spconcattransfer _w _h 8 _imatrix +[/_ds load /exec load /dup load dup]cvx +nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload +{/stuffclut{cmykindex 3 -1 roll put}def +/ftoint{1 exch sub 255 mul round cvi}def +/4compute { exch bit4x mul round cvi get 255 div } def +/computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def +/cmykclut bit4x 4 add string def +/cclut cmykclut def +/mclut cclut 1 bit4x defsubclut +/yclut cclut 2 bit4x defsubclut +/kclut cclut 3 bit4x defsubclut +/cmykindex 0 def +0 1 bit1x +{dup /cmykindex exch bit1x exch sub 4 mul def +3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div +exch bclut exch get 255 div +setrgbcolor currentcmykcolor +ftoint kclut stuffclut +ftoint yclut stuffclut +ftoint mclut stuffclut +ftoint cclut stuffclut +}for}def +/doclutimage{invalidcolortable?{computecmykclut}if +/4compute cclut mclut yclut kclut spconcattransfer +_w _h 8 _imatrix[/_ds load /exec load/dup load dup dup] cvx +nullproc nullproc nullproc true 4 colorimage}def}if +not endnoload +ncolors 0 eq +dup dup not startnoload +{/colorexpand{mystringexp 0 rgbclut +3 copy +7 -1 roll +{3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall +pop pop pop pop pop}def +/createexpandstr{/mystringexp exch st0 length mul string def}def +/doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix +[/_ds load /exec load /colorexpand load /exec load] cvx +false 3 colorimage}def}if not endnoload +level2 { dontloadlevel1 restore }if +level2 not{ save /dontloadlevel2 xdf }if +/rgbclut 0 def +/doclutimage{/hival 255 def +[/Indexed /DeviceRGB hival rgbclut] setcolorspace +_imagedict dup begin +/Width _w def +/Height _h def +/Decode [0 hival] def +/ImageMatrix _imatrix def +/BitsPerComponent 8 def +/Interpolate false def +end image}def +level2 not{ dontloadlevel2 restore }if +%%EndProcSet +/_imagedict 15 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def +/hascolor true def +/level2 systemdict /languagelevel known +{languagelevel 2 ge}{false}ifelse def +/foureq {4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +pop pop pop pop and and and} def +/setupbands{gsave +/_settransfer systemdict /settransfer get def +/_currenttransfer systemdict /currenttransfer get def +/_image systemdict /image get def +/blank 0 _currenttransfer exec +1 _currenttransfer exec eq def +/negative blank{0 _currenttransfer exec 0.5 lt} +{0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def +/inverted? negative def +/_currentgray systemdict /currentgray get def +hascolor{/$b 0 def} {/$b 5 def} ifelse +/setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub +0 1 0 0 setcmykcolor _currentgray 1 exch sub +0 0 1 0 setcmykcolor _currentgray 1 exch sub +0 0 0 1 setcmykcolor _currentgray 1 exch sub +4 {4 copy} repeat +1 0 0 0 foureq{/$b 1 store}if +0 1 0 0 foureq{/$b 2 store}if +0 0 1 0 foureq{/$b 3 store}if +0 0 0 1 foureq{/$b 4 store}if +0 0 0 0 foureq{/$b 6 store}if}if +blank{/$b 6 store}if +$b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def +/doimage{ +level2{image}{begin +Width Height BitsPerComponent ImageMatrix +Decode length 2 eq +{/DataSource load end image}{ +Decode length 6 eq +{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse +MultipleDataSources 3 end DSIColorImage}{ +Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get +DataSource 2 get DataSource 3 get}{/DataSource load}ifelse +MultipleDataSources 4 end DSIColorImage}if}ifelse +}ifelse +}ifelse}def +/dt{/rgbclut xdf /bwclut xdf}def +/cl{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +doclutimage} def +/cd{$i setmatrix translate scale +level2{_imagedict /ColorSpace get setcolorspace}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +/ImageMatrix _imatrix def +currentdict end +doimage} def +/im{$i setmatrix translate scale +level2{/DeviceGray setcolorspace 0 setgray}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +%/Decode [0 1] def +%/Width _w def +%/Height _h def +/ImageMatrix _imatrix def +currentdict end +doimage }def +/colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ +/$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf +$nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub +{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add +cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup +st0 exch get .3 mul +1 index 1 add st0 exch get .59 mul add +1 index 2 add st0 exch get .11 mul add +1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if +255.0 exch sub cvi +exch 4 div cvi exch +st1 3 1 roll put}for st1}}ifelse +image}def}ifelse +/OPIimage{image}def +/$i matrix def +/bi{scale translate $i currentmatrix pop translate scale setupbands}def +/ei{$i setmatrix}def +/ik{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_w _h false _imatrix 5 -1 roll _ds imagemask}def +/rs{2 copy translate +3 -1 roll neg rotate +neg exch neg exch translate +}def +/cs{_cvmtx astore concat} def +/fd{/_fd xdf}def +/_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def +/rf{/_encodindex xdf +FontDirectory exch known { +findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall +dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def +_encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} +{exch pop}ifelse}forall pop}if}if +currentdict end +definefont pop }{pop pop}ifelse} def +/cvsf{ findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore +makefont setfont}def +/mf{3 -1 roll findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch +_fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def +/wrk1 ( ) def +/wdict 16 dict def +/Work75 75 string def +/Nmk{Work75 cvs dup}def +/Npt{put cvn}def +/dhOdh{Nmk 2 79 Npt}def +/dhBdh{Nmk 2 111 Npt}def +/dhSdh{Nmk 2 83 Npt}def +/sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def +/fts{/ftSize exch def}def +/mkFT{dup/FontType get 0 eq{dup maxlength dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +currentdict end +dup/FDepVector 2 copy get +[exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def +tempFT begin +/FontMatrix [1 0 0 1 0 0] def +/FontType 3 def +dup/Encoding get/Encoding exch def +/proc2 2 index def +dup/ft1 exch def +/ft2 1 array def +/FontBBox [0 0 1 1] def +/BuildChar{CVDict begin +exch dup/ftdt xdf begin +wdict begin +/chr xdf/chrst wrk1 dup 0 chr put def +ft2 0 get null eq{ft1 dup length 2 add dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +/PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def +currentdict end +/q exch definefont +ft2 exch 0 exch put}if +ftdt/proc2 get exec end end end}def end tempFT}ifelse +3 index exch +definefont exch pop}def +/mb{dup dhBdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{boldproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/mo{dup dhOdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{outproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/ms{dup dhSdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{ShadChar} exch +mkFT pop pop pop exch pop +}ifelse }def +/outproc{ft1 setfont +gsave chrst sfWidth grestore +setcharwidth dblsh}def +/boldproc{ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +ft1 /PaintType get 2 ne +{0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} +{0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def +/dblsh{currentcolor currentcolorspace 1.0 setgray +chrst 0 0 moveto show setcolorspace setcolor +ft2 0 get setfont chrst 0 0 moveto show}def +/ShadChar{ +ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +chrst .06 0 moveto show +0 .05 translate dblsh}def +/ts{show}def +/ta{/_letterSpace xdf/_spaceXtra xdf}def +/tf{/_textWidth xdf +/_numSpaces xdf +/_stringLength xdf +dup stringwidth pop +_letterSpace _stringLength 1 sub mul add +_spaceXtra _numSpaces mul add +_textWidth exch sub /wdif xdf +_spaceXtra 0 32 +wdif _stringLength div +_letterSpace add +0 6 -1 roll awidthshow}def +/pb{/_PageSave save def/showpage{}def}def +/pe{_PageSave restore}def +/pm{_pm currentmatrix pop}def +/_pm matrix def +/ps{showpage _pm setmatrix}def +level2{/ef {gsave 6 array astore concat dup execform grestore}def} +{/ef{gsave 6 array astore concat newpath dup begin Matrix concat +BBox dup dup 0 get exch 1 get moveto +dup dup 0 get exch 3 get lineto +dup dup 2 get exch 3 get lineto +dup 2 get exch 1 get lineto closepath +clip newpath PaintProc end grestore}def}ifelse +/bc{save 11 1 roll 10 dict begin /BBox 5 1 roll 4 array astore def +/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def +/ec{pop restore}def +/printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def +/setoverprint where {pop}{/setoverprint{pop}def}ifelse +/sstr { /$sstrs exch def /$sstr exch def /$sstry exch def /$sstrx exch def /Courier findfont $sstrs +scalefont setfont $sstrx $sstry mt stm 1 -1 scale $sstr show rtm } def +/DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc + eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} + {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def +/DSIEndPaintProc{currentdict end}def +/DSIBeginPaintSubProc {save 11 1 roll 10 dict dup /pD exch def begin + /BBox 5 1 roll 4 array astore def /Matrix 7 1 roll 6 array astore def + /FormType 1 def /PaintProc { /pp get exec } def /pp} def +/DSIEndPaintSubProc {bind def end pD} def +/iaa {/ix ix ixn add def} def +/ias2 {/igap exch def /_st0 ia ix igap add get def _st0} def +/ias {/st0 ia ix get def iaa st0} def +/i85s {0 ias2 /ASCII85Decode filter st0 readstring pop iaa} def +/irles {0 ias2 /ASCII85Decode filter /RunLengthDecode filter st0 readstring pop iaa} def +initialize +end +%%EndResource +CVDict begin 0 id 1 fd +_encodArr 0 +[ +39/quotesingle 96/grave +128/bullet/bullet/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/bullet/bullet/bullet +/bullet/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark +/scaron +/guilsinglright/oe/bullet/bullet/Ydieresis +/bullet/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron +/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown +/Agrave/Aacute/Acircumflex/Atilde/Adieresis +/Aring +/AE/Ccedilla/Egrave/Eacute/Egrave/Edieresis/Igrave/Iacute/Icircumflex/Idieresis +/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls +/agrave/aacute/acircumflex/atilde/adieresis +/aring +/ae/ccedilla/egrave/eacute/ecircumflex/edieresis +/igrave +/iacute +/icircumflex +/idieresis +/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis +] put +_encodArr 1 +[ +39/quotesingle 96/grave +128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash +/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation +/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash +/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft +/guillemotright/ellipsis/blank/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge +/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex +/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +] put +end +%%EndProlog +%%IncludeFont: CourierNewPSMT +CVDict begin +/___FN2 /CourierNewPSMT dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { -1 }{-1} ifelse end} {pop -1} ifelse rf +end +0.25 0.25 translate +%%BeginSetup +CVDict begin + +0 id +1 fd +0 151 translate +300 -300 scli +%%EndSetup +%%BeginPageSetup +pm +%%EndPageSetup +/DeviceCMYK scs +0 0 0 1. f4 +0.1250 0.1250 sp + 0 lc + 2 lj +T sa +%page rect: 160 309 501 461 +stm +0 0 0 1. f4 +4.1667 4.1667 sp +11.5 lm + 0 lj +np +332.8565 43.7608 mt +207.8568 210.427 li +ds +rtm +stm +0 0 0 1. f4 +np +332.8565 43.7608 mt +457.8562 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Observe that both vertices with index 3 have the same key, and hence)56 W +7200 29110 MT +(the right hand vertex with index 2 is redundant.)SH +11 /Times-Bold AF +7200 32727 MT +(4.3. Apply)SH +10 /Times-Roman AF +8200 34104 MT +(The procedure)148 W +/Times-Italic SF +14549 XM +(Apply)SH +/Times-Roman SF +17280 XM +(provides the basic method for creating the representation of a function according) +148 W( to the)147 W +7200 35481 MT +(operators in a Boolean expression or logic gate network. 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With our representation, we can implement all of the operators with a single)12 W +7200 44508 MT +(algorithm. In) +972 W( contrast, many Boolean function manipulation programs) +361 W( [6]) +SH( require different) +361 W( algorithms for)362 W +7200 45885 MT +(complementing, intersecting \050) +58 W( =)SH +/Symbol SF +22479 XM +(\327)SH +/Times-Roman SF +(\051, and unioning \050) +58 W( = +\051) +SH( functions,) +58 W( and then implement other operators by)57 W +7200 47262 MT +(combining these operations.)SH +8200 49741 MT +(The algorithm proceeds from the roots of the two argument graphs downward, creating vertices in the) +5 W( result graph)6 W +7200 51118 MT +(at the branching points of the two arguments graphs. First,) +46 W( let us explain the basic idea of the algorithm. Then we)45 W +7200 52495 MT +(will describe two refinements to improve the efficiency. 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First,)10 W +7200 58972 MT +(suppose both)161 W +/Times-Italic SF +13022 XM +(v)SH +8 /Times-Roman AF +13466 59317 MT +(1)SH +10 SS +14277 58972 MT +(and)SH +/Times-Italic SF +16132 XM +(v)SH +8 /Times-Roman AF +16576 59317 MT +(2)SH +10 SS +17387 58972 MT +(are terminal vertices. Then the result graph consists of) +161 W( a terminal vertex having value)162 W +/Times-Italic SF +7200 60349 MT +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +10143 60694 MT +(1)SH +10 SS +10543 60349 MT +(\051 <)SH +/Times-Italic SF +(op)SH +/Times-Roman SF +(>)SH +/Times-Italic SF +13504 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +16447 60694 MT +(2)SH +10 SS +16847 60349 MT +(\051. Otherwise,) +1360 W( suppose at least one of the two is a nonterminal vertex. If)555 W +/Times-Italic SF +7200 61726 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +10143 62071 MT +(1)SH +10 SS +10543 61726 MT +(\051 =)SH +/Times-Italic SF +11940 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +14883 62071 MT +(2)SH +10 SS +15283 61726 MT +(\051 =)SH +/Times-Italic SF +16680 XM +(i)SH +/Times-Roman SF +(, we create a vertex)106 W +/Times-Italic SF +25484 XM +(u)SH +/Times-Roman SF +26340 XM +(having index)106 W +/Times-Italic SF +31997 XM +(i)SH +/Times-Roman SF +(, and apply the algorithm recursively on)107 W +/Times-Italic SF +49244 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +51466 62071 MT +(1)SH +10 SS +51866 61726 MT +(\051 and)107 W +/Times-Italic SF +7200 63103 MT +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +9422 63448 MT +(2)SH +10 SS +9822 63103 MT +(\051 to generate the subgraph whose root) +30 W( becomes)29 W +/Times-Italic SF +29111 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051, and on)29 W +/Times-Italic SF +35253 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +37808 63448 MT +(1)SH +10 SS +38208 63103 MT +(\051 and)29 W +/Times-Italic SF +40543 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +43098 63448 MT +(2)SH +10 SS +43498 63103 MT +(\051 to generate the subgraph)29 W +7200 64480 MT +(whose root becomes)112 W +/Times-Italic SF +15951 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051. Suppose,) +474 W( on the other hand,) +112 W( that)113 W +/Times-Italic SF +33653 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +36596 64825 MT +(1)SH +10 SS +36996 64480 MT +(\051 =)SH +/Times-Italic SF +38393 XM +(i)SH +/Times-Roman SF +(, but either)113 W +/Times-Italic SF +43565 XM +(v)SH +8 /Times-Roman AF +44009 64825 MT +(2)SH +10 SS +44772 64480 MT +(is a terminal vertex or)113 W +/Times-Italic SF +7200 65857 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +10143 66202 MT +(2)SH +10 SS +10543 65857 MT +(\051 >)SH +/Times-Italic SF +11940 XM +(i)SH +/Times-Roman SF +(, Then the function represented by the graph with root)SH +/Times-Italic SF +34075 XM +(v)SH +8 /Times-Roman AF +34519 66202 MT +(2)SH +10 SS +35169 65857 MT +(is independent of)SH +/Times-Italic SF +42307 XM +(x)SH +8 SS +42751 66202 MT +(i)SH +10 /Times-Roman AF +42973 65857 MT +(, i.e.)SH +/Times-Italic SF +9200 67762 MT +(f)SH +8 /Times-Roman AF +9478 68107 MT +(2)SH +10 /Symbol AF +10028 67762 MT +(|)SH +8 /Times-Italic AF +10228 68107 MT +(x)SH +10583 68452 MT +(i)SH +/Times-Roman SF +10805 68107 MT +(=0)SH +10 SS +12342 67762 MT +(=)SH +/Times-Italic SF +13592 XM +(f)SH +8 /Times-Roman AF +13870 68107 MT +(2)SH +10 /Symbol AF +14420 67762 MT +(|)SH +8 /Times-Italic AF +14620 68107 MT +(x)SH +14975 68452 MT +(i)SH +/Times-Roman SF +15197 68107 MT +(=1)SH +10 SS +16734 67762 MT +(=)SH +/Times-Italic SF +17984 XM +(f)SH +8 /Times-Roman AF +18262 68107 MT +(2)SH +10 SS +18662 67762 MT +(.)SH +10800 50 7200 70352 UL +6 SS +8000 71691 MT +(5)SH +8 SS +8300 72000 MT +(Alternatively, a function can be complemented by simply complementing the values of the terminal vertices.)SH +ES +%%Page: 13 13 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(13)SH +7200 7886 MT +(Hence we create a vertex)105 W +/Times-Italic SF +18025 XM +(u)SH +/Times-Roman SF +18880 XM +(having index)105 W +/Times-Italic SF +24534 XM +(i)SH +/Times-Roman SF +(, but recursively apply the algorithm on)105 W +/Times-Italic SF +41600 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +43822 8231 MT +(1)SH +10 SS +44222 7886 MT +(\051 and)104 W +/Times-Italic SF +46707 XM +(v)SH +8 /Times-Roman AF +47151 8231 MT +(2)SH +10 SS +47905 7886 MT +(to generate the)104 W +7200 9263 MT +(subgraph whose root becomes)25 W +/Times-Italic SF +19631 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051, and on)25 W +/Times-Italic SF +25761 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +28316 9608 MT +(1)SH +10 SS +28716 9263 MT +(\051 and)25 W +/Times-Italic SF +31043 XM +(v)SH +8 /Times-Roman AF +31487 9608 MT +(2)SH +10 SS +32162 9263 MT +(to generate the subgraph whose root becomes)25 W +/Times-Italic SF +50806 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051.)SH +7200 10640 MT +(A similar situation holds when the roles of the two) +55 W( vertices in the previous case are reversed. In general the graph)54 W +7200 12017 MT +(produced by this process will not be reduced, and we apply the reduction algorithm to it before returning.)SH +8200 14496 MT +(If we were to implement the technique described in) +16 W( the previous paragraph directly we would obtain an algorithm)17 W +7200 15873 MT +(of exponential \050in)99 W +/Times-Italic SF +14857 XM +(n)SH +/Times-Roman SF +(\051 time complexity, because every call for which one of the arguments is a nonterminal) +99 W( vertex)98 W +7200 17250 MT +(generates two recursive calls. This complexity can be reduced by two refinements.)SH +8200 19729 MT +(First, the algorithm need not evaluate a given pair of subgraphs more than once. Instead, we can) +25 W( maintain a table)26 W +7200 21106 MT +(containing entries of the form \050)12 W +/Times-Italic SF +(v)SH +8 /Times-Roman AF +20174 21451 MT +(1)SH +10 SS +20574 21106 MT +(,)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +21268 21451 MT +(2)SH +10 SS +21668 21106 MT +(,)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 indicating that the result of applying the algorithm to subgraphs with roots)12 W +/Times-Italic SF +53156 XM +(v)SH +8 /Times-Roman AF +53600 21451 MT +(1)SH +10 SS +7200 22483 MT +(and)SH +/Times-Italic SF +8925 XM +(v)SH +8 /Times-Roman AF +9369 22828 MT +(2)SH +10 SS +10050 22483 MT +(was a subgraph with root)31 W +/Times-Italic SF +20509 XM +(u)SH +/Times-Roman SF +(. Then) +312 W( before applying the algorithm to a pair) +31 W( of vertices, we first check whether)32 W +7200 23860 MT +(the table contains an entry for these two vertices. If so, we can immediately) +163 W( return the result. Otherwise, we)162 W +7200 25237 MT +(proceed as described in the previous paragraph and add a new entry to the table. This refinement alone drops) +78 W( the)79 W +7200 26614 MT +(time complexity to O) +150 82 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +17379 26959 MT +(1)SH +10 /Symbol AF +17779 26614 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +19151 26959 MT +(2)SH +10 /Symbol AF +19551 26614 MT +(|)SH +/Times-Roman SF +(\051, as we will show later. This refinement shows how we can exploit the) +82 W( sharing of)81 W +7200 27991 MT +(subgraphs in the data) +30 W( structures to gain efficiency in the algorithms. If the two argument graphs each contain many)31 W +7200 29368 MT +(shared subgraphs, we obtain a high "hit rate") +97 W( for our table. In practice we have found hit rates to range between)96 W +7200 30745 MT +(40% and 50%. Note that) +96 W( with a 50% hit rate, we obtain a speed improvement far better than the factor of 2 one)97 W +7200 32122 MT +(might first expect. Finding an entry \050)193 W +/Times-Italic SF +(v)SH +8 /Times-Roman AF +23994 32467 MT +(1)SH +10 SS +24394 32122 MT +(,)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +25088 32467 MT +(2)SH +10 SS +25488 32122 MT +(,)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 in the table counts) +193 W( as only one "hit", but avoids the potentially)192 W +7200 33499 MT +(numerous recursive calls required to construct the subgraph rooted by)SH +/Times-Italic SF +35361 XM +(u)SH +/Times-Roman SF +(.)SH +8200 35978 MT +(Second, suppose the algorithm is applied) +142 W( to two vertices where one, say)143 W +/Times-Italic SF +38907 XM +(v)SH +8 /Times-Roman AF +39351 36323 MT +(1)SH +10 SS +39751 35978 MT +(, is a terminal vertex, and for this)143 W +7200 37355 MT +(particular operator,)562 W +/Times-Italic SF +16238 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +19181 37700 MT +(1)SH +10 SS +19581 37355 MT +(\051 is a "controlling" value, i.e.) +562 W( either)1373 W +/Times-Italic SF +38639 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +41582 37700 MT +(1)SH +10 SS +41982 37355 MT +(\051 <)SH +/Times-Italic SF +(op)SH +/Times-Roman SF +(>)SH +/Times-Italic SF +44943 XM +(a)SH +/Times-Roman SF +45693 XM +(= 1) +SH( for all)561 W +/Times-Italic SF +51606 XM +(a)SH +/Times-Roman SF +(, or)561 W +/Times-Italic SF +7200 38732 MT +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +10143 39077 MT +(1)SH +10 SS +10543 38732 MT +(\051 <)SH +/Times-Italic SF +(op)SH +/Times-Roman SF +(>)SH +/Times-Italic SF +13504 XM +(a)SH +/Times-Roman SF +14254 XM +(= 0) +SH( for all)203 W +/Times-Italic SF +19093 XM +(a)SH +/Times-Roman SF +(. For) +656 W( example, 1 is a controlling value for either argument of)204 W +8 SS +46989 XM +(OR)SH +10 SS +(, while 0 is a)204 W +7200 40109 MT +(controlling value for) +41 W( either argument of)40 W +8 SS +23550 XM +(AND)SH +10 SS +(. In) +330 W( this case, there is no need to evaluate further. We simply create a)40 W +7200 41486 MT +(terminal vertex having the appropriate value.) +37 W( While) +326 W( this refinement does not improve the worst case complexity of)38 W +7200 42863 MT +(the algorithm, it certainly helps in many cases. In practice we have found this case occurs around 10% of the time.)SH +8200 45342 MT +(A sketch of the code is shown in Figure) +12 W( 6. For simplicity and to optimize the worst case performance, the table is)11 W +7200 46719 MT +(implemented as a two dimensional array indexed by the) +20 W( unique identifiers of the two vertices. In practice, this table)21 W +7200 48096 MT +(will be very sparse, and hence it is more efficient to use a hash table. To detect whether one) +97 W( of the two vertices)96 W +7200 49473 MT +(contains a controlling value for the operator,) +185 W( we evaluate the expression v1.value) +186 W( v2.value) +SH( using a three-)186 W +7200 50850 MT +(valued algebra where X \050the value at any nonterminal vertex\051) +514 W( represents "don't care". That is, if)513 W +/Times-Italic SF +7200 52227 MT +(b)SH +/Times-Roman SF +7950 XM +( 1 =)SH +/Times-Italic SF +11892 XM +(b)SH +/Times-Roman SF +12642 XM +( 0 =)SH +/Times-Italic SF +16584 XM +(a)SH +/Times-Roman SF +(, then)97 W +/Times-Italic SF +19750 XM +(b)SH +/Times-Roman SF +20500 XM +( X =)SH +/Times-Italic SF +24664 XM +(a)SH +/Times-Roman SF +(, otherwise)98 W +/Times-Italic SF +29998 XM +(b)SH +/Times-Roman SF +30748 XM +( X = X.) +SH( This) +446 W( evaluation technique is used in many)98 W +7200 53604 MT +(logic simulators.) +161 W( [14]) +SH( Before) +572 W( the final graph is returned, we apply the procedure)161 W +/Times-Italic SF +41785 XM +(Reduce)SH +/Times-Roman SF +45138 XM +(to transform it into a)160 W +7200 54981 MT +(reduced graph and to assign unique identifiers to the vertices.)SH +8200 57460 MT +(To analyze the time complexity of this algorithm when called) +46 W( on graphs with)47 W +/Symbol SF +39882 XM +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +40804 57805 MT +(1)SH +10 /Symbol AF +41204 57460 MT +(|)SH +/Times-Roman SF +41701 XM +(and)SH +/Symbol SF +43442 XM +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +44364 57805 MT +(2)SH +10 /Symbol AF +44764 57460 MT +(|)SH +/Times-Roman SF +45261 XM +(vertices, respectively,)47 W +7200 58837 MT +(observe that the procedure)94 W +/Times-Italic SF +18406 XM +(Apply-step)SH +/Times-Roman SF +23027 XM +(only generates recursive calls the first time it is invoked) +94 W( on a given pair of)93 W +7200 60214 MT +(vertices, hence the total number of recursive calls to this procedure cannot exceed) +53 W( 2)54 W +/Symbol SF +(\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +42553 60559 MT +(1)SH +10 /Symbol AF +42953 60214 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +44325 60559 MT +(2)SH +10 /Symbol AF +44725 60214 MT +(|)SH +/Times-Roman SF +(. Within) +358 W( a single call,)54 W +7200 61591 MT +(all operations \050including looking for an entry in the table\051 require constant time. Furthermore the initialization of the)SH +7200 62968 MT +(table requires at time proportional) +141 W( to its size, i.e. O) +150 142 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +30602 63313 MT +(1)SH +10 /Symbol AF +31002 62968 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +32374 63313 MT +(2)SH +10 /Symbol AF +32774 62968 MT +(|)SH +/Times-Roman SF +(\051. Hence,) +534 W( the total complexity of the algorithm is)142 W +7200 64382 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +9327 64727 MT +(1)SH +10 /Symbol AF +9727 64382 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +11099 64727 MT +(2)SH +10 /Symbol AF +11499 64382 MT +(|)SH +/Times-Roman SF +(\051.)SH +8 SS +12282 64037 MT +(6)SH +10 SS +13300 64382 MT +(In the worst case,) +59 W( the algorithm may actually require this much time, because the reduced graph for)58 W +7200 65759 MT +(the function)105 W +/Times-Italic SF +12465 XM +(f)SH +8 /Times-Roman AF +12743 66104 MT +(1)SH +10 SS +13393 65759 MT +()SH +/Times-Italic SF +15771 XM +(f)SH +8 /Times-Roman AF +16049 66104 MT +(2)SH +10 SS +16804 65759 MT +(can contain O) +150 105 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +23973 66104 MT +(1)SH +10 /Symbol AF +24373 65759 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +25745 66104 MT +(2)SH +10 /Symbol AF +26145 65759 MT +(|)SH +/Times-Roman SF +(\051 vertices.) +105 W( For) +462 W( example, choose any positive integers)106 W +/Times-Italic SF +48822 XM +(m)SH +/Times-Roman SF +49900 XM +(and)SH +/Times-Italic SF +51700 XM +(n)SH +/Times-Roman SF +52556 XM +(and)SH +7200 67136 MT +(define the functions)SH +/Times-Italic SF +15393 XM +(f)SH +8 /Times-Roman AF +15671 67481 MT +(1)SH +10 SS +16321 67136 MT +(and)SH +/Times-Italic SF +18015 XM +(f)SH +8 /Times-Roman AF +18293 67481 MT +(2)SH +10 SS +18943 67136 MT +(as:)SH +10800 50 7200 70352 UL +6 SS +8000 71691 MT +(6)SH +8 /Times-Bold AF +8300 72000 MT +(Update:)SH +/Times-Roman SF +11455 XM +(See the footnote for Table 1 discussing the inaccuracy of this complexity measure.)SH +ES +%%Page: 14 14 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(14)SH +/Times-Italic SF +9200 7886 MT +(f)SH +8 /Times-Roman AF +9478 8231 MT +(1)SH +10 SS +9878 7886 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +10655 8231 MT +(1)SH +10 SS +11055 7886 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +13749 8231 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +/Times-Italic SF +(m)SH +10 /Times-Roman AF +15978 7886 MT +(\051 =)436 W +/Times-Italic SF +18247 XM +(x)SH +8 /Times-Roman AF +18691 8231 MT +(1)SH +10 /Symbol AF +19091 7886 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +19785 8231 MT +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +/Times-Roman SF +(+1)SH +10 SS +22315 7886 MT +(+)SH +/Symbol SF +23379 XM +(\327 \327 \327)SH +/Times-Roman SF +25129 XM +(+)SH +/Times-Italic SF +25943 XM +(x)SH +8 SS +26387 8231 MT +(n)SH +10 /Symbol AF +26787 7886 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +27481 8231 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +10 SS +9200 9092 MT +(f)SH +8 /Times-Roman AF +9478 9437 MT +(2)SH +10 SS +9878 9092 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +10655 9437 MT +(1)SH +10 SS +11055 9092 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +13749 9437 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +/Times-Italic SF +(m)SH +10 /Times-Roman AF +15978 9092 MT +(\051 =)436 W +/Times-Italic SF +18247 XM +(x)SH +8 SS +18691 9437 MT +(n)SH +/Times-Roman SF +(+1)SH +10 /Symbol AF +19942 9092 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +20636 9437 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +/Times-Roman SF +(+1)SH +10 SS +23566 9092 MT +(+)SH +/Symbol SF +24630 XM +(\327 \327 \327)SH +/Times-Roman SF +26380 XM +(+)SH +/Times-Italic SF +27194 XM +(x)SH +8 SS +27638 9437 MT +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +10 /Symbol AF +29067 9092 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +29761 9437 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +/Times-Italic SF +(m)SH +10 /Times-Roman AF +7200 10820 MT +(These functions are represented) +64 W( by graphs containing 2)63 W +8 /Times-Italic AF +29722 10475 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +31286 10820 MT +(and 2)63 W +8 /Times-Italic AF +33543 10475 MT +(m)SH +/Times-Roman SF +(+1)SH +10 SS +35285 10820 MT +(vertices, respectively, as we saw in Section 3.)63 W +7200 12197 MT +(If we compute)SH +/Times-Italic SF +13226 XM +(f)SH +/Times-Roman SF +13754 XM +(=)SH +/Times-Italic SF +14568 XM +(f)SH +8 /Times-Roman AF +14846 12542 MT +(1)SH +10 SS +15396 12197 MT +(+)SH +/Times-Italic SF +16110 XM +(f)SH +8 /Times-Roman AF +16388 12542 MT +(2)SH +10 SS +16788 12197 MT +(, the resulting function is)SH +/Times-Italic SF +9200 13988 MT +(f)SH +/Times-Roman SF +9628 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +10405 14333 MT +(1)SH +10 SS +10805 13988 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +13499 14333 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +/Times-Italic SF +(m)SH +10 /Times-Roman AF +15728 13988 MT +(\051 =)436 W +/Times-Italic SF +17997 XM +(x)SH +8 /Times-Roman AF +18441 14333 MT +(1)SH +10 /Symbol AF +18841 13988 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +19535 14333 MT +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +/Times-Roman SF +(+1)SH +10 SS +22065 13988 MT +(+)SH +/Symbol SF +23129 XM +(\327 \327 \327)SH +/Times-Roman SF +24879 XM +(+)SH +/Times-Italic SF +25693 XM +(x)SH +8 SS +26137 14333 MT +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +10 /Symbol AF +27566 13988 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +28260 14333 MT +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2)SH +/Times-Italic SF +(m)SH +10 /Times-Roman AF +7200 15716 MT +(which is represented by a graph with 2)238 W +8 /Times-Italic AF +24335 15371 MT +(n)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(m)SH +/Times-Roman SF +(+1)SH +10 SS +26865 15716 MT +(= 0.5)SH +/Symbol SF +(\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +30101 16061 MT +(1)SH +10 /Symbol AF +30501 15716 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +31873 16061 MT +(2)SH +10 /Symbol AF +32273 15716 MT +(|)SH +/Times-Roman SF +32961 XM +(vertices. Hence) +726 W( the worst case) +238 W( efficiency of the)239 W +7200 17093 MT +(algorithm is limited by the size of the result graph, and we cannot expect) +227 W( any algorithm to achieve a better)226 W +7200 18470 MT +(performance than ours under) +181 W( these conditions. Empirically, we have never observed this worst case occurring)182 W +7200 19847 MT +(except when the result graph is large. We conjecture the algorithm) +46 W( could be refined to have worst case complexity)45 W +7200 21224 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +9327 21569 MT +(1)SH +10 /Symbol AF +9727 21224 MT +(|)SH +/Times-Roman SF +(+)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +11413 21569 MT +(2)SH +10 /Symbol AF +11813 21224 MT +(|)SH +/Times-Roman SF +(+)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +13499 21569 MT +(3)SH +10 /Symbol AF +13899 21224 MT +(|)SH +/Times-Roman SF +(\051, where)SH +/Times-Italic SF +17625 XM +(G)SH +8 /Times-Roman AF +18347 21569 MT +(3)SH +10 SS +18997 21224 MT +(is the resulting reduced graph.)SH +8 SS +31078 20879 MT +(7)SH +10 SS +9200 23015 MT +(function)SH +/Times-Italic SF +12783 XM +(Apply)SH +/Times-Roman SF +(\050v1, v2: vertex; : operator\051: vertex)SH +11718 24120 MT +(var T: array[1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +18693 24465 MT +(1)SH +10 /Symbol AF +19093 24120 MT +(|)SH +/Times-Roman SF +(, 1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +21715 24465 MT +(2)SH +10 /Symbol AF +22115 24120 MT +(|)SH +/Times-Roman SF +(] of vertex;)SH +11718 26330 MT +({Recursive routine to implement)SH +/Times-Italic SF +25086 XM +(Apply)SH +/Times-Roman SF +(})SH +11718 27435 MT +(function)SH +/Times-Italic SF +15301 XM +(Apply-step)SH +/Times-Roman SF +(\050v1, v2: vertex\051: vertex;)SH +11718 28540 MT +(begin)SH +14236 29645 MT +(u := T[v1.id, v2.id];)SH +14236 30750 MT +(if u)SH +/Symbol SF +15997 XM +(\271)SH +/Times-Roman SF +16946 XM +(null then return\050u\051; {have already evaluated})SH +14236 31855 MT +(u := new vertex record; u.mark := false;)SH +14236 32960 MT +(T[v1.id, v2.id] := u; {add vertex to table})SH +14236 34065 MT +(u.value := v1.value v2.value;)SH +14236 35170 MT +(if u.value)SH +/Symbol SF +18413 XM +(\271)SH +/Times-Roman SF +19362 XM +(X)SH +14236 36275 MT +(then begin {create terminal vertex})SH +16754 37380 MT +(u.index := n+1; u.low := null; u.high := null;)SH +14236 38485 MT +(end)SH +14236 39590 MT +(else begin {create nonterminal and evaluate further down})SH +16754 40695 MT +(u.index :=)SH +/Times-Italic SF +21068 XM +(Min)SH +/Times-Roman SF +(\050v1.index, v2.index\051;)SH +16754 41800 MT +(if v1.index = u.index)SH +19272 42905 MT +(then begin vlow1 := v1.low; vhigh1 := v1.high end)SH +19272 44010 MT +(else begin vlow1 := v1; vhigh1 := v1 end;)SH +16754 45115 MT +(if v2.index = u.index)SH +19272 46220 MT +(then begin vlow2 := v2.low; vhigh2 := v2.high end)SH +19272 47325 MT +(else begin vlow2 := v2; vhigh2 := v2 end;)SH +16754 48430 MT +(u.low :=)SH +/Times-Italic SF +20346 XM +(Apply-step)SH +/Times-Roman SF +(\050vlow1, vlow2\051;)SH +16754 49535 MT +(u.high :=)SH +/Times-Italic SF +20624 XM +(Apply-step)SH +/Times-Roman SF +(\050vhigh1, vhigh2\051;)SH +14236 50640 MT +(end;)SH +14236 51745 MT +(return\050u\051;)SH +11718 52850 MT +(end;)SH +9200 55060 MT +(begin {Main) +500 W( routine})SH +11718 56165 MT +(Initialize all elements of T to null;)SH +11718 57270 MT +(u :=)SH +/Times-Italic SF +13560 XM +(Apply-step)SH +/Times-Roman SF +(\050v1, v2\051;)SH +11718 58375 MT +(return\050)SH +/Times-Italic SF +(Reduce)SH +/Times-Roman SF +(\050u\051\051;)SH +9200 59480 MT +(end;)SH +23795 61090 MT +(Figure 6.Implementation of)SH +/Times-Italic SF +35072 XM +(Apply)SH +/Times-Roman SF +8200 63569 MT +(Figure 7 shows an example of how this algorithm would proceed in) +217 W( applying the "or" operation to graphs)216 W +7200 64946 MT +(representing the functions)246 W +/Symbol SF +18725 XM +(\330)SH +/Times-Roman SF +19588 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +20365 65291 MT +(1)SH +10 /Symbol AF +20765 64946 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +21459 65291 MT +(3)SH +10 SS +21859 64946 MT +(\051 and)246 W +/Times-Italic SF +24628 XM +(x)SH +8 /Times-Roman AF +25072 65291 MT +(2)SH +10 /Symbol AF +25472 64946 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +26166 65291 MT +(3)SH +10 SS +26566 64946 MT +(. This) +744 W( figure shows the graph created by the algorithm before)247 W +7200 66323 MT +(reduction. Next) +736 W( to each vertex in the resulting graph,) +243 W( we indicate the two vertices on which the procedure)242 W +10800 50 7200 69428 UL +6 SS +8000 70767 MT +(7)SH +8 /Times-Bold AF +8300 71076 MT +(Update:)SH +/Times-Roman SF +11273 XM +(No one) +18 W( has improved on the worst case complexity of the algorithm described here. Disproving a conjecture such as this one would)19 W +7200 72000 MT +(be difficult, however.)SH +ES +%%Page: 15 15 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(15)SH +1 SS +36020 7200 43220 PB +%%BeginDocument: figs/fig7.eps +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 345 361 +%%Title: (C:\\Shared Files\\Documents\\BDD Paper\\fig7.CNV) +%%Creator: Canvas (7.0.1) +%%CreationDate: (05/09/00 @ 03:51 PM) +%%DocumentProcSets: Deneba_CanvasCustomColors +%%+ CVDict +%%DocumentFonts: +%%DocumentNeededFonts: CourierNewPSMT + +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%EndComments +%%BeginProlog +%%BeginResource: procset CVDict +%%Title: Deneba standard graphics +%%Version: 1.02 +%%CreationDate: 10/14/99 +%%Copyright: (C) 1994, 1999 Deneba Systems. All Rights Reserved +userdict /CVDict 200 dict put +CVDict begin +/invalidcolortable? true def +/level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def +level2 not { /xshow { +0 currentpoint 5 -1 roll {pop pop moveto dup 1 add 3 1 roll exch dup 4 1 roll exch get 0 rmoveto currentpoint} exch kshow +pop pop pop pop} def } if +/initialize +{CVDict +{dup xcheck{bind}if pop pop}forall +currentscreen /_sp exch def /_sa exch def /_sf exch def +/DefaultScreen{_sf _sa /_sp load}def +/DefaultScreenProc{ /_sp load}def +/setstrokeadjust where{pop true}{false}ifelse +{/sa /setstrokeadjust load def +/li /lineto load def +/mt /moveto load def +/ct /curveto load def +/lh { currentpoint exch pop lineto}bind def +/lv { currentpoint pop exch lineto}bind def} +{/sa{{currentdict StrokeAdjustDict ne{StrokeAdjustDict begin}if}{ currentdict StrokeAdjustDict eq{end}if}ifelse} bind def +userdict /StrokeAdjustDict 100 dict put +StrokeAdjustDict begin +/_xl +{transform +.25 sub round .25 add exch +.25 sub round .25 add exch +itransform +}bind def +/li{ _xl lineto}bind def +/mt{ _xl moveto}bind def +/ct{ _xl curveto}bind def +/lh { dup _xl pop currentpoint exch pop lineto}bind def +/lv { dup _xl pop currentpoint pop exch lineto}bind def +/rp{_xl/cv_b xdf/cv_r xdf +_xl/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath +}bind def +end}ifelse}def +/terminate +{userdict /StrokeAdjustDict known {currentdict StrokeAdjustDict eq{end}if}if +currentdict CVDict eq{end}if}def +/mxMax 64 def +/mxStk [mxMax {matrix} repeat]def +/mxIdx 0 def +/stm{mxStk mxIdx 0 lt {/mxIdx 0 store}if mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def +/rtm{mxIdx 1 ge {/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}if }def +/scl { 72 div exch 72 div exch scale } def +/scli { 72 exch div exch 72 exch div exch scale } def +/_cvmtx matrix def +/_txtmtx matrix def +/_initmtx matrix def +/xdf{exch def}def +/min{2 copy gt{exch}if pop}def +/max{2 copy lt{exch}if pop}def +/CVvec 256 array def +/T true def +/F false def +/_id 0 def +/_encodindex 0 def +/_encodArr 3 array def +/_fd 0 def +/_textSave null def +/_letterSpace 0 def +/_spaceXtra 0 def +/_PageSave null def +/_penh 1.0 def +/_penv 1.0 def +/save0 null def +/_w 0 def +/_h 0 def +/_ds 0 def +/hct false def +/pw 0 def +/wdif 0 def +/temp 0 def +/currot 0 def +/savehandler null def +/ia 0 def +/ix 0 def +/ixn 1 def +/MultipleDataSources false def +/debug +{dup /$eps exch not def{/savehandler errordict /handleerror get def +errordict +/handleerror{$error begin +newerror{/newerror false def +showpage +72 72 scale +/Helvetica findfont .2 scalefont setfont +.25 9.6 moveto +(Error: )show +errorname +{dup type/stringtype ne +{(converted error name will end up in this string)cvs}if +show}exec +.25 9.4 moveto +(Offending Command: )show +/command load +{dup type/stringtype ne +{(converted command name will end up in this string)cvs}if +show}exec +.25 9.2 moveto +(Stack: )show +ostack +{.25 currentpoint .2 sub exch pop moveto +dup type/stringtype ne{dup type/integertype eq +1 index type/realtype eq or +1 index type/nametype eq or +not {type}if +(converted stack will end up in this string)cvs}if +show}forall +showpage +CVDict /savehandler get exec}if +end}put}if}def +/lc{setlinecap}def +/lj{setlinejoin}def +/lm{setmiterlimit}def +/lf{setflat}def +/cp{closepath}def +/np{newpath}def +/$psz 30 array def +/$pix 0 def +/gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def +/gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def +/li{lineto} def +/lh{currentpoint exch pop lineto}def +/lv{currentpoint pop exch lineto}def +/mt{moveto} def +/_cv_mh{currentpoint exch pop moveto}def +/_cv_mv{currentpoint pop exch moveto}def +/ct{curveto} def +/rt{4 -2 roll moveto +dup 0 exch rlineto +exch 0 rlineto +neg 0 exch rlineto +closepath}def +/_hr +{ +{/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} +{/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} +}def +/_cs null def +/fg{setgray}def +/setcmykcolor where {pop true}{false}ifelse +{/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse +/setcolorspace where {pop true}{false}ifelse +{/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def +}{/f3{ setrgbcolor}def/scs{ pop}def}ifelse +/$cc 5 array def +/findcmykcustomcolor where {pop true}{false}ifelse +not {/findcmykcustomcolor{$cc astore}def}if +/setcustomcolor where {pop true}{false}ifelse +not{/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def}if +/fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def +/setseparationgray where {pop true}{false}ifelse +not{/setseparationgray {systemdict /setgray get exec} def}if +/sg{ setseparationgray}def +/sp{/_penv xdf/_penh xdf}def +/rp +{/cv_b xdf/cv_r xdf +/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath}def +/rr{/yradius xdf +/xradius xdf +/cv_b xdf +/cv_r xdf +/cv_t xdf +/cv_l xdf +cv_l xradius add cv_t moveto +cv_r xradius sub cv_t lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +1 0 1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_r cv_b yradius sub lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 1 -1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l xradius add cv_b lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +-1 0 -1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l cv_t yradius add lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 -1 1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +closepath}def +/ep{_cvmtx currentmatrix pop +translate +scale +1 0 moveto +0 0 1 0 360 arc +closepath +_cvmtx setmatrix}def +/wd{_cvmtx currentmatrix pop +translate +scale +0 0 moveto +0 0 1 5 -2 roll arcn +closepath +_cvmtx setmatrix}def +/as{_cvmtx currentmatrix pop +translate +scale +0 0 1 5 -2 roll arcn +_cvmtx setmatrix}def +/ds +{_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def +/df{fill}def +/de{eofill}def +/_gstk 5 array def +/_gstkc 0 def +/_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def +/_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def +/hs{newpath gsave stm}def +/hc{/_hr load exch get exec newpath}def +/he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store +grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq +not{aload pop moveto}{pop}ifelse dup /_cf get setfont +dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def +/id{ /_id xdf}def +/_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def +% Copyright (C) 1990, 1991 by Adobe Systems Incorporated. +% All rights reserved. +level2 { save /dontloadlevel1 xdf }if +/startnoload{{/noload save def} if}def +/endnoload{{noload restore}if }def +/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def +/ncolors 1 def +/colorimage where{pop true}{false}ifelse{/ncolors 0 +statusdict begin +/processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known +{pop {deviceinfo /Colors get}}if}if}ifelse end def +ncolors 0 ne{/colorimage testsystemdict +/setcolortransfer testsystemdict +/currentcolortransfer testsystemdict +/currentcmykcolor testsystemdict +and and and not{/ncolors 0 def}if}if}if +ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if +ncolors 1 eq dup dup not startnoload +{/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] +cvx bind settransfer _w _h 8 _imatrix _ds image}def}if +not endnoload +ncolors dup 3 eq exch 4 eq or +dup dup not startnoload +{/nullproc{{}}def/concatutil{/exec load +7 -1 roll /exec load}def +/defsubclut +{1 add getinterval def}def +/spconcattransfer +{/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld +currentcolortransfer +[{Aclut ncompute}concatutil] cvx +[{Bclut ncompute}concatutil] cvx +[{Cclut ncompute}concatutil] cvx +[{Dclut ncompute}concatutil] cvx +setcolortransfer}def +/setuprgbcluts +{/bit3x rgbclut length 3 sub def +/bit1x bit3x 3 idiv def/rclut rgbclut def +/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not +endnoload ncolors 3 eq dup dup not startnoload +{/3compute{exch bit3x mul +round cvi get 255 div}def +/doclutimage{%/rgbclut xdf pop +setuprgbcluts/3compute rclut gclut bclut dup +spconcattransfer _w _h 8 _imatrix +[/_ds load /exec load /dup load dup]cvx +nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload +{/stuffclut{cmykindex 3 -1 roll put}def +/ftoint{1 exch sub 255 mul round cvi}def +/4compute { exch bit4x mul round cvi get 255 div } def +/computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def +/cmykclut bit4x 4 add string def +/cclut cmykclut def +/mclut cclut 1 bit4x defsubclut +/yclut cclut 2 bit4x defsubclut +/kclut cclut 3 bit4x defsubclut +/cmykindex 0 def +0 1 bit1x +{dup /cmykindex exch bit1x exch sub 4 mul def +3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div +exch bclut exch get 255 div +setrgbcolor currentcmykcolor +ftoint kclut stuffclut +ftoint yclut stuffclut +ftoint mclut stuffclut +ftoint cclut stuffclut +}for}def +/doclutimage{invalidcolortable?{computecmykclut}if +/4compute cclut mclut yclut kclut spconcattransfer +_w _h 8 _imatrix[/_ds load /exec load/dup load dup dup] cvx +nullproc nullproc nullproc true 4 colorimage}def}if +not endnoload +ncolors 0 eq +dup dup not startnoload +{/colorexpand{mystringexp 0 rgbclut +3 copy +7 -1 roll +{3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall +pop pop pop pop pop}def +/createexpandstr{/mystringexp exch st0 length mul string def}def +/doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix +[/_ds load /exec load /colorexpand load /exec load] cvx +false 3 colorimage}def}if not endnoload +level2 { dontloadlevel1 restore }if +level2 not{ save /dontloadlevel2 xdf }if +/rgbclut 0 def +/doclutimage{/hival 255 def +[/Indexed /DeviceRGB hival rgbclut] setcolorspace +_imagedict dup begin +/Width _w def +/Height _h def +/Decode [0 hival] def +/ImageMatrix _imatrix def +/BitsPerComponent 8 def +/Interpolate false def +end image}def +level2 not{ dontloadlevel2 restore }if +%%EndProcSet +/_imagedict 15 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def +/hascolor true def +/level2 systemdict /languagelevel known +{languagelevel 2 ge}{false}ifelse def +/foureq {4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +pop pop pop pop and and and} def +/setupbands{gsave +/_settransfer systemdict /settransfer get def +/_currenttransfer systemdict /currenttransfer get def +/_image systemdict /image get def +/blank 0 _currenttransfer exec +1 _currenttransfer exec eq def +/negative blank{0 _currenttransfer exec 0.5 lt} +{0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def +/inverted? negative def +/_currentgray systemdict /currentgray get def +hascolor{/$b 0 def} {/$b 5 def} ifelse +/setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub +0 1 0 0 setcmykcolor _currentgray 1 exch sub +0 0 1 0 setcmykcolor _currentgray 1 exch sub +0 0 0 1 setcmykcolor _currentgray 1 exch sub +4 {4 copy} repeat +1 0 0 0 foureq{/$b 1 store}if +0 1 0 0 foureq{/$b 2 store}if +0 0 1 0 foureq{/$b 3 store}if +0 0 0 1 foureq{/$b 4 store}if +0 0 0 0 foureq{/$b 6 store}if}if +blank{/$b 6 store}if +$b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def +/doimage{ +level2{image}{begin +Width Height BitsPerComponent ImageMatrix +Decode length 2 eq +{/DataSource load end image}{ +Decode length 6 eq +{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse +MultipleDataSources 3 end DSIColorImage}{ +Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get +DataSource 2 get DataSource 3 get}{/DataSource load}ifelse +MultipleDataSources 4 end DSIColorImage}if}ifelse +}ifelse +}ifelse}def +/dt{/rgbclut xdf /bwclut xdf}def +/cl{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +doclutimage} def +/cd{$i setmatrix translate scale +level2{_imagedict /ColorSpace get setcolorspace}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +/ImageMatrix _imatrix def +currentdict end +doimage} def +/im{$i setmatrix translate scale +level2{/DeviceGray setcolorspace 0 setgray}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +%/Decode [0 1] def +%/Width _w def +%/Height _h def +/ImageMatrix _imatrix def +currentdict end +doimage }def +/colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ +/$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf +$nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub +{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add +cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup +st0 exch get .3 mul +1 index 1 add st0 exch get .59 mul add +1 index 2 add st0 exch get .11 mul add +1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if +255.0 exch sub cvi +exch 4 div cvi exch +st1 3 1 roll put}for st1}}ifelse +image}def}ifelse +/OPIimage{image}def +/$i matrix def +/bi{scale translate $i currentmatrix pop translate scale setupbands}def +/ei{$i setmatrix}def +/ik{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_w _h false _imatrix 5 -1 roll _ds imagemask}def +/rs{2 copy translate +3 -1 roll neg rotate +neg exch neg exch translate +}def +/cs{_cvmtx astore concat} def +/fd{/_fd xdf}def +/_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def +/rf{/_encodindex xdf +FontDirectory exch known { +findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall +dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def +_encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} +{exch pop}ifelse}forall pop}if}if +currentdict end +definefont pop }{pop pop}ifelse} def +/cvsf{ findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore +makefont setfont}def +/mf{3 -1 roll findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch +_fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def +/wrk1 ( ) def +/wdict 16 dict def +/Work75 75 string def +/Nmk{Work75 cvs dup}def +/Npt{put cvn}def +/dhOdh{Nmk 2 79 Npt}def +/dhBdh{Nmk 2 111 Npt}def +/dhSdh{Nmk 2 83 Npt}def +/sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def +/fts{/ftSize exch def}def +/mkFT{dup/FontType get 0 eq{dup maxlength dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +currentdict end +dup/FDepVector 2 copy get +[exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def +tempFT begin +/FontMatrix [1 0 0 1 0 0] def +/FontType 3 def +dup/Encoding get/Encoding exch def +/proc2 2 index def +dup/ft1 exch def +/ft2 1 array def +/FontBBox [0 0 1 1] def +/BuildChar{CVDict begin +exch dup/ftdt xdf begin +wdict begin +/chr xdf/chrst wrk1 dup 0 chr put def +ft2 0 get null eq{ft1 dup length 2 add dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +/PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def +currentdict end +/q exch definefont +ft2 exch 0 exch put}if +ftdt/proc2 get exec end end end}def end tempFT}ifelse +3 index exch +definefont exch pop}def +/mb{dup dhBdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{boldproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/mo{dup dhOdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{outproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/ms{dup dhSdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{ShadChar} exch +mkFT pop pop pop exch pop +}ifelse }def +/outproc{ft1 setfont +gsave chrst sfWidth grestore +setcharwidth dblsh}def +/boldproc{ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +ft1 /PaintType get 2 ne +{0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} +{0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def +/dblsh{currentcolor currentcolorspace 1.0 setgray +chrst 0 0 moveto show setcolorspace setcolor +ft2 0 get setfont chrst 0 0 moveto show}def +/ShadChar{ +ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +chrst .06 0 moveto show +0 .05 translate dblsh}def +/ts{show}def +/ta{/_letterSpace xdf/_spaceXtra xdf}def +/tf{/_textWidth xdf +/_numSpaces xdf +/_stringLength xdf +dup stringwidth pop +_letterSpace _stringLength 1 sub mul add +_spaceXtra _numSpaces mul add +_textWidth exch sub /wdif xdf +_spaceXtra 0 32 +wdif _stringLength div +_letterSpace add +0 6 -1 roll awidthshow}def +/pb{/_PageSave save def/showpage{}def}def +/pe{_PageSave restore}def +/pm{_pm currentmatrix pop}def +/_pm matrix def +/ps{showpage _pm setmatrix}def +level2{/ef {gsave 6 array astore concat dup execform grestore}def} +{/ef{gsave 6 array astore concat newpath dup begin Matrix concat +BBox dup dup 0 get exch 1 get moveto +dup dup 0 get exch 3 get lineto +dup dup 2 get exch 3 get lineto +dup 2 get exch 1 get lineto closepath +clip newpath PaintProc end grestore}def}ifelse +/bc{save 11 1 roll 10 dict begin /BBox 5 1 roll 4 array astore def +/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def +/ec{pop restore}def +/printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def +/setoverprint where {pop}{/setoverprint{pop}def}ifelse +/sstr { /$sstrs exch def /$sstr exch def /$sstry exch def /$sstrx exch def /Courier findfont $sstrs +scalefont setfont $sstrx $sstry mt stm 1 -1 scale $sstr show rtm } def +/DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc + eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} + {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def +/DSIEndPaintProc{currentdict end}def +/DSIBeginPaintSubProc {save 11 1 roll 10 dict dup /pD exch def begin + /BBox 5 1 roll 4 array astore def /Matrix 7 1 roll 6 array astore def + /FormType 1 def /PaintProc { /pp get exec } def /pp} def +/DSIEndPaintSubProc {bind def end pD} def +/iaa {/ix ix ixn add def} def +/ias2 {/igap exch def /_st0 ia ix igap add get def _st0} def +/ias {/st0 ia ix get def iaa st0} def +/i85s {0 ias2 /ASCII85Decode filter st0 readstring pop iaa} def +/irles {0 ias2 /ASCII85Decode filter /RunLengthDecode filter st0 readstring pop iaa} def +initialize +end +%%EndResource +CVDict begin 0 id 1 fd +_encodArr 0 +[ +39/quotesingle 96/grave +128/bullet/bullet/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/bullet/bullet/bullet +/bullet/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark +/scaron +/guilsinglright/oe/bullet/bullet/Ydieresis +/bullet/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron +/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown +/Agrave/Aacute/Acircumflex/Atilde/Adieresis +/Aring +/AE/Ccedilla/Egrave/Eacute/Egrave/Edieresis/Igrave/Iacute/Icircumflex/Idieresis +/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls +/agrave/aacute/acircumflex/atilde/adieresis +/aring +/ae/ccedilla/egrave/eacute/ecircumflex/edieresis +/igrave +/iacute +/icircumflex +/idieresis +/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis +] put +_encodArr 1 +[ +39/quotesingle 96/grave +128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute +/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave +/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute +/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis +/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls +/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash +/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation +/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash +/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft +/guillemotright/ellipsis/blank/Agrave/Atilde/Otilde/OE/oe +/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge +/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl +/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex/Aacute +/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex +/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde +/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron +] put +end +%%EndProlog +%%IncludeFont: CourierNewPSMT +%%IncludeFont: TimesNewRomanPS-ItalicMT +%%IncludeFont: TimesNewRomanPSMT +CVDict begin +/___FN2 /CourierNewPSMT dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { -1 }{-1} ifelse end} {pop -1} ifelse rf +/___FN9-2 /Times-Italic dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { 0 }{-1} ifelse end} {pop -1} ifelse rf +/___FN9 /Times-Roman dup dup findfont dup /CharStrings known {begin CharStrings /Adieresis known { 0 }{-1} ifelse end} {pop -1} ifelse rf +end +0.25 0.25 translate +%%BeginSetup +CVDict begin 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f4 +np +939.3654 606.2497 1022.6986 689.5829 rp +de +0 0 0 1. f4 +np +939.3654 606.2497 1022.6986 689.5829 rp +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +967.4459 656.2491 mt +(0) show +T sa +rtm +rtm +stm +0 0 0 0 f4 +np +1187.2815 606.2497 1270.6146 689.5829 rp +de +0 0 0 1. f4 +np +1187.2815 606.2497 1270.6146 689.5829 rp +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +1217.4453 658.3324 mt +(1) show +T sa +rtm +rtm +stm +stm +37.4999 fts +/___FN2 cvsf +0 0 0 1. f4 +F sa +1195.5212 550.8327 mt +(1) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +966.3551 384.1665 mt +(0) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1028.855 550.8327 mt +(0) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1091.3548 384.1665 mt +(1) show +T sa +rtm +rtm +stm +stm +41.6666 fts +/___FN2 cvsf +0 0 0 1. f4 +F sa +1080.7713 304.1666 mt +(b1) [25.00 0 ] xshow +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1205.771 470.8329 mt +(b2) [25.00 0 ] xshow +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1289.1042 637.4991 mt +(b4) [25.00 0 ] xshow +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1039.1048 637.4991 mt +(b3) [25.00 0 ] xshow +T sa +rtm +rtm +stm +0 0 0 1. f4 +np +1103.9484 895.8324 mt +1062.2818 1062.4986 li +ds +rtm +stm +0 0 0 1. f4 +np +1103.9484 895.8324 mt +1228.9481 1395.8311 li +ds +rtm +stm +0 0 0 1. f4 +np +1228.9481 1395.8311 mt +1020.6152 1229.1649 li +ds +rtm +stm +0 0 0 1. f4 +np +1062.2818 1062.4986 mt +1020.6152 1229.1649 li +ds +rtm +stm +0 0 0 1. f4 +np +1062.2818 1062.4986 mt +1228.9481 1395.8311 li +ds +rtm +stm +0 0 0 0 f4 +np +41.6666 41.6666 1106.2921 897.9266 ep +de +0 0 0 1. f4 +np +41.6666 41.6666 1106.2921 897.9266 ep +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +1094.7894 908.3427 mt +(1) show +T sa +rtm +rtm +stm +0 0 0 0 f4 +np +41.6666 41.6666 1064.6256 1064.5928 ep +de +0 0 0 1. f4 +np +41.6666 41.6666 1064.6256 1064.5928 ep +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +1053.1228 1075.009 mt +(2) show +T sa +rtm +rtm +stm +0 0 0 0 f4 +np +1189.6253 1356.2479 1272.9584 1439.581 rp +de +0 0 0 1. f4 +np +1189.6253 1356.2479 1272.9584 1439.581 rp +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +1219.789 1408.3306 mt +(1) show +T sa +rtm +rtm +stm +stm +37.4999 fts +/___FN2 cvsf +0 0 0 1. f4 +F sa +1156.1984 967.4984 mt +(0) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1031.1987 967.4984 mt +(1) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +947.8656 1300.8309 mt +(1) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +989.5322 1134.1646 mt +(0) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1114.5319 1300.8309 mt +(0) show +T sa +rtm +rtm +stm +stm +0 0 0 1. f4 +F sa +1114.5319 1134.1646 mt +(1) show +T sa +rtm +rtm +stm +0 0 0 1. f4 +np +1020.6152 1229.1649 mt +978.9487 1395.8311 li +ds +rtm +stm +0 0 0 0 f4 +np +41.6666 41.6666 1022.959 1231.2591 ep +de +0 0 0 1. f4 +np +41.6666 41.6666 1022.959 1231.2591 ep +ds +rtm +stm +stm +41.6666 fts +/___FN2 cvsf +0 0 0 1. f4 +F sa +1011.4562 1241.6752 mt +(3) show +T sa +rtm +rtm +stm +0 0 0 0 f4 +np +941.7092 1356.2479 1025.0423 1439.581 rp +de +0 0 0 1. f4 +np +941.7092 1356.2479 1025.0423 1439.581 rp +ds +rtm +stm +stm +0 0 0 1. f4 +F sa +969.7897 1406.2473 mt +(0) show +T sa +rtm +rtm +stm +stm +49.9999 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +940.1071 815.8322 mt +(After Reduction) [36.00 16.50 14.00 22.00 16.50 12.50 33.50 22.00 25.00 25.00 22.00 14.00 14.00 25.00 0 ] xshow +T sa +rtm +rtm +stm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +293.2316 61.6674 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +315.2309 77.6674 mt +(1) show +T sa +rtm +stm +49.9999 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +336.0638 61.6674 mt +( ) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +348.5634 44.6674 mt +(.) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +361.063 61.6674 mt +( ) show +T sa +rtm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +373.5626 61.6674 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +395.562 77.6674 mt +(3) show +T sa +rtm +rtm +stm +0 0 0 1. f4 +np +292.2256 20.8669 mt +414.1063 20.8345 li +ds +rtm +stm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +980.724 61.6998 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1002.7233 77.6998 mt +(1) show +T sa +rtm +stm +49.9999 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1023.5561 61.6998 mt +( ) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1036.0557 44.6998 mt +(.) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1048.5553 61.6998 mt +( ) show +T sa +rtm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +1061.055 61.6998 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1083.0543 77.6998 mt +(2) show +T sa +rtm +rtm +stm +stm +74.9998 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +668.2247 67.533 mt +(+) show +T sa +rtm +rtm +stm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +1230.7293 915.832 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1252.7287 931.832 mt +(1) show +T sa +rtm +stm +49.9999 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1273.5615 915.832 mt +( ) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1286.0611 898.832 mt +(.) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1298.5607 915.832 mt +( ) show +T sa +rtm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +1311.0603 915.832 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1333.0597 931.832 mt +(2) show +T sa +rtm +stm +49.9999 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1353.8925 915.832 mt +( ) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1366.3921 898.832 mt +(.) show +T sa +rtm +stm +0 0 0 1. f4 +F sa +1378.8917 915.832 mt +( ) show +T sa +rtm +stm +49.9999 fts +/___FN9-2 cvsf +0 0 0 1. f4 +F sa +1391.3913 915.832 mt +(x) show +T sa +rtm +stm +41.6666 fts +/___FN9 cvsf +0 0 0 1. f4 +F sa +1413.3907 931.832 mt +(3) show +T sa +rtm +rtm +stm +0 0 0 1. f4 +np +1309.9375 874.9991 mt +1351.604 874.9991 li +ds +rtm +stm +0 0 0 1. f4 +np +1226.6043 854.1658 mt +1434.9372 854.1658 li +ds +rtm +terminate + +%%EndDocument + PE 10 SS +25184 44830 MT +(Figure 7.Example of)SH +/Times-Italic SF +33683 XM +(Apply)SH +7200 46621 MT +(Apply-step)SH +/Times-Roman SF +11757 XM +(was invoked) +30 W( in creating this vertex. Each of our two refinements is applied once: when the procedure is)31 W +7200 47998 MT +(invoked on vertices a3 and b1) +62 W( \050because 1 is a controlling value for this operator\051, and on the second invocation on)61 W +7200 49375 MT +(vertices a3 and b3. For larger graphs, we would expect these refinements to be applied) +150 W( more often. After the)151 W +7200 50752 MT +(reduction algorithm has been applied, we see that the resulting graph indeed represents the function)SH +/Symbol SF +47398 XM +(\330)SH +/Times-Roman SF +48261 XM +(\050)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +49038 51097 MT +(1)SH +10 /Symbol AF +49438 50752 MT +(\327)SH +/Times-Italic SF +(x)SH +/Symbol SF +49688 50371 MT +(-)SH +8 /Times-Roman AF +50237 51097 MT +(2)SH +10 /Symbol AF +50637 50752 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +51331 51097 MT +(3)SH +10 SS +51731 50752 MT +(\051.)SH +11 /Times-Bold AF +7200 54369 MT +(4.4. Restriction)SH +10 /Times-Roman AF +8200 55746 MT +(The restriction algorithm transforms the graph representing a function)75 W +/Times-Italic SF +37120 XM +(f)SH +/Times-Roman SF +37723 XM +(into one representing the) +75 W( function)74 W +/Times-Italic SF +51844 XM +(f)SH +/Symbol SF +52372 XM +(|)SH +8 /Times-Italic AF +52572 56091 MT +(x)SH +52927 56436 MT +(i)SH +/Times-Roman SF +53149 56091 MT +(=)SH +/Times-Italic SF +(b)SH +10 /Times-Roman AF +7200 57297 MT +(for specified values of)191 W +/Times-Italic SF +17128 XM +(i)SH +/Times-Roman SF +17847 XM +(and)SH +/Times-Italic SF +19733 XM +(b)SH +/Times-Roman SF +(. This) +634 W( algorithm proceeds by traversing the graph in the manner shown in the)192 W +7200 58674 MT +(procedure)SH +/Times-Italic SF +11520 XM +(Traverse)SH +/Times-Roman SF +15397 XM +(looking for every pointer \050either to the root of the graph) +72 W( or from some vertex to its child\051 to a)71 W +7200 60051 MT +(vertex)SH +/Times-Italic SF +9970 XM +(v)SH +/Times-Roman SF +10685 XM +(such that)21 W +/Times-Italic SF +14561 XM +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +18901 XM +(i)SH +/Times-Roman SF +(. When) +294 W( such a pointer is encountered, it is changed to point either to)22 W +/Times-Italic SF +47255 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 \050for)22 W +/Times-Italic SF +51853 XM +(b)SH +/Times-Roman SF +52603 XM +(=0\051)SH +7200 61428 MT +(or to)21 W +/Times-Italic SF +9353 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 \050for)21 W +/Times-Italic SF +14282 XM +(b)SH +/Times-Roman SF +15032 XM +(= 1.\051) +SH( Finally,) +292 W( the procedure)21 W +/Times-Italic SF +26588 XM +(Reduce)SH +/Times-Roman SF +29802 XM +(is called to) +21 W( reduce the graph and to assign unique identifiers)20 W +7200 62805 MT +(to the vertices. The amount computation required for) +140 W( each vertex is constant, and hence the complexity of this)141 W +7200 64182 MT +(algorithm is O) +150 4 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +(|)SH +/Times-Roman SF +(\051. Note) +258 W( that this algorithm could simultaneously restrict several) +4 W( of the function arguments without)3 W +7200 65559 MT +(changing the complexity.)SH +ES +%%Page: 16 16 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(16)SH +11 /Times-Bold AF +7200 7937 MT +(4.5. Composition)SH +10 /Times-Roman AF +8200 9314 MT +(The composition algorithm constructs the graph for the) +153 W( function obtained by composing two functions. This)154 W +7200 10691 MT +(algorithm allows) +159 W( us to more quickly derive the functions for a logic network or expression containing repeated)158 W +7200 12068 MT +(structures, a common occurrence in structured designs.) +95 W( Composition) +442 W( can be expressed in terms of restriction and)96 W +7200 13445 MT +(Boolean operations, according to the following expansion, derived) +86 W( directly from the Shannon expansion \050equation)85 W +7200 14822 MT +(1\051:)SH +52834 16613 MT +(\0502\051)SH +/Times-Italic SF +11200 17718 MT +(f)SH +8 /Times-Roman AF +11478 18063 MT +(1)SH +10 /Symbol AF +12028 17718 MT +(|)SH +8 /Times-Italic AF +12228 18063 MT +(x)SH +12583 18408 MT +(i)SH +/Times-Roman SF +12805 18063 MT +(=)SH +/Times-Italic SF +(f)SH +/Times-Roman SF +13478 18408 MT +(2)SH +10 SS +14564 17718 MT +(=)SH +/Times-Italic SF +15814 XM +(f)SH +8 /Times-Roman AF +16092 18063 MT +(2)SH +10 /Symbol AF +16492 17718 MT +(\327)SH +/Times-Italic SF +(f)SH +8 /Times-Roman AF +17020 18063 MT +(1)SH +10 /Symbol AF +17570 17718 MT +(|)SH +8 /Times-Italic AF +17770 18063 MT +(x)SH +18125 18408 MT +(i)SH +/Times-Roman SF +18347 18063 MT +(=1)SH +10 SS +19448 17718 MT +(+ \050)SH +/Symbol SF +20745 XM +(\330)SH +/Times-Italic SF +21608 XM +(f)SH +8 /Times-Roman AF +21886 18063 MT +(2)SH +10 SS +22286 17718 MT +(\051)SH +/Symbol SF +(\327)SH +/Times-Italic SF +(f)SH +8 /Times-Roman AF +23147 18063 MT +(1)SH +10 /Symbol AF +23697 17718 MT +(|)SH +8 /Times-Italic AF +23897 18063 MT +(x)SH +24252 18408 MT +(i)SH +/Times-Roman SF +24474 18063 MT +(=0)SH +10 SS +7200 19269 MT +(Thus, our algorithms for restriction and application are sufficient to implement composition.) +83 W( However,) +418 W( if the two)84 W +7200 20646 MT +(functions are represented by graphs)293 W +/Times-Italic SF +23132 XM +(G)SH +8 /Times-Roman AF +23854 20991 MT +(1)SH +10 SS +24796 20646 MT +(and)SH +/Times-Italic SF +26782 XM +(G)SH +8 /Times-Roman AF +27504 20991 MT +(2)SH +10 SS +27904 20646 MT +(, respectively, we would obtain a worst case complexity of)292 W +7200 22060 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +9327 22405 MT +(1)SH +10 /Symbol AF +9727 22060 MT +(|)SH +8 /Times-Roman AF +9927 21715 MT +(2)SH +10 /Symbol AF +10327 22060 MT +(\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +11499 22405 MT +(2)SH +10 /Symbol AF +11899 22060 MT +(|)SH +8 /Times-Roman AF +12099 21715 MT +(2)SH +10 SS +12499 22060 MT +(\051. We) +386 W( can improve this complexity to O) +150 68 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +30585 22405 MT +(1)SH +10 /Symbol AF +30985 22060 MT +(|)SH +8 /Times-Roman AF +31185 21715 MT +(2)SH +10 /Symbol AF +31585 22060 MT +(\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +32757 22405 MT +(2)SH +10 /Symbol AF +33157 22060 MT +(|)SH +/Times-Roman SF +(\051 by observing) +68 W( that equation 2 can be expressed in)69 W +7200 23437 MT +(terms of a ternary Boolean operation)SH +/Times-Italic SF +22141 XM +(ITE)SH +/Times-Roman SF +23891 XM +(\050short for if-then-else\051)SH +/Times-Italic SF +9200 25342 MT +(ITE)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(a)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +11933 XM +(b)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +12833 XM +(c)SH +/Times-Roman SF +(\051 =)436 W +/Times-Italic SF +15546 XM +(a)SH +/Symbol SF +(\327)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +17046 XM +(+\050)SH +/Symbol SF +18093 XM +(\330)SH +/Times-Italic SF +18956 XM +(a)SH +/Times-Roman SF +(\051)SH +/Symbol SF +(\327)SH +/Times-Italic SF +(c)SH +/Times-Roman SF +7200 26976 MT +(This operation can be applied to the three functions)39 W +/Times-Italic SF +28353 XM +(f)SH +8 /Times-Roman AF +28631 27321 MT +(2)SH +10 SS +29031 26976 MT +(,)SH +/Times-Italic SF +29570 XM +(f)SH +8 /Times-Roman AF +29848 27321 MT +(1)SH +10 /Symbol AF +30398 26976 MT +(|)SH +8 /Times-Italic AF +30598 27321 MT +(x)SH +30953 27666 MT +(i)SH +/Times-Roman SF +31175 27321 MT +(=1)SH +10 SS +32026 26976 MT +(, and)39 W +/Times-Italic SF +34298 XM +(f)SH +8 /Times-Roman AF +34576 27321 MT +(1)SH +10 /Symbol AF +35126 26976 MT +(|)SH +8 /Times-Italic AF +35326 27321 MT +(x)SH +35681 27666 MT +(i)SH +/Times-Roman SF +35903 27321 MT +(=0)SH +10 SS +37043 26976 MT +(by an extension of the)38 W +/Times-Italic SF +46315 XM +(Apply)SH +/Times-Roman SF +48936 XM +(procedure to)38 W +7200 28527 MT +(ternary operations. The procedure)23 W +/Times-Italic SF +21366 XM +(Compose)SH +/Times-Roman SF +25361 XM +(shown in Figure 8 utilizes this technique to compose two) +23 W( functions. In)24 W +7200 29904 MT +(this code the recursive routine)35 W +/Times-Italic SF +19678 XM +(Compose-step)SH +/Times-Roman SF +25629 XM +(both applies the operation)35 W +/Times-Italic SF +36379 XM +(ITE)SH +/Times-Roman SF +38164 XM +(and computes the restrictions of)35 W +/Times-Italic SF +51365 XM +(f)SH +8 /Times-Roman AF +51643 30249 MT +(1)SH +10 SS +52327 29904 MT +(as it)34 W +7200 31281 MT +(traverses the graphs.)SH +8200 33760 MT +(It is unclear whether the efficiency of this algorithm truly has a quadratic dependence on the size) +155 W( of its first)156 W +7200 35137 MT +(argument, or whether this indicates a weakness in our performance analysis. We have found no cases) +110 W( for which)109 W +7200 36514 MT +(composition requires time greater than O) +150 0 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +25019 36859 MT +(1)SH +10 /Symbol AF +25419 36514 MT +(|\327|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +26791 36859 MT +(2)SH +10 /Symbol AF +27191 36514 MT +(|)SH +/Times-Roman SF +(\051.)SH +8 SS +27974 36169 MT +(8)SH +10 SS +8200 38993 MT +(In many instances, two functions can) +163 W( be composed in a simpler and more efficient way by a more syntactic)164 W +7200 40370 MT +(technique.)SH +8 SS +11338 40025 MT +(9)SH +10 SS +12256 40370 MT +(That is suppose functions)9 W +/Times-Italic SF +22736 XM +(f)SH +8 /Times-Roman AF +23014 40715 MT +(1)SH +10 SS +23673 40370 MT +(and)SH +/Times-Italic SF +25376 XM +(f)SH +8 /Times-Roman AF +25654 40715 MT +(2)SH +10 SS +26313 40370 MT +(are represented by) +9 W( graphs)8 W +/Times-Italic SF +36843 XM +(G)SH +8 /Times-Roman AF +37565 40715 MT +(1)SH +10 SS +38223 40370 MT +(and)SH +/Times-Italic SF +39925 XM +(G)SH +8 /Times-Roman AF +40647 40715 MT +(2)SH +10 SS +41047 40370 MT +(, respectively. We can compose)8 W +7200 41747 MT +(the functions by replacing each) +107 W( vertex)108 W +/Times-Italic SF +23340 XM +(v)SH +/Times-Roman SF +24142 XM +(in graph)108 W +/Times-Italic SF +27913 XM +(G)SH +8 /Times-Roman AF +28635 42092 MT +(1)SH +10 SS +29393 41747 MT +(having index)108 W +/Times-Italic SF +35053 XM +(i)SH +/Times-Roman SF +35689 XM +(by a copy of)108 W +/Times-Italic SF +41342 XM +(G)SH +8 /Times-Roman AF +42064 42092 MT +(2)SH +10 SS +42464 41747 MT +(, replacing each branch to a)108 W +7200 43124 MT +(terminal vertex in)9 W +/Times-Italic SF +14587 XM +(G)SH +8 /Times-Roman AF +15309 43469 MT +(2)SH +10 SS +15967 43124 MT +(by a branch to)8 W +/Times-Italic SF +21942 XM +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 or)8 W +/Times-Italic SF +25846 XM +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051 depending on the value of the terminal vertex. We can do this)8 W +7200 44501 MT +(however, only if the) +2 W( resulting graph would not violate our index ordering restriction. That is, there can be no indices)3 W +/Times-Italic SF +7200 45878 MT +(j)SH +/Symbol SF +7778 XM +(\316)SH +/Times-Italic SF +8791 XM +(I)SH +8 SS +9124 46223 MT +(f)SH +/Times-Roman SF +9346 46568 MT +(1)SH +10 SS +9746 45878 MT +(,)SH +/Times-Italic SF +10315 XM +(k)SH +/Symbol SF +11059 XM +(\316)SH +/Times-Italic SF +12072 XM +(I)SH +8 SS +12405 46223 MT +(f)SH +/Times-Roman SF +12627 46568 MT +(2)SH +10 SS +13027 45878 MT +(, such) +69 W( that)68 W +/Times-Italic SF +17565 XM +(i)SH +/Times-Roman SF +17993 XM +(<)SH +/Times-Italic SF +18707 XM +(j)SH +/Symbol SF +19135 XM +(\243)SH +/Times-Italic SF +19834 XM +(k)SH +/Times-Roman SF +20596 XM +(or)SH +/Times-Italic SF +21747 XM +(i)SH +/Times-Roman SF +22175 XM +(>)SH +/Times-Italic SF +22889 XM +(j)SH +/Symbol SF +23317 XM +(\263)SH +/Times-Italic SF +24016 XM +(k)SH +/Times-Roman SF +(. Assuming) +386 W( both)68 W +/Times-Italic SF +31816 XM +(G)SH +8 /Times-Roman AF +32538 46223 MT +(1)SH +10 SS +33256 45878 MT +(and)SH +/Times-Italic SF +35018 XM +(G)SH +8 /Times-Roman AF +35740 46223 MT +(2)SH +10 SS +36458 45878 MT +(are reduced, the graph resulting from these)68 W +7200 47429 MT +(replacements is also reduced, and we can even avoid applying the reduction algorithm.) +13 W( While) +277 W( this technique applies)14 W +7200 48806 MT +(only under restricted conditions, we have found it a worthwhile optimization.)SH +11 /Times-Bold AF +7200 52423 MT +(4.6. Satisfy)SH +10 /Times-Roman AF +8200 53800 MT +(There are many questions) +187 W( one could ask about the satisfying set)186 W +/Times-Italic SF +35995 XM +(S)SH +8 SS +36495 54145 MT +(f)SH +10 /Times-Roman AF +37303 53800 MT +(for a function, including the number of)186 W +7200 55177 MT +(elements, a listing of the elements, or perhaps just a single element.) +42 W( As) +335 W( can be seen from Table 1, these operations)43 W +7200 56554 MT +(are performed by algorithms of widely varying complexity. A single element can be found in time proportional to)3 W +/Times-Italic SF +53250 XM +(n)SH +/Times-Roman SF +(,)SH +7200 57931 MT +(the number of function arguments, assuming the graph is reduced. Considering that the value of) +27 W( an element in)28 W +/Times-Italic SF +52183 XM +(S)SH +8 SS +52683 58276 MT +(f)SH +10 /Times-Roman AF +53333 57931 MT +(is)SH +7200 59308 MT +(specified by a) +18 W( bit sequence of length)17 W +/Times-Italic SF +22179 XM +(n)SH +/Times-Roman SF +(, this algorithm is optimal. We can list all elements of)17 W +/Times-Italic SF +44827 XM +(S)SH +8 SS +45327 59653 MT +(f)SH +10 /Times-Roman AF +45966 59308 MT +(in time proportional)17 W +7200 60685 MT +(to)SH +/Times-Italic SF +8239 XM +(n)SH +/Times-Roman SF +9000 XM +(times the) +11 W( number of elements, which again is optimal. However, this is generally not a wise thing to do---many)12 W +7200 62062 MT +(functions that are represented by small graphs have very large satisfying sets. For example, the) +187 W( function)186 W +/Times-Bold SF +52397 XM +(1)SH +/Times-Roman SF +53333 XM +(is)SH +7200 63439 MT +(represented by a graph with one vertex, yet) +31 W( all 2)32 W +8 /Times-Italic AF +26754 63094 MT +(n)SH +10 /Times-Roman AF +27436 63439 MT +(possible combinations of argument values are in its satisfying set.)32 W +10800 50 7200 66366 UL +6 SS +8000 67705 MT +(8)SH +8 /Times-Bold AF +8300 68014 MT +(Update:)SH +/Times-Roman SF +11401 XM +(Jain)SH +/Times-Italic SF +13035 XM +(et al.)146 W +/Times-Roman SF +15126 XM +(\050J. Jain,)146 W +/Times-Italic SF +18083 XM +(et al.)146 W +/Times-Roman SF +(, "Analysis of Composition Complexity) +146 W( and How to Obtain Smaller Canonical Graphs",)147 W +/Times-Italic SF +49965 XM +(37th Design)147 W +7200 69083 MT +(Automation Conference)44 W +/Times-Roman SF +(, 2000, pp. 681-686.\051 have shown an example for which the resulting graph is of size O) +150 44 WX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Times-Roman SF +44864 69428 MT +(1)SH +/Symbol SF +45264 69083 MT +(|)SH +/Times-Roman SF +45424 68738 MT +(2)SH +/Symbol SF +45824 69083 MT +(\327|)SH +/Times-Italic SF +(G)SH +/Times-Roman SF +46762 69428 MT +(2)SH +/Symbol SF +47162 69083 MT +(|)SH +/Times-Roman SF +(\051. In) +287 W( the worst case,)43 W +7200 70152 MT +(the algorithm)SH +/Times-Italic SF +11686 XM +(does)SH +/Times-Roman SF +13352 XM +(have a quadratic dependence on the size of its first argument.)SH +6 SS +8000 71691 MT +(9)SH +8 /Times-Bold AF +8300 72000 MT +(Update:)SH +/Times-Roman SF +11255 XM +(Elena Dubrova \050private correspondence, Feb., 2000\051 pointed out that this approach can yield a non-reduced graph.)SH +ES +%%Page: 17 17 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(17)SH +9200 7886 MT +(function)SH +/Times-Italic SF +12783 XM +(Compose)SH +/Times-Roman SF +(\050v1, v2: vertex; i:integer\051: vertex)SH +11718 8991 MT +(var T: array[1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +18693 9336 MT +(1)SH +10 /Symbol AF +19093 8991 MT +(|)SH +/Times-Roman SF +(, 1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +21715 9336 MT +(1)SH +10 /Symbol AF +22115 8991 MT +(|)SH +/Times-Roman SF +(, 1..)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +8 /Times-Roman AF +24737 9336 MT +(2)SH +10 /Symbol AF +25137 8991 MT +(|)SH +/Times-Roman SF +(] of vertex;)SH +11718 11201 MT +({Recursive routine to implement)SH +/Times-Italic SF +25086 XM +(Compose)SH +/Times-Roman SF +(})SH +11718 12306 MT +(function)SH +/Times-Italic SF +15301 XM +(Compose-step)SH +/Times-Roman SF +(\050vlow1, vhigh1, v2: vertex\051: vertex;)SH +11718 13411 MT +(begin)SH +14236 14516 MT +({Perform restrictions})SH +14236 15621 MT +(if vlow1.index = i then vlow1 := vlow1.low;)SH +14236 16726 MT +(if vhigh1.index = i then vhigh1 := vhigh1.high;)SH +14236 17831 MT +({Apply operation)SH +/Times-Italic SF +21493 XM +(ITE)SH +/Times-Roman SF +23143 XM +(})SH +14236 18936 MT +(u := T[vlow1.id, vhigh1.id, v2.id];)SH +14236 20041 MT +(if u)SH +/Symbol SF +15997 XM +(\271)SH +/Times-Roman SF +16946 XM +(null then return\050u\051; {have already evaluated})SH +14236 21146 MT +(u := new vertex record; u.mark := false;)SH +14236 22251 MT +(T[vlow1.id, vhigh1.id, v2.id] := u; {add vertex to table})SH +14236 23356 MT +(u.value := \050)SH +/Symbol SF +18977 XM +(\330)SH +/Times-Roman SF +20090 XM +(v2.value)SH +/Symbol SF +23756 XM +(\327)SH +/Times-Roman SF +24256 XM +(vlow1.value\051 + \050v2.value)SH +/Symbol SF +34568 XM +(\327)SH +/Times-Roman SF +35068 XM +(vhigh1.value\051;)SH +14236 24461 MT +(if u.value)SH +/Symbol SF +18413 XM +(\271)SH +/Times-Roman SF +19362 XM +(X)SH +14236 25566 MT +(then begin {create terminal vertex})SH +16754 26671 MT +(u.index := n+1; u.low := null; u.high := null;)SH +14236 27776 MT +(end)SH +14236 28881 MT +(else begin {create nonterminal and evaluate further down})SH +16754 29986 MT +(u.index :=)SH +/Times-Italic SF +21068 XM +(Min)SH +/Times-Roman SF +(\050vlow1.index, vhigh1.index, v2.index\051;)SH +16754 31091 MT +(if vlow1.index = u.index)SH +19272 32196 MT +(then begin vll1 := vlow1.low; vlh1 := vlow1.high end)SH +19272 33301 MT +(else begin vll1 := vlow1; vlh1 := vlow1 end;)SH +16754 34406 MT +(if vhigh1.index = u.index)SH +19272 35511 MT +(then begin vhl1 := vhigh1.low; vhh1 := vhigh1.high end)SH +19272 36616 MT +(else begin vhl1 := vhigh1; vhh1 := vhigh1 end;)SH +16754 37721 MT +(if v2.index = u.index)SH +19272 38826 MT +(then begin vlow2 := v2.low; vhigh2 := v2.high end)SH +19272 39931 MT +(else begin vlow2 := v2; vhigh2 := v2 end;)SH +16754 41036 MT +(u.low :=)SH +/Times-Italic SF +20346 XM +(Compose-step)SH +/Times-Roman SF +(\050vll1, vhl1, vlow2\051;)SH +16754 42141 MT +(u.high :=)SH +/Times-Italic SF +20624 XM +(Compose-step)SH +/Times-Roman SF +(\050vlh1, vhh1, vhigh2\051;)SH +14236 43246 MT +(end;)SH +14236 44351 MT +(return\050u\051;)SH +11718 45456 MT +(end;)SH +9200 47666 MT +(begin {Main) +500 W( routine})SH +11718 48771 MT +(Initialize all elements of T to null;)SH +11718 49876 MT +(u :=)SH +/Times-Italic SF +13560 XM +(Compose-step)SH +/Times-Roman SF +(\050v1, v1, v2\051;)SH +11718 50981 MT +(return\050)SH +/Times-Italic SF +(Reduce)SH +/Times-Roman SF +(\050u\051\051;)SH +9200 52086 MT +(end;)SH +23100 53696 MT +(Figure 8.Implementation of)SH +/Times-Italic SF +34377 XM +(Compose)SH +/Times-Roman SF +7200 55487 MT +(Hence, care) +1 W( must be exercised in invoking this algorithm. If we wish to find an element of the satisfying set obeying)SH +7200 56864 MT +(some property, it can be very inefficient to enumerate all elements of the satisfying) +22 W( set and then pick out an element)23 W +7200 58241 MT +(with the desired characteristics.) +81 W( Instead,) +410 W( we should specify this property in terms of a Boolean function, compute)80 W +7200 59618 MT +(the Boolean product of this function and the original function,) +86 W( and then use the procedure)87 W +/Times-Italic SF +44546 XM +(Satisfy-one)SH +/Times-Roman SF +49327 XM +(to select an)87 W +7200 60995 MT +(element. Finally,) +388 W( we can compute the size of the satisfying set by an algorithm of time) +69 W( proportional to the size of)68 W +7200 62372 MT +(the graph \050assuming integer operations of sufficient precision) +69 W( can be performed in constant time.\051 In general, it is)70 W +7200 63749 MT +(much faster to apply this algorithm than to enumerate all elements of the satisfying set and count them.)SH +8200 66228 MT +(The procedure)11 W +/Times-Italic SF +14275 XM +(Satisfy-one)SH +/Times-Roman SF +18980 XM +(shown in Figure 9 is called with the root of the graph) +11 W( and an array x initialized to some)10 W +7200 67605 MT +(arbitrary pattern of 0's and 1's. It returns the value false) +35 W( if the function is unsatisfiable \050)36 W +/Times-Italic SF +(S)SH +8 SS +43627 67950 MT +(f)SH +10 /Times-Roman AF +44249 67605 MT +(=)SH +/Symbol SF +45063 XM +(\306)SH +/Times-Roman SF +(\051, and the value true)36 W +7200 68982 MT +(if it is. In the latter case, the entries) +5 W( in the array are set to a set of values denoting some element in the satisfying set.)4 W +7200 70359 MT +(This procedure utilizes) +68 W( a classic depth-first search with backtracking scheme to find a terminal vertex in the graph)69 W +7200 71736 MT +(having value 1. This procedure will work on any function graph.) +43 W( However,) +335 W( when called on an unreduced function)42 W +ES +%%Page: 18 18 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(18)SH +7200 7886 MT +(graph, this could require time exponential in)31 W +/Times-Italic SF +25415 XM +(n)SH +/Times-Roman SF +26196 XM +(\050consider a complete binary tree where only the final terminal) +31 W( vertex)32 W +7200 9263 MT +(in the search has value 1\051. For a reduced graph, however, we can assume the following property:)SH +/Times-Bold SF +10200 11507 MT +(Lemma 3:)8 W +/Times-Roman SF +14826 XM +(Every nonterminal vertex in a reduced function graph has a terminal vertex with value 1 as) +8 W( a)7 W +9200 12612 MT +(descendant.)SH +7200 14132 MT +(The procedure will only backtrack at some vertex when) +2 W( the first child it tries is a terminal vertex with value 0, and in)3 W +7200 15509 MT +(this case it is guaranteed to succeed for the second child. Thus, the complexity of the algorithm is O) +150 0 WX(\050)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051.)SH +9200 17300 MT +(function)SH +/Times-Italic SF +12783 XM +(Satisfy-one)SH +/Times-Roman SF +(\050v: vertex; var x: array[1..n] of integer\051: boolean)SH +9200 18405 MT +(begin)SH +11718 19510 MT +(if v.value = 0 then return\050false\051; {failure})SH +11718 20615 MT +(if v.value = 1 then return\050true\051; {success})SH +11718 21720 MT +(x[i] := 0;)SH +11718 22825 MT +(if)SH +/Times-Italic SF +12579 XM +(Satisfy-one)SH +/Times-Roman SF +(\050v.low, x\051 then return\050true\051;)SH +11718 23930 MT +(x[i] := 1;)SH +11718 25035 MT +(return\050Satisfy-one\050v.high, x\051\051)SH +9200 26140 MT +(end;)SH +22739 27750 MT +(Figure 9.Implementation of)SH +/Times-Italic SF +34016 XM +(Satisfy-one)SH +/Times-Roman SF +8200 30229 MT +(To enumerate) +7 W( all elements of the satisfying set, we can perform an exhaustive search of the graph, printing out the)6 W +7200 31606 MT +(element corresponding to the current path every time we reach a terminal vertex with value 1. The procedure)167 W +/Times-Italic SF +7200 32983 MT +(Satisfy-all)SH +/Times-Roman SF +11688 XM +(shown in Figure 10) +182 W( implements this method. This procedure has three arguments: the index of the)181 W +7200 34360 MT +(current function argument in the enumeration, the root vertex of the subgraph being searched, and an array)259 W +7200 35737 MT +(describing the state of the search. It) +22 W( is called at the top level with index 1, the root vertex of the graph, and an array)21 W +7200 37114 MT +(with arbitrary) +106 W( initialization. The effect of the procedure when invoked with index)107 W +/Times-Italic SF +41639 XM +(i)SH +/Times-Roman SF +(, vertex)107 W +/Times-Italic SF +45380 XM +(v)SH +/Times-Roman SF +(, and with the array)107 W +7200 38491 MT +(having its first)SH +/Times-Italic SF +13228 XM +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1 elements equal to)SH +/Times-Italic SF +22054 XM +(b)SH +8 /Times-Roman AF +22554 38836 MT +(1)SH +10 SS +22954 38491 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +25704 38836 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +27015 38491 MT +(is to enumerate all elements in the set)SH +9200 40396 MT +({\050)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +10513 40741 MT +(1)SH +10 SS +10913 40396 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +13663 40741 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +14724 40396 MT +(,)SH +/Times-Italic SF +(x)SH +8 SS +15418 40741 MT +(i)SH +10 /Times-Roman AF +15640 40396 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +18334 40741 MT +(n)SH +10 /Times-Roman AF +18734 40396 MT +(\051)SH +/Symbol SF +19217 XM +(|)SH +/Times-Italic SF +19567 XM +(f)SH +8 SS +19845 40741 MT +(v)SH +10 /Times-Roman AF +20200 40396 MT +(\050)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +21033 40741 MT +(1)SH +10 SS +21433 40396 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +24183 40741 MT +(i)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +25244 40396 MT +(,)SH +/Times-Italic SF +(x)SH +8 SS +25938 40741 MT +(i)SH +10 /Times-Roman AF +26160 40396 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +28854 40741 MT +(n)SH +10 /Times-Roman AF +29254 40396 MT +(\051 = 1}.)SH +8200 42875 MT +(As with the previous algorithm, this procedure will work for any function) +216 W( graph, but it could require time)215 W +7200 44252 MT +(exponential in)50 W +/Times-Italic SF +13244 XM +(n)SH +/Times-Roman SF +14044 XM +(for an unreduced graph regardless of the size of the satisfying set \050consider a complete binary tree)50 W +7200 45629 MT +(with all terminal vertices having value 0.\051) +1 W( For) +250 W( a reduced graph, however, we are guaranteed that the search will only)SH +7200 47006 MT +(fail when the procedure is called on a terminal vertex with value 0, and in this case the recursive call to the other)81 W +7200 48383 MT +(child will succeed.) +40 W( Hence) +329 W( at least half of the recursive calls to)39 W +/Times-Italic SF +32926 XM +(Satisfy-all)SH +/Times-Roman SF +37271 XM +(generate at least one new argument value)39 W +7200 49760 MT +(to some element in the satisfying set, and the overall complexity is O) +150 0 WX(\050)SH +/Times-Italic SF +(n)SH +/Symbol SF +(\327|)SH +/Times-Italic SF +(S)SH +8 SS +36770 50105 MT +(f)SH +10 /Symbol AF +37142 49760 MT +(|)SH +/Times-Roman SF +(\051.)SH +8200 52239 MT +(Finally, to) +49 W( compute the size of the satisfying set, we assign a value)50 W +/Symbol SF +35763 XM +(a)SH +8 /Times-Italic AF +36394 52584 MT +(v)SH +10 /Times-Roman AF +37049 52239 MT +(to each vertex)50 W +/Times-Italic SF +43058 XM +(v)SH +/Times-Roman SF +43802 XM +(in the graph according to)50 W +7200 53616 MT +(the following recursive formula:)SH +9200 54955 MT +(1.)SH +10200 XM +(If)SH +/Times-Italic SF +11116 XM +(v)SH +/Times-Roman SF +11810 XM +(is a terminal vertex:)SH +/Symbol SF +20031 XM +(a)SH +8 /Times-Italic AF +20662 55300 MT +(v)SH +10 /Times-Roman AF +21267 54955 MT +(=)SH +/Times-Italic SF +22081 XM +(value)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +9200 56713 MT +(2.)SH +10200 XM +(If)SH +/Times-Italic SF +11116 XM +(v)SH +/Times-Roman SF +11810 XM +(is a nonterminal vertex)SH +8 SS +21003 56368 MT +(10)SH +10 SS +21803 56713 MT +(:)SH +/Symbol SF +12200 58712 MT +(a)SH +8 /Times-Italic AF +12831 59057 MT +(v)SH +10 /Times-Roman AF +13872 58712 MT +(=)SH +/Symbol SF +15122 XM +(a)SH +8 /Times-Italic AF +15753 59057 MT +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 /Symbol AF +17796 58712 MT +(\327)SH +/Times-Roman SF +(2)SH +8 /Times-Italic AF +18546 58367 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051)SH +/Symbol SF +(-)SH +/Times-Italic SF +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 SS +26161 58712 MT +(+)SH +/Symbol SF +26975 XM +(a)SH +8 /Times-Italic AF +27606 59057 MT +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 /Symbol AF +29915 58712 MT +(\327)SH +/Times-Roman SF +(2)SH +8 /Times-Italic AF +30665 58367 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051)SH +/Symbol SF +(-)SH +/Times-Italic SF +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +10 SS +9200 60470 MT +(3.)SH +10200 XM +(where a terminal vertex has index)SH +/Times-Italic SF +23974 XM +(n)SH +/Times-Roman SF +(+1.)SH +8200 62949 MT +(This computation can) +189 W( be performed by a procedure that traverses the graph in the manner of the procedure)190 W +/Times-Italic SF +7200 64326 MT +(Traverse)SH +/Times-Roman SF +(. The) +476 W( formula is applied only once for each vertex in the) +113 W( graph, and hence the total time complexity is)112 W +10800 50 7200 67610 UL +6 SS +8000 68949 MT +(10)SH +8 /Times-Bold AF +8600 69258 MT +(Update:)SH +/Times-Roman SF +11865 XM +(Adnan Darwiche of UCLA \050private correspondence, 2001\051 observed that the formula given in the paper) +55 W( is incorrect. Here is the)54 W +7200 70182 MT +(correct one:)SH +/Symbol SF +8800 72000 MT +(a)SH +/Times-Italic SF +9305 72345 MT +(v)SH +/Times-Roman SF +10209 72000 MT +(=)SH +/Symbol SF +11209 XM +(a)SH +/Times-Italic SF +11714 72345 MT +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +13757 72000 MT +(\327)SH +/Times-Roman SF +(2)SH +/Times-Italic SF +14357 71655 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(low)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051)SH +/Symbol SF +(-)SH +/Times-Italic SF +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +22761 72000 MT +(+)SH +/Symbol SF +23412 XM +(a)SH +/Times-Italic SF +23917 72345 MT +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +26226 72000 MT +(\327)SH +/Times-Roman SF +(2)SH +/Times-Italic SF +26826 71655 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(high)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051\051)SH +/Symbol SF +(-)SH +/Times-Italic SF +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +ES +%%Page: 19 19 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(19)SH +9200 7886 MT +(procedure)SH +/Times-Italic SF +13448 XM +(Satisfy-all)SH +/Times-Roman SF +(\050i: integer; v: vertex; x: array[1..n] of integer\051:)SH +9200 8991 MT +(begin)SH +11718 10096 MT +(if v.value = 0 then return; {failure})SH +11718 11201 MT +(if i = n+1 and v.value = 1)SH +11718 12306 MT +(then begin {success})SH +14236 13411 MT +(Print element x[1],...,x[n];)SH +14236 14516 MT +(return;)SH +11718 15621 MT +(end;)SH +11718 16726 MT +(if v.index > i)SH +11718 17831 MT +(then begin {function independent of)SH +/Times-Italic SF +26446 XM +(x)SH +8 SS +26890 18176 MT +(i)SH +10 /Times-Roman AF +27112 17831 MT +(})SH +14236 19037 MT +(x[i] := 0;)SH +/Times-Italic SF +18050 XM +(Satisfy-all)SH +/Times-Roman SF +(\050i+1, v, x\051;)SH +14236 20142 MT +(x[i] := 1;)SH +/Times-Italic SF +18050 XM +(Satisfy-all)SH +/Times-Roman SF +(\050i+1, v, x\051;)SH +11718 21247 MT +(end)SH +11718 22352 MT +(else begin {function depends on)SH +/Times-Italic SF +24835 XM +(x)SH +8 SS +25279 22697 MT +(i)SH +10 /Times-Roman AF +25501 22352 MT +(})SH +14236 23558 MT +(x[i] := 0;)SH +/Times-Italic SF +18050 XM +(Satisfy-all)SH +/Times-Roman SF +(\050i+1, v.low, x\051;)SH +14236 24663 MT +(x[i] := 1;)SH +/Times-Italic SF +18050 XM +(Satisfy-all)SH +/Times-Roman SF +(\050i+1, v.high, x\051;)SH +11718 25768 MT +(end;)SH +9200 26873 MT +(end;)SH +22683 28483 MT +(Figure 10.Implementation of)SH +/Times-Italic SF +34460 XM +(Satisfy-all)SH +/Times-Roman SF +7200 30388 MT +(O) +150 MX(\050)SH +/Symbol SF +(|)SH +/Times-Italic SF +(G)SH +/Symbol SF +(|)SH +/Times-Roman SF +(\051 Once we have computed these values for a graph with root)SH +/Times-Italic SF +33854 XM +(v)SH +/Times-Roman SF +(, we compute the size of the satisfying set as)SH +/Symbol SF +9200 32387 MT +(|)SH +/Times-Italic SF +(S)SH +8 SS +9900 32732 MT +(f)SH +10 /Symbol AF +10272 32387 MT +(|)SH +/Times-Roman SF +11158 XM +(=)SH +/Symbol SF +12408 XM +(a)SH +8 /Times-Italic AF +13039 32732 MT +(v)SH +10 /Symbol AF +13394 32387 MT +(\327)SH +/Times-Roman SF +(2)SH +8 /Times-Italic AF +14144 32042 MT +(index)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(v)SH +/Times-Roman SF +(\051)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +17602 32387 MT +(.)SH +12 /Times-Bold AF +7200 36071 MT +(5. Experimental Results)SH +10 /Times-Roman AF +8200 37448 MT +(As with all other known algorithms for solving NP-hard problems, our algorithms have) +24 W( a worst-case performance)25 W +7200 38825 MT +(that is unacceptable for all but the smallest problems. We hope that our approach will be practical for) +42 W( a reasonable)41 W +7200 40202 MT +(class of applications, but this can only be demonstrated experimentally. We have) +29 W( already shown that the size of the)30 W +7200 41579 MT +(graph representing a function can depend heavily) +41 W( on the ordering of the input variables, and that our algorithms are)40 W +7200 42956 MT +(quite efficient as) +52 W( long as the functions are represented by graphs of reasonable size. Hence, the major questions to)53 W +7200 44333 MT +(be answered by our) +69 W( experimental investigation are: how can an appropriate input ordering be chosen, and given a)68 W +7200 45710 MT +(good ordering how large are the graphs encountered in typical applications.)SH +8200 48189 MT +(We have implemented the algorithms described in this paper and have applied) +65 W( them to problems in logic design)66 W +7200 49566 MT +(verification, test) +29 W( pattern generation, and combinatorics. On the whole, our experience has been quite favorable. By)28 W +7200 50943 MT +(analyzing the problem domain, we can generally develop strategies for choosing a good ordering of) +158 W( the inputs.)159 W +7200 52320 MT +(Furthermore, it is not necessary to find)181 W +/Times-Italic SF +24243 XM +(the)SH +/Times-Roman SF +25896 XM +(optimal ordering.) +181 W( Many) +611 W( orderings will produce acceptable results.)180 W +7200 53697 MT +(Functions rarely require graphs of size exponential in the number of inputs, as long as) +54 W( a reasonable ordering of the)55 W +7200 55074 MT +(inputs has been chosen. In addition, the algorithms) +84 W( are quite fast, remaining practical for graphs with as many as)83 W +7200 56451 MT +(20,000 vertices.)SH +8200 58930 MT +(For this paper,) +100 W( we consider the problem of verifying that the implementation of a logic function \050in terms of a)101 W +7200 60307 MT +(combinational logic gate network\051 satisfies its specification) +25 W( \050in terms of Boolean expressions.\051 As examples we use)24 W +7200 61684 MT +(a family of Arithmetic Logic Unit \050ALU\051 designs constructed) +70 W( from 74181 and 74182 TTL integrated circuits) +71 W( [15].)SH +7200 63061 MT +(The '181 implements a 4 bit ALU slice, while the '182 implements a lookahead carry generator. These chips) +10 W( can be)9 W +7200 64438 MT +(combined to create an ALU with any word size that is a) +85 W( multiple of 4 bits. An ALU with an)86 W +/Times-Italic SF +46157 XM +(n)SH +/Times-Roman SF +46993 XM +(bit word size has)86 W +7200 65815 MT +(6+2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +9566 XM +(inputs: 5) +354 W( control inputs labeled)52 W +/Times-Italic SF +22767 XM +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +24128 66160 MT +(0)SH +10 SS +24528 65815 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +25167 66160 MT +(1)SH +10 SS +25567 65815 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +26206 66160 MT +(2)SH +10 SS +26606 65815 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +27245 66160 MT +(3)SH +10 SS +27947 65815 MT +(to select the) +52 W( ALU function, a carry input labeled)51 W +/Times-Italic SF +48015 XM +(cin)SH +/Times-Roman SF +(, and 2 data)51 W +7200 67192 MT +(words of)130 W +/Times-Italic SF +11237 XM +(n)SH +/Times-Roman SF +12117 XM +(bits each, labeled)130 W +/Times-Italic SF +19672 XM +(a)SH +8 /Times-Roman AF +20172 67537 MT +(0)SH +10 SS +20572 67192 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +23322 67537 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +24941 67192 MT +(and)SH +/Times-Italic SF +26765 XM +(b)SH +8 /Times-Roman AF +27265 67537 MT +(0)SH +10 SS +27665 67192 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +30415 67537 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +31654 67192 MT +(. It) +510 W( produces)130 W +/Times-Italic SF +37645 XM +(n)SH +/Times-Roman SF +(+2 outputs:)130 W +/Times-Italic SF +43192 XM +(n)SH +/Times-Roman SF +44072 XM +(function outputs labeled)131 W +/Times-Italic SF +7200 68569 MT +(f)SH +8 /Times-Roman AF +7478 68914 MT +(0)SH +10 SS +7878 68569 MT +(, . . . ,)SH +/Times-Italic SF +(f)SH +8 SS +10406 68914 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +11645 68569 MT +(, a carry output labeled)163 W +/Times-Italic SF +21901 XM +(cout)SH +/Times-Roman SF +(, and a comparison output labeled)162 W +/Times-Italic SF +38343 XM +(A)SH +/Times-Roman SF +(=)SH +/Times-Italic SF +(B)SH +/Times-Roman SF +40541 XM +(\050the logical)162 W +8 SS +45642 XM +(AND)SH +10 SS +47788 XM +(of the function)162 W +7200 69946 MT +(outputs.\051)SH +ES +%%Page: 20 20 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(20)SH +/Times-Bold SF +9644 7886 MT +(Word Size)SH +20018 XM +(Gates)SH +28795 XM +(Patterns)SH +37057 XM +(CPU Minutes)SH +/Times-Italic SF +46885 XM +(A)SH +/Times-Roman SF +(=)SH +/Times-Italic SF +(B)SH +/Times-Bold SF +48921 XM +(Graph)SH +/Times-Roman SF +11630 10135 MT +(4)SH +20740 XM +(52)SH +28850 XM +(1.6)SH +/Symbol SF +30250 XM +(\264)SH +/Times-Roman SF +30949 XM +(10)SH +8 SS +31949 9790 MT +(4)SH +10 SS +39335 10135 MT +(1.1)SH +48570 XM +(197)SH +11630 11512 MT +(8)SH +20490 XM +(123)SH +28850 XM +(4.2)SH +/Symbol SF +30250 XM +(\264)SH +/Times-Roman SF +30949 XM +(10)SH +8 SS +31949 11167 MT +(6)SH +10 SS +39335 11512 MT +(2.3)SH +48570 XM +(377)SH +11380 12889 MT +(16)SH +20490 XM +(227)SH +28650 XM +(2.7)SH +/Symbol SF +30050 XM +(\264)SH +/Times-Roman SF +30749 XM +(10)SH +8 SS +31749 12544 MT +(11)SH +10 SS +39335 12889 MT +(6.3)SH +48570 XM +(737)SH +11380 14266 MT +(32)SH +20490 XM +(473)SH +28650 XM +(1.2)SH +/Symbol SF +30050 XM +(\264)SH +/Times-Roman SF +30749 XM +(10)SH +8 SS +31749 13921 MT +(21)SH +10 SS +39085 14266 MT +(22.8)SH +48320 XM +(1457)SH +11380 15643 MT +(64)SH +20490 XM +(927)SH +28650 XM +(2.2)SH +/Symbol SF +30050 XM +(\264)SH +/Times-Roman SF +30749 XM +(10)SH +8 SS +31749 15298 MT +(40)SH +10 SS +39085 15643 MT +(95.8)SH +48320 XM +(2897)SH +23296 17253 MT +(Table 2.ALU Verification Examples)SH +8200 19732 MT +(For our experiments, we derived the functions for) +194 W( the two chips from their gate-level descriptions and then)195 W +7200 21109 MT +(composed these functions to form the different ALU's according to the chip-level interconnections) +130 W( in the circuit)129 W +7200 22486 MT +(manual. We) +594 W( then compared these circuit functions to functions derived) +172 W( from Boolean expressions obtained by)173 W +7200 23863 MT +(encoding the behavioral) +15 W( specification in the circuit manual. We succeeded in verifying ALU's with word sizes of 4,)14 W +7200 25240 MT +(8, 16, 32, and 64 bits. The performance of our program for this task is summarized Table 2. These data were)134 W +7200 26617 MT +(measured with the best ordering we were able to find, which happened to be the first one we tried:) +4 W( first) +256 W( the 5 control)3 W +7200 27994 MT +(inputs, then the carry input, and then) +35 W( an interleaving of the two data words from the least significant to the most. In)36 W +7200 29371 MT +(this table, the number of gates is defined as the number of) +68 W( logic gates in the schematic diagrams for the two chips)67 W +7200 30748 MT +(times the number of each chip used.) +109 W( The) +470 W( number of patterns equals the number of different input combinations.)110 W +7200 32125 MT +(CPU time is expressed in minutes as measured on a Digital Equipment Corporation VAX 11/780 \050a 1) +245 W( MIP)244 W +7200 33502 MT +(machine.\051 The) +296 W( times given are for complete verification, i.e.) +23 W( to construct the functions from both the circuit and the)24 W +7200 34879 MT +(behavioral descriptions and to establish their equivalence. The final column shows the size) +19 W( of the reduced graph for)18 W +7200 36256 MT +(the)SH +/Times-Italic SF +8672 XM +(A)SH +/Times-Roman SF +(=)SH +/Times-Italic SF +(B)SH +/Times-Roman SF +10708 XM +(output. In) +250 W( all cases, this was the largest graph generated.)SH +8200 38735 MT +(As can be seen, the time required to verify these circuits is quite reasonable,) +38 W( in part because the basic procedures)39 W +7200 40112 MT +(are fast. Amortizing the time used for memory management, for the user interface, and for reducing the) +113 W( graphs,)112 W +7200 41489 MT +(each call to the evaluation routines)71 W +/Times-Italic SF +21790 XM +(Apply-step)SH +/Times-Roman SF +26388 XM +(and)SH +/Times-Italic SF +28153 XM +(Compose-step)SH +/Times-Roman SF +34140 XM +(requires around 3 milliseconds.) +71 W( For) +394 W( example, in)72 W +7200 42866 MT +(verifying the 64) +52 W( bit ALU, these two procedures were called over 1.6)51 W +/Symbol SF +35212 XM +(\264)SH +/Times-Roman SF +35911 XM +(10)SH +8 SS +36911 42521 MT +(6)SH +10 SS +37612 42866 MT +(times. The) +352 W( total verification time grows)51 W +7200 44243 MT +(as the square of the word size. This is as good as can) +83 W( be expected: both the number of gates and the sizes of the)84 W +7200 45620 MT +(graphs being operated on grow linearly with the word size, and) +119 W( the total execution time grows as the product of)118 W +7200 46997 MT +(these two factors. This quadratic growth is far superior) +191 W( to the exponential growth that would be required for)192 W +7200 48374 MT +(exhaustive analysis.) +1 W( For) +250 W( example, suppose that at the time the universe first formed \050about 20 billion years ago [16]\051)SH +7200 49751 MT +(we started analyzing the 32 bit ALU exhaustively at) +30 W( a rate of one pattern every microsecond. By now we would be)31 W +7200 51128 MT +(about half way through! For the 64 bit ALU, the advantage over exhaustive analysis is even greater.)SH +8200 53607 MT +(These ALU circuits provide an interesting test case for evaluating different input) +310 W( orderings, because the)309 W +7200 54984 MT +(successive bits of the function output word are) +55 W( functions of increasingly more variables. Figure 11 shows how the)56 W +7200 56361 MT +(sizes of these graphs) +213 W( depend on the ordering of circuit inputs. The best case was obtained for the ordering:)212 W +/Times-Italic SF +7200 57738 MT +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +8561 58083 MT +(0)SH +10 SS +8961 57738 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +9600 58083 MT +(1)SH +10 SS +10000 57738 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +10639 58083 MT +(2)SH +10 SS +11039 57738 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +11678 58083 MT +(3)SH +10 SS +12078 57738 MT +(,)SH +/Times-Italic SF +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +14300 58083 MT +(0)SH +10 SS +14700 57738 MT +(,)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +15450 58083 MT +(0)SH +10 SS +15850 57738 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +18600 58083 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Times-Italic AF +19839 57738 MT +(b)SH +8 SS +20339 58083 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +21578 57738 MT +(. This) +290 W( ordering is also what one) +20 W( would choose for a bit-serial implementation of)21 W +7200 59115 MT +(the ALU: first read) +11 W( in the bits describing the function to be computed, and then read in the successive bits of the two)10 W +7200 60492 MT +(data words starting with the least significant bits. Hence, our) +114 W( bit-serial computer analogy presented in Section 3)115 W +7200 61869 MT +(guides us to the best solution. The next best case) +891 W( tested occurred with the ordering:)890 W +/Times-Italic SF +7200 63246 MT +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +8561 63591 MT +(0)SH +10 SS +8961 63246 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +9600 63591 MT +(1)SH +10 SS +10000 63246 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +10639 63591 MT +(2)SH +10 SS +11039 63246 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +11678 63591 MT +(3)SH +10 SS +12078 63246 MT +(,)SH +/Times-Italic SF +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 SS +14300 63591 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +15539 63246 MT +(,)SH +/Times-Italic SF +(b)SH +8 SS +16289 63591 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +17528 63246 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +20278 63591 MT +(0)SH +10 /Times-Italic AF +20678 63246 MT +(b)SH +8 /Times-Roman AF +21178 63591 MT +(0)SH +10 SS +21578 63246 MT +(, i.e. the same as before) +87 W( but with the data ordered with the most significant bit)88 W +7200 64623 MT +(first. This) +562 W( ordering represents an alternative but often) +156 W( successful strategy: order the bits in decreasing order of)155 W +7200 66000 MT +(importance. The)302 W +/Times-Italic SF +14332 XM +(i)SH +/Times-Roman SF +14760 XM +(th bit of the output) +26 W( word depends more strongly on the)27 W +/Times-Italic SF +37135 XM +(i)SH +/Times-Roman SF +37563 XM +(th bits of the input words than any lower)27 W +7200 67377 MT +(order bits. As can be seen, this strategy also works quite well. The next case shown) +103 W( is for an ordering with the)102 W +7200 68754 MT +(control inputs last:)65 W +/Times-Italic SF +15405 XM +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +17377 69099 MT +(0)SH +10 SS +17777 68754 MT +(,)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +18527 69099 MT +(0)SH +10 SS +18927 68754 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +21677 69099 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Times-Italic AF +22916 68754 MT +(b)SH +8 SS +23416 69099 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +24655 68754 MT +(,)SH +/Times-Italic SF +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +26266 69099 MT +(0)SH +10 SS +26666 68754 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +27305 69099 MT +(1)SH +10 SS +27705 68754 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +28344 69099 MT +(2)SH +10 SS +28744 68754 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +29383 69099 MT +(3)SH +10 SS +29783 68754 MT +(. This) +380 W( ordering) +65 W( could be expected to produce a rather poor)66 W +7200 70131 MT +(result, since) +78 W( the outputs depend strongly on the control inputs. However, the complexity of the graphs still grows)77 W +7200 71508 MT +(linearly, due to the fact that the number of control inputs is a constant. To explain this linear growth in) +24 W( terms of the)25 W +ES +%%Page: 21 21 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(21)SH +1 SS +25444 7200 32644 PB +%%BeginDocument: figs/alu.eps +%!PS-Adobe-3.0 EPSF-3.0 +%%BoundingBox: 0 0 401 256 +%%Title: (F:\\txt\\papers\\tc86\\figs\\alu.CNV) +%%Creator: Canvas (7.0.1) +%%CreationDate: (05/08/00 @ 04:42 PM) +%%DocumentProcSets: Deneba_CanvasCustomColors +%%+ CVDict +%%DocumentFonts: +%%DocumentNeededFonts: Arial-BoldMT +%%+ ArialMT + +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%EndComments +%%BeginProlog +%%BeginResource: procset CVDict +%%Title: Deneba standard graphics +%%Version: 1.02 +%%CreationDate: 10/14/99 +%%Copyright: (C) 1994, 1999 Deneba Systems. All Rights Reserved +userdict /CVDict 200 dict put +CVDict begin +/invalidcolortable? true def +/level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def +level2 not { /xshow { +0 currentpoint 5 -1 roll {pop pop moveto dup 1 add 3 1 roll exch dup 4 1 roll exch get 0 rmoveto currentpoint} exch kshow +pop pop pop pop} def } if +/initialize +{CVDict +{dup xcheck{bind}if pop pop}forall +currentscreen /_sp exch def /_sa exch def /_sf exch def +/DefaultScreen{_sf _sa /_sp load}def +/DefaultScreenProc{ /_sp load}def +/setstrokeadjust where{pop true}{false}ifelse +{/sa /setstrokeadjust load def +/li /lineto load def +/mt /moveto load def +/ct /curveto load def +/lh { currentpoint exch pop lineto}bind def +/lv { currentpoint pop exch lineto}bind def} +{/sa{{currentdict StrokeAdjustDict ne{StrokeAdjustDict begin}if}{ currentdict StrokeAdjustDict eq{end}if}ifelse} bind def +userdict /StrokeAdjustDict 100 dict put +StrokeAdjustDict begin +/_xl +{transform +.25 sub round .25 add exch +.25 sub round .25 add exch +itransform +}bind def +/li{ _xl lineto}bind def +/mt{ _xl moveto}bind def +/ct{ _xl curveto}bind def +/lh { dup _xl pop currentpoint exch pop lineto}bind def +/lv { dup _xl pop currentpoint pop exch lineto}bind def +/rp{_xl/cv_b xdf/cv_r xdf +_xl/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath +}bind def +end}ifelse}def +/terminate +{userdict /StrokeAdjustDict known {currentdict StrokeAdjustDict eq{end}if}if +currentdict CVDict eq{end}if}def +/mxMax 64 def +/mxStk [mxMax {matrix} repeat]def +/mxIdx 0 def +/stm{mxStk mxIdx 0 lt {/mxIdx 0 store}if mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def +/rtm{mxIdx 1 ge {/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}if }def +/scl { 72 div exch 72 div exch scale } def +/scli { 72 exch div exch 72 exch div exch scale } def +/_cvmtx matrix def +/_txtmtx matrix def +/_initmtx matrix def +/xdf{exch def}def +/min{2 copy gt{exch}if pop}def +/max{2 copy lt{exch}if pop}def +/CVvec 256 array def +/T true def +/F false def +/_id 0 def +/_encodindex 0 def +/_encodArr 3 array def +/_fd 0 def +/_textSave null def +/_letterSpace 0 def +/_spaceXtra 0 def +/_PageSave null def +/_penh 1.0 def +/_penv 1.0 def +/save0 null def +/_w 0 def +/_h 0 def +/_ds 0 def +/hct false def +/pw 0 def +/wdif 0 def +/temp 0 def +/currot 0 def +/savehandler null def +/ia 0 def +/ix 0 def +/ixn 1 def +/MultipleDataSources false def +/debug +{dup /$eps exch not def{/savehandler errordict /handleerror get def +errordict +/handleerror{$error begin +newerror{/newerror false def +showpage +72 72 scale +/Helvetica findfont .2 scalefont setfont +.25 9.6 moveto +(Error: )show +errorname +{dup type/stringtype ne +{(converted error name will end up in this string)cvs}if +show}exec +.25 9.4 moveto +(Offending Command: )show +/command load +{dup type/stringtype ne +{(converted command name will end up in this string)cvs}if +show}exec +.25 9.2 moveto +(Stack: )show +ostack +{.25 currentpoint .2 sub exch pop moveto +dup type/stringtype ne{dup type/integertype eq +1 index type/realtype eq or +1 index type/nametype eq or +not {type}if +(converted stack will end up in this string)cvs}if +show}forall +showpage +CVDict /savehandler get exec}if +end}put}if}def +/lc{setlinecap}def +/lj{setlinejoin}def +/lm{setmiterlimit}def +/lf{setflat}def +/cp{closepath}def +/np{newpath}def +/$psz 30 array def +/$pix 0 def +/gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def +/gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def +/li{lineto} def +/lh{currentpoint exch pop lineto}def +/lv{currentpoint pop exch lineto}def +/mt{moveto} def +/_cv_mh{currentpoint exch pop moveto}def +/_cv_mv{currentpoint pop exch moveto}def +/ct{curveto} def +/rt{4 -2 roll moveto +dup 0 exch rlineto +exch 0 rlineto +neg 0 exch rlineto +closepath}def +/_hr +{ +{/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} +{/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} +}def +/_cs null def +/fg{setgray}def +/setcmykcolor where {pop true}{false}ifelse +{/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse +/setcolorspace where {pop true}{false}ifelse +{/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def +}{/f3{ setrgbcolor}def/scs{ pop}def}ifelse +/$cc 5 array def +/findcmykcustomcolor where {pop true}{false}ifelse +not {/findcmykcustomcolor{$cc astore}def}if +/setcustomcolor where {pop true}{false}ifelse +not{/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def}if +/fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def +/setseparationgray where {pop true}{false}ifelse +not{/setseparationgray {systemdict /setgray get exec} def}if +/sg{ setseparationgray}def +/sp{/_penv xdf/_penh xdf}def +/rp +{/cv_b xdf/cv_r xdf +/cv_t xdf/cv_l xdf +cv_l cv_t moveto +cv_r cv_t lineto +cv_r cv_b lineto +cv_l cv_b lineto +closepath}def +/rr{/yradius xdf +/xradius xdf +/cv_b xdf +/cv_r xdf +/cv_t xdf +/cv_l xdf +cv_l xradius add cv_t moveto +cv_r xradius sub cv_t lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +1 0 1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_r cv_b yradius sub lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 1 -1 1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l xradius add cv_b lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +-1 0 -1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +cv_l cv_t yradius add lineto +_cvmtx currentmatrix pop +currentpoint translate +xradius yradius scale +0 -1 1 -1 1 arcto 4{pop}repeat +_cvmtx setmatrix +closepath}def +/ep{_cvmtx currentmatrix pop +translate +scale +1 0 moveto +0 0 1 0 360 arc +closepath +_cvmtx setmatrix}def +/wd{_cvmtx currentmatrix pop +translate +scale +0 0 moveto +0 0 1 5 -2 roll arcn +closepath +_cvmtx setmatrix}def +/as{_cvmtx currentmatrix pop +translate +scale +0 0 1 5 -2 roll arcn +_cvmtx setmatrix}def +/ds +{_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def +/df{fill}def +/de{eofill}def +/_gstk 5 array def +/_gstkc 0 def +/_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def +/_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def +/hs{newpath gsave stm}def +/hc{/_hr load exch get exec newpath}def +/he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store +grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq +not{aload pop moveto}{pop}ifelse dup /_cf get setfont +dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def +/id{ /_id xdf}def +/_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def +% Copyright (C) 1990, 1991 by Adobe Systems Incorporated. +% All rights reserved. +level2 { save /dontloadlevel1 xdf }if +/startnoload{{/noload save def} if}def +/endnoload{{noload restore}if }def +/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def +/ncolors 1 def +/colorimage where{pop true}{false}ifelse{/ncolors 0 +statusdict begin +/processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known +{pop {deviceinfo /Colors get}}if}if}ifelse end def +ncolors 0 ne{/colorimage testsystemdict +/setcolortransfer testsystemdict +/currentcolortransfer testsystemdict +/currentcmykcolor testsystemdict +and and and not{/ncolors 0 def}if}if}if +ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if +ncolors 1 eq dup dup not startnoload +{/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] +cvx bind settransfer _w _h 8 _imatrix _ds image}def}if +not endnoload +ncolors dup 3 eq exch 4 eq or +dup dup not startnoload +{/nullproc{{}}def/concatutil{/exec load +7 -1 roll /exec load}def +/defsubclut +{1 add getinterval def}def +/spconcattransfer +{/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld +currentcolortransfer +[{Aclut ncompute}concatutil] cvx +[{Bclut ncompute}concatutil] cvx +[{Cclut ncompute}concatutil] cvx +[{Dclut ncompute}concatutil] cvx +setcolortransfer}def +/setuprgbcluts +{/bit3x rgbclut length 3 sub def +/bit1x bit3x 3 idiv def/rclut rgbclut def +/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not +endnoload ncolors 3 eq dup dup not startnoload +{/3compute{exch bit3x mul +round cvi get 255 div}def +/doclutimage{%/rgbclut xdf pop +setuprgbcluts/3compute rclut gclut bclut dup +spconcattransfer _w _h 8 _imatrix +[/_ds load /exec load /dup load dup]cvx +nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload +{/stuffclut{cmykindex 3 -1 roll put}def +/ftoint{1 exch sub 255 mul round cvi}def +/4compute { exch bit4x mul round cvi get 255 div } def +/computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def +/cmykclut bit4x 4 add string def +/cclut cmykclut def +/mclut cclut 1 bit4x defsubclut +/yclut cclut 2 bit4x defsubclut +/kclut cclut 3 bit4x defsubclut +/cmykindex 0 def +0 1 bit1x +{dup /cmykindex exch bit1x exch sub 4 mul def +3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div +exch bclut exch get 255 div +setrgbcolor currentcmykcolor +ftoint kclut stuffclut +ftoint yclut stuffclut +ftoint mclut stuffclut +ftoint cclut stuffclut +}for}def +/doclutimage{invalidcolortable?{computecmykclut}if +/4compute cclut mclut yclut kclut spconcattransfer +_w _h 8 _imatrix[/_ds load /exec load/dup load dup dup] cvx +nullproc nullproc nullproc true 4 colorimage}def}if +not endnoload +ncolors 0 eq +dup dup not startnoload +{/colorexpand{mystringexp 0 rgbclut +3 copy +7 -1 roll +{3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall +pop pop pop pop pop}def +/createexpandstr{/mystringexp exch st0 length mul string def}def +/doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix +[/_ds load /exec load /colorexpand load /exec load] cvx +false 3 colorimage}def}if not endnoload +level2 { dontloadlevel1 restore }if +level2 not{ save /dontloadlevel2 xdf }if +/rgbclut 0 def +/doclutimage{/hival 255 def +[/Indexed /DeviceRGB hival rgbclut] setcolorspace +_imagedict dup begin +/Width _w def +/Height _h def +/Decode [0 hival] def +/ImageMatrix _imatrix def +/BitsPerComponent 8 def +/Interpolate false def +end image}def +level2 not{ dontloadlevel2 restore }if +%%EndProcSet +/_imagedict 15 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def +/hascolor true def +/level2 systemdict /languagelevel known +{languagelevel 2 ge}{false}ifelse def +/foureq {4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +4 index eq 8 1 roll +pop pop pop pop and and and} def +/setupbands{gsave +/_settransfer systemdict /settransfer get def +/_currenttransfer systemdict /currenttransfer get def +/_image systemdict /image get def +/blank 0 _currenttransfer exec +1 _currenttransfer exec eq def +/negative blank{0 _currenttransfer exec 0.5 lt} +{0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def +/inverted? negative def +/_currentgray systemdict /currentgray get def +hascolor{/$b 0 def} {/$b 5 def} ifelse +/setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub +0 1 0 0 setcmykcolor _currentgray 1 exch sub +0 0 1 0 setcmykcolor _currentgray 1 exch sub +0 0 0 1 setcmykcolor _currentgray 1 exch sub +4 {4 copy} repeat +1 0 0 0 foureq{/$b 1 store}if +0 1 0 0 foureq{/$b 2 store}if +0 0 1 0 foureq{/$b 3 store}if +0 0 0 1 foureq{/$b 4 store}if +0 0 0 0 foureq{/$b 6 store}if}if +blank{/$b 6 store}if +$b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def +/doimage{ +level2{image}{begin +Width Height BitsPerComponent ImageMatrix +Decode length 2 eq +{/DataSource load end image}{ +Decode length 6 eq +{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse +MultipleDataSources 3 end DSIColorImage}{ +Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get +DataSource 2 get DataSource 3 get}{/DataSource load}ifelse +MultipleDataSources 4 end DSIColorImage}if}ifelse +}ifelse +}ifelse}def +/dt{/rgbclut xdf /bwclut xdf}def +/cl{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +doclutimage} def +/cd{$i setmatrix translate scale +level2{_imagedict /ColorSpace get setcolorspace}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +/ImageMatrix _imatrix def +currentdict end +doimage} def +/im{$i setmatrix translate scale +level2{/DeviceGray setcolorspace 0 setgray}if +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_imagedict begin +%/Decode [0 1] def +%/Width _w def +%/Height _h def +/ImageMatrix _imatrix def +currentdict end +doimage }def +/colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ +/$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf +$nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub +{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add +cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup +st0 exch get .3 mul +1 index 1 add st0 exch get .59 mul add +1 index 2 add st0 exch get .11 mul add +1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if +255.0 exch sub cvi +exch 4 div cvi exch +st1 3 1 roll put}for st1}}ifelse +image}def}ifelse +/OPIimage{image}def +/$i matrix def +/bi{scale translate $i currentmatrix pop translate scale setupbands}def +/ei{$i setmatrix}def +/ik{$i setmatrix translate scale +/_w _imagedict /Width get def +/_h _imagedict /Height get def +/_ds _imagedict /DataSource get def +_w _h false _imatrix 5 -1 roll _ds imagemask}def +/rs{2 copy translate +3 -1 roll neg rotate +neg exch neg exch translate +}def +/cs{_cvmtx astore concat} def +/fd{/_fd xdf}def +/_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def +/rf{/_encodindex xdf +FontDirectory exch known { +findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall +dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def +_encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} +{exch pop}ifelse}forall pop}if}if +currentdict end +definefont pop }{pop pop}ifelse} def +/cvsf{ findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore +makefont setfont}def +/mf{3 -1 roll findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch +_fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def +/wrk1 ( ) def +/wdict 16 dict def +/Work75 75 string def +/Nmk{Work75 cvs dup}def +/Npt{put cvn}def +/dhOdh{Nmk 2 79 Npt}def +/dhBdh{Nmk 2 111 Npt}def +/dhSdh{Nmk 2 83 Npt}def +/sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def +/fts{/ftSize exch def}def +/mkFT{dup/FontType get 0 eq{dup maxlength dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +currentdict end +dup/FDepVector 2 copy get +[exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def +tempFT begin +/FontMatrix [1 0 0 1 0 0] def +/FontType 3 def +dup/Encoding get/Encoding exch def +/proc2 2 index def +dup/ft1 exch def +/ft2 1 array def +/FontBBox [0 0 1 1] def +/BuildChar{CVDict begin +exch dup/ftdt xdf begin +wdict begin +/chr xdf/chrst wrk1 dup 0 chr put def +ft2 0 get null eq{ft1 dup length 2 add dict begin +{1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall +/PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def +currentdict end +/q exch definefont +ft2 exch 0 exch put}if +ftdt/proc2 get exec end end end}def end tempFT}ifelse +3 index exch +definefont exch pop}def +/mb{dup dhBdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{boldproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/mo{dup dhOdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{outproc} exch +mkFT pop pop pop exch pop +}ifelse }def +/ms{dup dhSdh FontDirectory 1 index known +{exch pop}{dup 2 index +findfont{ShadChar} exch +mkFT pop pop pop exch pop +}ifelse }def +/outproc{ft1 setfont +gsave chrst sfWidth grestore +setcharwidth dblsh}def +/boldproc{ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +ft1 /PaintType get 2 ne +{0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} +{0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def +/dblsh{currentcolor currentcolorspace 1.0 setgray +chrst 0 0 moveto show setcolorspace setcolor +ft2 0 get setfont chrst 0 0 moveto show}def +/ShadChar{ +ft1 setfont +gsave +chrst sfWidth 1 index 0 ne{exch .05 add exch}if +grestore +setcharwidth +chrst .06 0 moveto show +0 .05 translate dblsh}def +/ts{show}def +/ta{/_letterSpace xdf/_spaceXtra xdf}def +/tf{/_textWidth xdf +/_numSpaces xdf +/_stringLength xdf +dup stringwidth pop +_letterSpace _stringLength 1 sub mul add +_spaceXtra _numSpaces mul add +_textWidth exch sub /wdif xdf +_spaceXtra 0 32 +wdif _stringLength div +_letterSpace add +0 6 -1 roll awidthshow}def +/pb{/_PageSave save def/showpage{}def}def +/pe{_PageSave restore}def +/pm{_pm currentmatrix pop}def +/_pm matrix def +/ps{showpage _pm setmatrix}def +level2{/ef {gsave 6 array astore concat dup execform grestore}def} +{/ef{gsave 6 array astore concat newpath dup begin Matrix concat +BBox dup dup 0 get exch 1 get moveto +dup dup 0 get exch 3 get lineto +dup dup 2 get exch 3 get lineto +dup 2 get exch 1 get lineto closepath +clip newpath PaintProc end grestore}def}ifelse +/bc{save 11 1 roll 10 dict begin /BBox 5 1 roll 4 array astore def +/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def +/ec{pop restore}def +/printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def +/setoverprint where {pop}{/setoverprint{pop}def}ifelse +/sstr { /$sstrs exch def /$sstr exch def /$sstry exch def /$sstrx exch def /Courier findfont $sstrs +scalefont setfont $sstrx $sstry mt stm 1 -1 scale $sstr show rtm } def +/DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc + eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} + {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def +/DSIEndPaintProc{currentdict end}def +/DSIBeginPaintSubProc {save 11 1 roll 10 dict dup /pD exch def begin + /BBox 5 1 roll 4 array astore def /Matrix 7 1 roll 6 array astore def + /FormType 1 def /PaintProc { /pp get exec } def /pp} def +/DSIEndPaintSubProc {bind def end pD} def +/iaa {/ix ix ixn add def} def +/ias2 {/igap exch def /_st0 ia ix igap add get def _st0} def +/ias {/st0 ia ix get def iaa st0} def +/i85s {0 ias2 /ASCII85Decode filter st0 readstring pop iaa} def +/irles {0 ias2 /ASCII85Decode filter /RunLengthDecode filter st0 readstring pop iaa} def +initialize +end +%%EndResource +CVDict begin 0 id 1 fd +_encodArr 0 +[ +39/quotesingle 96/grave 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+(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +10525 35743 MT +(0)SH +10 SS +10925 35398 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +11564 35743 MT +(1)SH +10 SS +11964 35398 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +12603 35743 MT +(2)SH +10 SS +13003 35398 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +13642 35743 MT +(3)SH +10 SS +14042 35398 MT +(,)SH +/Times-Italic SF +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +16264 35743 MT +(0)SH +10 SS +16664 35398 MT +(,)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +17414 35743 MT +(0)SH +10 SS +17814 35398 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +20564 35743 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Times-Italic AF +21803 35398 MT +(b)SH +8 SS +22303 35743 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +30600 35398 MT +(2:)SH +/Times-Italic SF +31628 XM +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH 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AF +34028 37120 MT +(1)SH +10 SS +34428 36775 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +35067 37120 MT +(2)SH +10 SS +35467 36775 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +36106 37120 MT +(3)SH +10 SS +36506 36775 MT +(,)SH +/Times-Italic SF +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +38728 37120 MT +(0)SH +10 SS +39128 36775 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +41878 37120 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Times-Italic AF +43117 36775 MT +(b)SH +8 /Times-Roman AF +43617 37120 MT +(0)SH +10 SS +44017 36775 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +46767 37120 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +17477 39762 MT +(Figure 11.ALU Output Graph Sizes for Different Input Orderings)SH +7200 41553 MT +(bit-serial processor analogy, we could implement the ALU with the control inputs read last) +126 W( by computing all 32)125 W +7200 42930 MT +(possible ALU functions and) +24 W( then selecting the appropriate result once the desired function is known. The final case)25 W +7200 44307 MT +(shows what happens if a poor ordering is) +31 W( chosen, in this case the ordering)30 W +/Times-Italic SF +37372 XM +(m)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +38733 44652 MT +(0)SH +10 SS +39133 44307 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +39772 44652 MT +(1)SH +10 SS +40172 44307 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +40811 44652 MT +(2)SH +10 SS +41211 44307 MT +(,)SH +/Times-Italic SF +(s)SH +8 /Times-Roman AF +41850 44652 MT +(3)SH +10 SS +42250 44307 MT +(,)SH +/Times-Italic SF +(cin)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +44472 44652 MT +(0)SH +10 SS +44872 44307 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +47622 44652 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 /Times-Italic AF +48861 44307 MT +(b)SH +8 /Times-Roman AF +49361 44652 MT +(0)SH +10 SS +49761 44307 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +52511 44652 MT +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +53750 44307 MT +(.)SH +7200 45684 MT +(This ordering requires the program to represent functions similar to) +72 W( the function)73 W +/Times-Italic SF +40340 XM +(x)SH +8 /Times-Roman AF +40784 46029 MT +(1)SH +10 /Symbol AF +41184 45684 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +41878 46029 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +43379 45684 MT +(+)SH +/Symbol SF +44443 XM +(\327 \327 \327)SH +/Times-Roman SF +46193 XM +(+)SH +/Times-Italic SF +47007 XM +(x)SH +8 SS +47451 46029 MT +(n)SH +10 /Symbol AF +47851 45684 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +48545 46029 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +49668 45684 MT +(considered)SH +7200 47061 MT +(in Section 3, with the same exponential growth characteristics.)SH +8200 49540 MT +(These experimental results indicate that our representation works quite well for functions representing addition)107 W +7200 50917 MT +(and logical operations on words of data, as) +51 W( long as we choose an ordering in which the successive bits of the input)52 W +7200 52294 MT +(words are interleaved. Our) +152 W( representation seems especially efficient when compared to other representations of)151 W +7200 53671 MT +(Boolean functions. For example, a truth table representation would be totally) +3 W( impractical for ALU's with word sizes)4 W +7200 55048 MT +(greater than 8 bits. A reduced sum-of-products representation of the most) +106 W( significant bit in the sum of two)105 W +/Times-Italic SF +52089 XM +(n)SH +/Times-Roman SF +52944 XM +(bit)SH +7200 56425 MT +(numbers requires about 2)188 W +8 /Times-Italic AF +17901 56080 MT +(n)SH +/Times-Roman SF +(+2)SH +10 SS +19590 56425 MT +(product terms, and hence a reduced sum-of-products representation of this circuit)189 W +7200 57802 MT +(would be equally impractical.)SH +12 /Times-Bold AF +7200 61486 MT +(6. Conclusion)SH +10 /Times-Roman AF +8200 62863 MT +(We have shown that by taking a well-known graphical representation of Boolean functions) +204 W( and imposing a)203 W +7200 64240 MT +(restriction on the vertex labels, the minimum size) +270 W( graph representing a function becomes a canonical form.)271 W +7200 65617 MT +(Furthermore, given any graph representing a function, we) +112 W( can reduce it to a canonical form graph in linear time.)111 W +7200 66994 MT +(Thus our reduction algorithm not only) +7 W( minimizes the amount of storage required to represent a function and the time)8 W +7200 68371 MT +(required to perform symbolic operations on the function, it also makes such tasks as testing for equivalence,)224 W +7200 69748 MT +(satisfiability, or tautology very simple. We have found this property valuable in many applications.)SH +ES +%%Page: 22 22 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(22)SH +8200 7886 MT +(We have presented a set of algorithms for performing a variety of) +13 W( operations on Boolean functions represented by)14 W +7200 9263 MT +(our data structure. Each of these algorithms) +98 W( obeys an important closure property---if the argument graphs satisfy)97 W +7200 10640 MT +(our ordering restrictions, so does the result graph. By combining concepts from Boolean algebra with) +92 W( techniques)93 W +7200 12017 MT +(from graph algorithms, we achieve a high degree of efficiency. That is, the performance is limited) +6 W( more by the sizes)5 W +7200 13394 MT +(of the data structures rather than by the algorithms that operate on them.)SH +8200 15873 MT +(Akers [17]) +SH( has devised a variety of coding techniques to reduce the size of the binary) +264 W( decision diagrams)265 W +7200 17250 MT +(representing the output functions of a system. For example, he can represent the functions for all 8 outputs) +71 W( of the)70 W +7200 18627 MT +(74181 ALU slice by a total of 35 vertices, whereas our representation requires 918. Several of these techniques)116 W +7200 20004 MT +(could be applied to our representation without) +48 W( violating the properties required by our algorithms. We will discuss)47 W +7200 21381 MT +(two such refinements briefly.)SH +8200 23860 MT +(Most digital systems contain multiple outputs.) +64 W( In) +380 W( our current implementation we represent each output function)65 W +7200 25237 MT +(by a separate graph, even though these function may be closely) +216 W( related and therefore have graphs containing)215 W +7200 26614 MT +(isomorphic subgraphs. Alternatively,) +107 W( we could represent a set of functions by a single graph with multiple roots)108 W +7200 27991 MT +(\050one for each function.\051)167 W +8 SS +17142 27646 MT +(11)SH +10 SS +18776 27991 MT +(Our reduction algorithm could be applied to such graphs to eliminate) +167 W( any duplicate)168 W +7200 29368 MT +(subgraphs and) +268 W( to guarantee that the subgraph consisting of a root and all of its descendants is a canonical)267 W +7200 30745 MT +(representation of the corresponding) +25 W( function. For example, we could represent the)26 W +/Times-Italic SF +40835 XM +(n)SH +/Times-Roman SF +(+1 functions for the addition of)26 W +7200 32122 MT +(two)SH +/Times-Italic SF +8983 XM +(n)SH +/Times-Roman SF +(-bit numbers by a single graph containing 9)33 W +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1 vertices \050assuming the inputs are ordered most significant bits)33 W +7200 33499 MT +(first\051, whereas representing them by separate) +89 W( graphs requires a total of 3)90 W +/Times-Italic SF +(n)SH +8 /Times-Roman AF +37566 33154 MT +(2)SH +10 SS +37966 33499 MT +(+6)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+2 vertices. Taking this idea to an)90 W +7200 34876 MT +(extreme, we could manage our entire set of graphs as a single shared data structure, using an extension) +167 W( of the)166 W +7200 36253 MT +(reduction algorithm to merge a newly created) +20 W( graph into this structure. With such a structure, we could enhance the)21 W +7200 37630 MT +(performance of) +56 W( the)55 W +/Times-Italic SF +15224 XM +(Apply)SH +/Times-Roman SF +17862 XM +(procedure by maintaining a table containing entries of the form \050)55 W +/Times-Italic SF +(v)SH +8 /Times-Roman AF +44740 37975 MT +(1)SH +10 SS +45140 37630 MT +(,)SH +/Times-Italic SF +(v)SH +8 /Times-Roman AF +45834 37975 MT +(2)SH +10 SS +46234 37630 MT +(,<)SH +/Times-Italic SF +(op)SH +/Times-Roman SF +(>,)SH +/Times-Italic SF +(u)SH +/Times-Roman SF +(\051 indicating)55 W +7200 39007 MT +(that the result of applying operation <)72 W +/Times-Italic SF +(op)SH +/Times-Roman SF +(> to graphs with roots)72 W +/Times-Italic SF +33146 XM +(v)SH +8 /Times-Roman AF +33590 39352 MT +(1)SH +10 SS +34312 39007 MT +(and)SH +/Times-Italic SF +36078 XM +(v)SH +8 /Times-Roman AF +36522 39352 MT +(2)SH +10 SS +37245 39007 MT +(was a graph with root)73 W +/Times-Italic SF +46525 XM +(u)SH +/Times-Roman SF +(. In) +396 W( this way we)73 W +7200 40384 MT +(would exploit the information generated by previous invocations of)136 W +/Times-Italic SF +35671 XM +(Apply)SH +/Times-Roman SF +38390 XM +(as well as by the current one.) +136 W( These)520 W +7200 41761 MT +(savings in overall storage) +129 W( requirements and algorithm efficiencies would be offset somewhat by a more difficult)130 W +7200 43138 MT +(memory management problem, however.)SH +8200 45617 MT +(Akers also saves storage by representing functions in decomposed form. That) +79 W( is, we can represent the function)78 W +/Times-Italic SF +7200 46994 MT +(f)SH +/Symbol SF +7728 XM +(|)SH +8 /Times-Italic AF +7928 47339 MT +(x)SH +8283 47684 MT +(i)SH +/Times-Roman SF +8505 47339 MT +(=)SH +/Times-Italic SF +(g)SH +10 /Times-Roman AF +9759 46994 MT +(in terms of)153 W +/Times-Italic SF +14801 XM +(f)SH +/Times-Roman SF +15482 XM +(and)SH +/Times-Italic SF +17329 XM +(g)SH +/Times-Roman SF +18232 XM +(\050which may themselves) +153 W( be represented in decomposed form\051. Unfortunately, a given)154 W +7200 48545 MT +(function can be decomposed in many different ways,) +61 W( and hence this technique would not lead to a canonical form.)60 W +7200 49922 MT +(However, as noted on page 16 there are certain instances in) +220 W( which functions)221 W +/Times-Italic SF +40722 XM +(f)SH +/Times-Roman SF +41621 XM +(and)SH +/Times-Italic SF +43536 XM +(g)SH +/Times-Roman SF +44507 XM +(can be composed in a)221 W +7200 51299 MT +(straightforward way by simply replacing each vertex representing the composition variable)63 W +/Times-Italic SF +44580 XM +(x)SH +8 SS +45024 51644 MT +(i)SH +10 /Times-Roman AF +45559 51299 MT +(with the graph for)62 W +/Times-Italic SF +53250 XM +(g)SH +/Times-Roman SF +(.)SH +7200 52676 MT +(For such decompositions, functions) +277 W( could be stored in decomposed form and expanded into canonical form)278 W +7200 54053 MT +(dynamically as operations are performed on) +112 W( them. In some instances, the storage savings could be considerable.)111 W +7200 55430 MT +(For example the graph for the function)SH +/Times-Italic SF +9200 57221 MT +(x)SH +8 /Times-Roman AF +9644 57566 MT +(1)SH +10 /Symbol AF +10044 57221 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +10738 57566 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +11788 57221 MT +(+)SH +/Times-Italic SF +12602 XM +(x)SH +8 /Times-Roman AF +13046 57566 MT +(2)SH +10 /Symbol AF +13446 57221 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +14140 57566 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +10 SS +16029 57221 MT +(+ .) +250 W( . . +)SH +/Times-Italic SF +19657 XM +(x)SH +8 SS +20101 57566 MT +(n)SH +10 /Symbol AF +20501 57221 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +21195 57566 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +7200 58949 MT +(requires a total of 2)SH +8 /Times-Italic AF +14976 58604 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +16477 58949 MT +(vertices. This) +250 W( function can be decomposed as a series of functions where)SH +/Times-Italic SF +46081 XM +(f)SH +8 SS +46359 59294 MT +(n)SH +10 /Times-Roman AF +47009 58949 MT +(=)SH +/Times-Italic SF +47823 XM +(x)SH +8 SS +48267 59294 MT +(n)SH +10 /Symbol AF +48667 58949 MT +(\327)SH +/Times-Italic SF +(x)SH +8 SS +49361 59294 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +50612 58949 MT +(,)SH +/Times-Italic SF +9200 60854 MT +(f)SH +8 SS +9478 61199 MT +(i)SH +10 /Times-Roman AF +10386 60854 MT +(= \050)436 W +/Times-Italic SF +(x)SH +8 SS +12413 61199 MT +(i)SH +10 /Symbol AF +12635 60854 MT +(\327)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +13329 61199 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(+1)SH +10 SS +15891 60854 MT +(+)SH +/Times-Italic SF +16705 XM +(x)SH +8 SS +17149 61199 MT +(i)SH +/Times-Roman SF +(+1)SH +10 SS +18222 60854 MT +(\051)SH +/Symbol SF +18705 XM +(|)SH +8 /Times-Italic AF +18905 61199 MT +(x)SH +19260 61544 MT +(i)SH +/Times-Roman SF +(+1)SH +20333 61199 MT +(=)SH +/Times-Italic SF +(f)SH +21006 61544 MT +(i)SH +/Times-Roman SF +(+1)SH +10 SS +7200 62519 MT +(for)SH +/Times-Italic SF +8624 XM +(n)SH +/Times-Roman SF +9274 XM +(>)SH +/Times-Italic SF +9988 XM +(i)SH +/Symbol SF +10416 XM +(\263)SH +/Times-Roman SF +11115 XM +(1, and)8 W +/Times-Italic SF +13825 XM +(f)SH +8 /Times-Roman AF +14103 62864 MT +(1)SH +10 SS +14761 62519 MT +(equals the desired function. Each of these) +8 W( functions can be represented by graphs with 6 vertices.)9 W +7200 63896 MT +(It is unclear, however,) +42 W( how often such decompositions occur, how easy they are to find, and how they would affect)41 W +7200 65273 MT +(the efficiency of the algorithms.)SH +10800 50 7200 69428 UL +6 SS +8000 70767 MT +(11)SH +8 /Times-Bold AF +8600 71076 MT +(Update:)SH +/Times-Roman SF +11584 XM +(This idea was subsequently termed a "Shared BDD" \050S. Minato, N. Ishiura, and) +29 W( S. Yajima, "Shared Binary Decision Diagram with)28 W +7200 72000 MT +(Attributed Edges,")SH +/Times-Italic SF +13368 XM +(27th Design Automation Conference)SH +/Times-Roman SF +(, 1990, pp. 52-57.\051 It has now become the most common implementation method.)SH +ES +%%Page: 23 23 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(23)SH +12 /Times-Bold AF +7200 8004 MT +(Appendix: The Complexity of Integer Multiplication)SH +10 /Times-Roman AF +8200 9381 MT +(In this appendix, we prove that the functions representing the outputs of an integer multiplier provide a difficult)67 W +7200 10758 MT +(case for our representation, i.e. the graph sizes grow exponentially in the word size regardless of the ordering of the)31 W +7200 12135 MT +(input variables. Given that there are \0502)18 W +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051! possible orderings of the input variables, we could not hope) +18 W( to derive this)19 W +7200 13512 MT +(result experimentally, and hence we must provide a detailed proof.)SH +8200 15991 MT +(Our proof is based on) +102 W( principles similar to those used in proving area-time lower bounds on multiplier circuits)101 W +7450 17368 MT +([18, 19].) +SH( However,) +394 W( we must show not just that a large amount of information must) +72 W( be transferred from the set of)73 W +7200 18745 MT +(inputs to the set of outputs in performing multiplication, but that certain individual outputs require high) +31 W( information)30 W +7200 20122 MT +(transfer.)SH +8200 22601 MT +(Consider a multiplier with inputs)9 W +/Times-Italic SF +21718 XM +(a)SH +8 /Times-Roman AF +22218 22946 MT +(1)SH +10 SS +22618 22601 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +25368 22946 MT +(n)SH +10 /Times-Roman AF +26027 22601 MT +(and)SH +/Times-Italic SF +27730 XM +(b)SH +8 /Times-Roman AF +28230 22946 MT +(1)SH +10 SS +28630 22601 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +31380 22946 MT +(n)SH +10 /Times-Roman AF +32039 22601 MT +(corresponding to the binary encoding) +9 W( of integers)10 W +/Times-Italic SF +51796 XM +(a)SH +/Times-Roman SF +52556 XM +(and)SH +/Times-Italic SF +7200 23978 MT +(b)SH +/Times-Roman SF +7972 XM +(with)SH +/Times-Italic SF +10022 XM +(a)SH +8 /Times-Roman AF +10522 24323 MT +(1)SH +10 SS +11194 23978 MT +(and)SH +/Times-Italic SF +12910 XM +(b)SH +8 /Times-Roman AF +13410 24323 MT +(1)SH +10 SS +14082 23978 MT +(being the least) +22 W( significant bits. This circuit has 2)21 W +/Times-Italic SF +(n)SH +/Times-Roman SF +34654 XM +(outputs corresponding to the binary encoding of)21 W +7200 25355 MT +(the product)356 W +/Times-Italic SF +12689 XM +(a)SH +/Symbol SF +13339 XM +(\327)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(, described by functions)356 W +/Times-Italic SF +25317 XM +(mul)SH +8 SS +26817 25700 MT +(i)SH +10 /Times-Roman AF +27189 25355 MT +(\050)SH +/Times-Italic SF +(a)SH +8 /Times-Roman AF +28022 25700 MT +(1)SH +10 SS +28422 25355 MT +(, . . . ,)SH +/Times-Italic SF +(a)SH +8 SS +31172 25700 MT +(n)SH +10 /Times-Roman AF +31572 25355 MT +(,)SH +/Times-Italic SF +(b)SH +8 /Times-Roman AF +32322 25700 MT +(1)SH +10 SS +32722 25355 MT +(, . . . ,)SH +/Times-Italic SF +(b)SH +8 SS +35472 25700 MT +(n)SH +10 /Times-Roman AF +35872 25355 MT +(\051 for) +356 W( 1)357 W +/Symbol SF +39234 XM +(\243)SH +/Times-Italic SF +39933 XM +(i)SH +/Symbol SF +40361 XM +(\243)SH +/Times-Roman SF +41060 XM +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(. For) +964 W( a permutation)357 W +/Symbol SF +52011 XM +(p)SH +/Times-Roman SF +53167 XM +(of)SH +7200 26732 MT +({1, . . . ,2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(}, let)SH +/Times-Italic SF +13660 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 be a graph which for inputs)SH +/Times-Italic SF +27595 XM +(x)SH +8 /Times-Roman AF +28039 27077 MT +(1)SH +10 SS +28439 26732 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +31133 27077 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +32183 26732 MT +(denotes the function)SH +/Times-Italic SF +40543 XM +(mul)SH +8 SS +42043 27077 MT +(i)SH +10 /Times-Roman AF +42415 26732 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Symbol AF +43192 27077 MT +(p)SH +/Times-Roman SF +(\0501\051)SH +10 SS +44563 26732 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Symbol AF +47257 27077 MT +(p)SH +/Times-Roman SF +(\0502)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051)SH +10 SS +49028 26732 MT +(\051)SH +/Times-Bold SF +10200 29184 MT +(Theorem 2:)SH +/Times-Roman SF +15421 XM +(For any)SH +/Symbol SF +18754 XM +(p)SH +/Times-Roman SF +19553 XM +(there exists an)SH +/Times-Italic SF +25524 XM +(i)SH +/Times-Roman SF +(, 1)SH +/Symbol SF +26952 XM +(\243)SH +/Times-Italic SF +27651 XM +(i)SH +/Symbol SF +28079 XM +(\243)SH +/Times-Roman SF +28778 XM +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +30028 XM +(such that)SH +/Times-Italic SF +33861 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 contains at least 2)SH +8 /Times-Italic AF +43714 28839 MT +(n)SH +/Times-Roman SF +(/8)SH +10 SS +44986 29184 MT +(vertices.)SH +/Times-Italic SF +8200 31663 MT +(Proof)SH +/Times-Roman SF +10741 XM +(: Informally,) +276 W( our proof proceeds as follows. If one input \050the "control"\051 to a multiplier is a power) +13 W( of 2 then)12 W +7200 33040 MT +(the circuit acts as a shifter, transferring the bits of the other input \050the "data"\051 to) +75 W( the output with some offset. For)76 W +7200 34417 MT +(example, if)SH +/Times-Italic SF +11949 XM +(b)SH +/Times-Roman SF +12699 XM +(= 2)SH +8 /Times-Italic AF +14013 34072 MT +(j)SH +10 /Times-Roman AF +14235 34417 MT +(, then)SH +9200 37026 MT +([)SH +/Times-Italic SF +(a)SH +/Symbol SF +(\327)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(])SH +8 /Times-Italic AF +11116 37371 MT +(i)SH +10 /Times-Roman AF +12024 37026 MT +(=)SH +24 /Symbol AF +13274 37442 MT +({)SH +10 /Times-Italic AF +14426 36735 MT +(a)SH +8 SS +14926 37080 MT +(i)SH +/Symbol SF +(-)SH +/Times-Italic SF +(j)SH +10 SS +16495 36735 MT +(j)SH +/Times-Roman SF +16923 XM +(<)SH +/Times-Italic SF +17637 XM +(i)SH +/Symbol SF +18065 XM +(\243)SH +/Times-Italic SF +18764 XM +(j)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +14426 37941 MT +(0 else)1319 W +7200 39461 MT +(Graph)SH +/Times-Italic SF +10018 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 must contain enough vertices to encode all values of)69 W +/Times-Italic SF +34613 XM +(a)SH +8 SS +35113 39806 MT +(i)SH +/Symbol SF +(-)SH +/Times-Italic SF +(j)SH +10 /Times-Roman AF +36314 39461 MT +(for which)68 W +/Times-Italic SF +40560 XM +(a)SH +8 SS +41060 39806 MT +(i)SH +/Symbol SF +(-)SH +/Times-Italic SF +(j)SH +10 /Times-Roman AF +42261 39461 MT +(occurs in the first half of the)68 W +7200 40838 MT +(input sequence while)8 W +/Times-Italic SF +15917 XM +(b)SH +8 SS +16417 41183 MT +(j)SH +10 /Times-Roman AF +16897 40838 MT +(occurs in the second. Furthermore,) +8 W( we can show that for any ordering of input variables, we)9 W +7200 42215 MT +(can choose which input \050)78 W +/Times-Italic SF +(a)SH +/Times-Roman SF +18338 XM +(or)SH +/Times-Italic SF +19499 XM +(b)SH +/Times-Roman SF +(\051 is control and which is data such that for some output)78 W +/Times-Italic SF +43155 XM +(i)SH +/Times-Roman SF +(, this undesirable) +78 W( splitting)77 W +7200 43592 MT +(occurs for at least)SH +/Times-Italic SF +14531 XM +(n)SH +/Times-Roman SF +(/8 values of)SH +/Times-Italic SF +19947 XM +(j)SH +/Times-Roman SF +(.)SH +8200 46071 MT +(More formally, for permutation)SH +/Symbol SF +21059 XM +(p)SH +/Times-Roman SF +21858 XM +(let)SH +/Times-Italic SF +9200 47976 MT +(t)SH +/Times-Roman SF +10164 XM +(=)SH +/Symbol SF +11414 XM +(|)SH +/Times-Roman SF +11764 XM +({)SH +/Times-Italic SF +12394 XM +(j)SH +/Symbol SF +12822 XM +(|)SH +/Times-Roman SF +13172 XM +(1)SH +/Symbol SF +13822 XM +(\243)SH +/Times-Italic SF +14521 XM +(j)SH +/Symbol SF +14949 XM +(\243)SH +/Times-Italic SF +15648 XM +(n)SH +/Times-Roman SF +(,)SH +/Symbol SF +16648 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +17958 XM +(\051)SH +/Symbol SF +18441 XM +(\243)SH +/Times-Italic SF +19140 XM +(n)SH +/Times-Roman SF +19790 XM +(})SH +/Symbol SF +20420 XM +(|)SH +/Times-Roman SF +(,)SH +7200 49610 MT +(i.e. the number of bits) +37 W( of argument)38 W +/Times-Italic SF +21599 XM +(a)SH +/Times-Roman SF +22387 XM +(occurring in the first half of the input sequence. If)38 W +/Times-Italic SF +43245 XM +(t)SH +/Symbol SF +43823 XM +(\263)SH +/Times-Italic SF +44672 XM +(n)SH +/Times-Roman SF +(/2 define sets)38 W +/Times-Italic SF +50813 XM +(F)SH +/Times-Roman SF +51712 XM +(and)SH +/Times-Italic SF +53444 XM +(L)SH +/Times-Roman SF +7200 50987 MT +(as)SH +/Times-Italic SF +9200 52892 MT +(F)SH +/Times-Roman SF +10497 XM +(= {)436 W +/Symbol SF +12377 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +13687 XM +(\051)SH +/Symbol SF +14170 XM +(|)SH +/Times-Roman SF +14520 XM +(1)SH +/Symbol SF +15170 XM +(\243)SH +/Times-Italic SF +15869 XM +(j)SH +/Symbol SF +16297 XM +(\243)SH +/Times-Italic SF +16996 XM +(n)SH +/Times-Roman SF +(,)SH +/Symbol SF +17996 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +19306 XM +(\051)SH +/Symbol SF +19789 XM +(\243)SH +/Times-Italic SF +20488 XM +(n)SH +/Times-Roman SF +21138 XM +(})SH +/Times-Italic SF +9200 55102 MT +(L)SH +/Times-Roman SF +10442 XM +(= {)436 W +/Symbol SF +12322 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +13632 XM +(\051)SH +/Symbol SF +14115 XM +(|)SH +/Times-Italic SF +14465 XM +(n)SH +/Times-Roman SF +(+1)SH +/Symbol SF +16179 XM +(\243)SH +/Times-Italic SF +16878 XM +(j)SH +/Symbol SF +17306 XM +(\243)SH +/Times-Roman SF +18005 XM +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(,)SH +/Symbol SF +19505 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +20815 XM +(\051) +150 MX(>)SH +/Times-Italic SF +22012 XM +(n)SH +/Times-Roman SF +22662 XM +(})SH +7200 56622 MT +(That is,)166 W +/Times-Italic SF +10782 XM +(F)SH +/Times-Roman SF +11809 XM +(represents those the indices of) +166 W( argument)165 W +/Times-Italic SF +29133 XM +(a)SH +/Times-Roman SF +30048 XM +(occurring in the first half of the input sequence, while)165 W +/Times-Italic SF +53444 XM +(L)SH +/Times-Roman SF +7200 57999 MT +(represents those indices of)SH +/Times-Italic SF +18031 XM +(b)SH +/Times-Roman SF +18781 XM +(\050with)SH +/Times-Italic SF +21142 XM +(n)SH +/Times-Roman SF +21892 XM +(added to them\051 occurring in the second half. If)SH +/Times-Italic SF +40971 XM +(t)SH +/Times-Roman SF +41549 XM +(<)SH +/Times-Italic SF +42413 XM +(n)SH +/Times-Roman SF +(/2 then define)SH +/Times-Italic SF +48662 XM +(F)SH +/Times-Roman SF +49523 XM +(and)SH +/Times-Italic SF +51217 XM +(L)SH +/Times-Roman SF +52023 XM +(as)SH +/Times-Italic SF +9200 59904 MT +(F)SH +/Times-Roman SF +10497 XM +(= {)436 W +/Symbol SF +12377 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +13687 XM +(\051)SH +/Symbol SF +14170 XM +(|)SH +/Times-Roman SF +14520 XM +(1)SH +/Symbol SF +15170 XM +(\243)SH +/Times-Italic SF +15869 XM +(j)SH +/Symbol SF +16297 XM +(\243)SH +/Times-Italic SF +16996 XM +(n)SH +/Times-Roman SF +(,)SH +/Symbol SF +17996 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +19306 XM +(\051) +150 MX(>)SH +/Times-Italic SF +20503 XM +(n)SH +/Times-Roman SF +21153 XM +(})SH +/Times-Italic SF +9200 62114 MT +(L)SH +/Times-Roman SF +10442 XM +(= {)436 W +/Symbol SF +12322 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +13632 XM +(\051)SH +/Symbol SF +14115 XM +(|)SH +/Times-Italic SF +14465 XM +(n)SH +/Times-Roman SF +(+1)SH +/Symbol SF +16179 XM +(\243)SH +/Times-Italic SF +16878 XM +(j)SH +/Symbol SF +17306 XM +(\243)SH +/Times-Roman SF +18005 XM +(2)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(,)SH +/Symbol SF +19505 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +20815 XM +(\051)SH +/Symbol SF +21298 XM +(\243)SH +/Times-Italic SF +21997 XM +(n)SH +/Times-Roman SF +22647 XM +(})SH +7200 63634 MT +(That is,)86 W +/Times-Italic SF +10622 XM +(F)SH +/Times-Roman SF +11569 XM +(represents those indices of)86 W +/Times-Italic SF +22744 XM +(b)SH +/Times-Roman SF +23580 XM +(\050with)SH +/Times-Italic SF +26027 XM +(n)SH +/Times-Roman SF +26863 XM +(added to them\051 occurring in the first half while)86 W +/Times-Italic SF +46606 XM +(L)SH +/Times-Roman SF +47498 XM +(represents those)87 W +7200 65011 MT +(indices of)18 W +/Times-Italic SF +11402 XM +(a)SH +/Times-Roman SF +12170 XM +(occurring in the second. In either case the sets)18 W +/Times-Italic SF +31262 XM +(F)SH +/Times-Roman SF +32141 XM +(and)SH +/Times-Italic SF +33853 XM +(L)SH +/Times-Roman SF +34677 XM +(will each contain at least)18 W +/Times-Italic SF +44904 XM +(n)SH +/Times-Roman SF +(/2 elements.) +18 W( We) +284 W( will)17 W +7200 66388 MT +(consider the elements of)58 W +/Times-Italic SF +17430 XM +(F)SH +/Times-Roman SF +18349 XM +(to be data inputs and those of)58 W +/Times-Italic SF +30726 XM +(L)SH +/Times-Roman SF +31590 XM +(to be control. Since multiplication is) +58 W( commutative, we)59 W +7200 67765 MT +(are free to choose which argument is considered the control input and which is considered the data in our proof.)SH +8200 70244 MT +(For 1)SH +/Symbol SF +10489 XM +(\243)SH +/Times-Italic SF +11188 XM +(i)SH +/Symbol SF +11616 XM +(\243)SH +/Times-Roman SF +12315 XM +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1 define the set)SH +/Times-Italic SF +20196 XM +(F)SH +8 SS +20807 70589 MT +(i)SH +10 /Times-Roman AF +21279 70244 MT +(as)SH +ES +%%Page: 24 24 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(24)SH +/Times-Italic SF +9200 8000 MT +(F)SH +8 SS +9811 8345 MT +(i)SH +10 /Times-Roman AF +10719 8000 MT +(= {)436 W +/Times-Italic SF +12599 XM +(j)SH +/Symbol SF +13177 XM +(|)SH +/Times-Italic SF +13627 XM +(j)SH +/Symbol SF +14205 XM +(\316)SH +/Times-Italic SF +15218 XM +(F)SH +/Times-Roman SF +(,)SH +/Symbol SF +16579 XM +($)SH +/Times-Italic SF +(k)SH +/Symbol SF +17872 XM +(\316)SH +/Times-Italic SF +18885 XM +(L)SH +/Times-Roman SF +19591 XM +(\050)SH +/Times-Italic SF +20074 XM +(j)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(k)SH +/Times-Roman SF +21510 XM +(=)SH +/Times-Italic SF +22224 XM +(i)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+1\051) +150 MX(})SH +7200 9634 MT +(and let)76 W +/Times-Italic SF +10295 XM +(q)SH +8 SS +10795 9979 MT +(i)SH +10 /Times-Roman AF +11167 9634 MT +(=)SH +/Symbol SF +11881 XM +(|)SH +/Times-Italic SF +(F)SH +8 SS +12692 9979 MT +(i)SH +10 /Symbol AF +13064 9634 MT +(|)SH +/Times-Roman SF +(. That) +400 W( is, for output)75 W +/Times-Italic SF +21936 XM +(i)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +22789 XM +(F)SH +8 SS +23400 9979 MT +(i)SH +10 /Times-Roman AF +23947 9634 MT +(represents those indices of the data input occurring in the first half of the)75 W +7200 11011 MT +(input sequence such that the corresponding bits of the control input occur in the second half.)SH +8200 13490 MT +(Now consider the set of sequences)SH +/Times-Italic SF +9200 15395 MT +(S)SH +8 SS +9700 15740 MT +(i)SH +10 /Times-Roman AF +10608 15395 MT +(= {)436 W +/Times-Italic SF +(x)SH +8 /Times-Roman AF +12782 15740 MT +(1)SH +10 SS +13182 15395 MT +(, . . .)SH +/Times-Italic SF +15182 XM +(x)SH +8 SS +15626 15740 MT +(n)SH +10 /Symbol AF +16176 15395 MT +(|)SH +/Times-Italic SF +16526 XM +(x)SH +8 SS +16970 15740 MT +(j)SH +10 /Times-Roman AF +17192 15395 MT +(=0 if)150 W +/Symbol SF +19667 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +(\051)SH +/Symbol SF +21460 XM +(\317)SH +/Times-Italic SF +22473 XM +(F)SH +8 SS +23084 15740 MT +(i)SH +10 /Times-Roman AF +23456 15395 MT +(})SH +7200 17123 MT +(This set contains 2)89 W +8 /Times-Italic AF +14939 16778 MT +(q)SH +15339 17123 MT +(i)SH +10 /Times-Roman AF +15900 XM +(possible values for) +89 W( the first)90 W +/Times-Italic SF +27430 XM +(n)SH +/Times-Roman SF +28270 XM +(inputs. We) +430 W( claim that)90 W +/Times-Italic SF +37775 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 must contain a unique vertex for)90 W +7200 18500 MT +(each element of)167 W +/Times-Italic SF +14282 XM +(S)SH +8 SS +14782 18845 MT +(i)SH +10 /Times-Roman AF +15004 18500 MT +(. If) +584 W( this were not the case, then we could) +167 W( choose two sequences)166 W +/Times-Italic SF +43054 XM +(x)SH +8 /Times-Roman AF +43498 18845 MT +(1)SH +10 SS +43898 18500 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 SS +46592 18845 MT +(n)SH +10 /Times-Roman AF +47408 18500 MT +(and)SH +/Times-Italic SF +49268 XM +(x)SH +/Symbol SF +49862 XM +(\242)SH +8 /Times-Roman AF +50109 18845 MT +(1)SH +10 SS +50509 18500 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +/Symbol SF +53353 XM +(\242)SH +8 /Times-Italic AF +53600 18845 MT +(n)SH +10 /Times-Roman AF +7200 19877 MT +(leading to the same vertex in)69 W +/Times-Italic SF +19390 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 such that for some value)69 W +/Times-Italic SF +32545 XM +(j)SH +/Times-Roman SF +(,)SH +/Symbol SF +33392 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +34702 XM +(\051)SH +/Symbol SF +35185 XM +(\316)SH +/Times-Italic SF +36048 XM +(F)SH +8 SS +36659 20222 MT +(i)SH +10 /Times-Roman AF +37200 19877 MT +(and)SH +/Times-Italic SF +38964 XM +(x)SH +8 SS +39408 20222 MT +(j)SH +10 /Symbol AF +39780 19877 MT +(\271)SH +/Times-Italic SF +40479 XM +(x)SH +/Symbol SF +41073 XM +(\242)SH +8 /Times-Italic AF +41320 20222 MT +(j)SH +10 /Times-Roman AF +41542 19877 MT +(. Now) +390 W( consider the sequences)70 W +/Times-Italic SF +7200 21254 MT +(x)SH +8 SS +7644 21599 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +8895 21254 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +11589 21599 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +12639 21254 MT +(and)SH +/Times-Italic SF +14333 XM +(x)SH +/Symbol SF +14927 XM +(\242)SH +8 /Times-Italic AF +15174 21599 MT +(n)SH +/Times-Roman SF +(+1)SH +10 SS +16425 21254 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +/Symbol SF +19269 XM +(\242)SH +8 /Times-Roman AF +19516 21599 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +20566 21254 MT +(defined as)SH +/Times-Italic SF +9200 23863 MT +(x)SH +8 SS +9644 24208 MT +(k)SH +10 /Times-Roman AF +10685 23863 MT +(=)SH +/Times-Italic SF +11935 XM +(x)SH +/Symbol SF +12529 XM +(\242)SH +8 /Times-Italic AF +12776 24208 MT +(k)SH +10 /Times-Roman AF +13817 23863 MT +(=)SH +24 /Symbol AF +15067 24279 MT +({)SH +10 /Times-Roman AF +16219 23572 MT +(1,)SH +/Symbol SF +17655 XM +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(j)SH +/Times-Roman SF +(\051+)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(k)SH +/Times-Roman SF +(\051 =)SH +/Times-Italic SF +22435 XM +(i)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(+1)SH +16219 24477 MT +(0, else)436 W +7200 25997 MT +(Note that)15 W +/Times-Italic SF +11174 XM +(x)SH +8 SS +11618 26342 MT +(k)SH +10 /Times-Roman AF +12238 25997 MT +(and)SH +/Times-Italic SF +13947 XM +(x)SH +/Symbol SF +14541 XM +(\242)SH +8 /Times-Italic AF +14788 26342 MT +(k)SH +10 /Times-Roman AF +15408 25997 MT +(equal 1 for exactly) +15 W( one value of)14 W +/Times-Italic SF +28422 XM +(k)SH +/Times-Roman SF +(. The) +278 W( sequences)14 W +/Times-Italic SF +35781 XM +(x)SH +8 /Times-Roman AF +36225 26342 MT +(1)SH +10 SS +36625 25997 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Times-Roman AF +39319 26342 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +40383 25997 MT +(and)SH +/Times-Italic SF +42091 XM +(x)SH +/Symbol SF +42685 XM +(\242)SH +8 /Times-Roman AF +42932 26342 MT +(1)SH +10 SS +43332 25997 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +/Symbol SF +46176 XM +(\242)SH +8 /Times-Roman AF +46423 26342 MT +(2)SH +/Times-Italic SF +(n)SH +10 /Times-Roman AF +47487 25997 MT +(lead to the same)14 W +7200 27374 MT +(terminal vertex in)SH +/Times-Italic SF +14560 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051, but)SH +/Times-Italic SF +9200 29165 MT +(mul)SH +8 SS +10700 29510 MT +(i)SH +10 /Times-Roman AF +11072 29165 MT +(\050)SH +/Times-Italic SF +(x)SH +8 /Symbol AF +11849 29510 MT +(p)SH +/Times-Roman SF +(\0501\051)SH +10 SS +13220 29165 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +8 /Symbol AF +15914 29510 MT +(p)SH +/Times-Roman SF +(\0502)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051)SH +10 SS +17685 29165 MT +(\051 =)436 W +/Times-Italic SF +19954 XM +(x)SH +8 SS +20398 29510 MT +(j)SH +10 /Times-Roman AF +7200 30685 MT +(while)SH +/Times-Italic SF +9200 32590 MT +(mul)SH +8 SS +10700 32935 MT +(i)SH +10 /Times-Roman AF +11072 32590 MT +(\050)SH +/Times-Italic SF +(x)SH +/Symbol SF +11999 XM +(\242)SH +8 SS +12246 32935 MT +(p)SH +/Times-Roman SF +(\0501\051)SH +10 SS +13617 32590 MT +(, . . . ,)SH +/Times-Italic SF +(x)SH +/Symbol SF +16461 XM +(\242)SH +8 SS +16708 32935 MT +(p)SH +/Times-Roman SF +(\0502)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +(\051)SH +10 SS +18479 32590 MT +(\051 =)436 W +/Times-Italic SF +20748 XM +(x)SH +/Symbol SF +21342 XM +(\242)SH +8 /Times-Italic AF +21589 32935 MT +(j)SH +10 /Symbol AF +22647 32590 MT +(\271)SH +/Times-Italic SF +24032 XM +(x)SH +8 SS +24476 32935 MT +(j)SH +10 /Times-Roman AF +24698 32590 MT +(.)SH +7200 34318 MT +(This contradiction forces us to conclude that graph)SH +/Times-Italic SF +27752 XM +(G)SH +/Times-Roman SF +(\050)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(,)SH +/Symbol SF +(p)SH +/Times-Roman SF +(\051 must contain at least 2)SH +8 /Times-Italic AF +39411 33973 MT +(q)SH +39811 34318 MT +(i)SH +10 /Times-Roman AF +40283 XM +(vertices.)SH +8200 36797 MT +(To get the final result, we need only show that for) +45 W( some value of)46 W +/Times-Italic SF +34913 XM +(i)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +35737 XM +(q)SH +8 SS +36237 37142 MT +(i)SH +10 /Symbol AF +36859 36797 MT +(\263)SH +/Times-Italic SF +37808 XM +(n)SH +/Times-Roman SF +(/8. This) +342 W( involves a counting argument)46 W +7200 38174 MT +(expressed by the following lemma.)SH +/Times-Bold SF +10200 40532 MT +(Lemma 4:)SH +/Times-Roman SF +14810 XM +(Suppose)SH +/Times-Italic SF +18449 XM +(A)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(B)SH +/Symbol SF +20321 XM +(\315)SH +/Times-Roman SF +21434 XM +({1, . . . ,)SH +/Times-Italic SF +(n)SH +/Times-Roman SF +25314 XM +(} each contain at least)SH +/Times-Italic SF +34375 XM +(n)SH +/Times-Roman SF +(/2 elements. For 1)SH +/Symbol SF +42497 XM +(\243)SH +/Times-Italic SF +43196 XM +(i)SH +/Symbol SF +43624 XM +(\243)SH +/Times-Roman SF +44323 XM +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1 let)SH +/Times-Italic SF +11200 42437 MT +(q)SH +8 SS +11700 42782 MT +(i)SH +10 /Times-Roman AF +12608 42437 MT +(=)SH +/Symbol SF +13858 XM +(|)SH +/Times-Roman SF +14208 XM +({) +150 MX(<)SH +/Times-Italic SF +(a)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(>)SH +/Symbol SF +17366 XM +(|)SH +/Times-Italic SF +17716 XM +(a)SH +/Symbol SF +18366 XM +(\316)SH +/Times-Italic SF +19229 XM +(A)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +20340 XM +(b)SH +/Symbol SF +20990 XM +(\316)SH +/Times-Italic SF +21853 XM +(B)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +22964 XM +(a)SH +/Times-Roman SF +(+)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(=)SH +/Times-Italic SF +(i)SH +/Times-Roman SF +(+1})SH +/Symbol SF +27064 XM +(|)SH +/Times-Roman SF +9200 44071 MT +(Then there is some)SH +/Times-Italic SF +17032 XM +(i)SH +/Times-Roman SF +17560 XM +(such that)SH +/Times-Italic SF +21393 XM +(q)SH +8 SS +21893 44416 MT +(i)SH +10 /Symbol AF +22415 44071 MT +(\263)SH +/Times-Italic SF +23264 XM +(n)SH +/Times-Roman SF +(/8.)SH +/Times-Italic SF +8200 46550 MT +(Proof)SH +/Times-Roman SF +10744 XM +(: Observe) +282 W( that since the sets)16 W +/Times-Italic SF +22493 XM +(A)SH +/Times-Roman SF +23370 XM +(and)SH +/Times-Italic SF +25079 XM +(B)SH +/Times-Roman SF +25955 XM +(each contain at least)15 W +/Times-Italic SF +34346 XM +(n)SH +/Times-Roman SF +(/2 elements, there are at least)15 W +/Times-Italic SF +46794 XM +(n)SH +8 /Times-Roman AF +47294 46205 MT +(2)SH +10 SS +47694 46550 MT +(/4 ordered pairs)15 W +7200 47927 MT +(<)SH +/Times-Italic SF +(a)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +(b)SH +/Times-Roman SF +(> with)SH +/Times-Italic SF +11856 XM +(a)SH +/Symbol SF +12756 XM +(\316)SH +/Times-Italic SF +13869 XM +(A)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +14980 XM +(b)SH +/Symbol SF +15880 XM +(\316)SH +/Times-Italic SF +16993 XM +(B)SH +/Times-Roman SF +(. Hence)250 W +/Times-Italic SF +11239 50745 MT +(q)SH +8 SS +11739 51090 MT +(j)SH +10 /Symbol AF +12797 50745 MT +(\263)SH +8 /Times-Roman AF +9200 49581 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +16 /Symbol AF +9449 51036 MT +(\345)SH +8 /Times-Italic AF +9483 51830 MT +(j)SH +/Times-Roman SF +(=1)SH +10 /Times-Italic AF +14182 50081 MT +(n)SH +8 /Times-Roman AF +14682 49736 MT +(2)SH +900 50 14182 50500 UL +10 SS +14382 51405 MT +(4)SH +7200 52925 MT +(For some value of)SH +/Times-Italic SF +14699 XM +(i)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +15477 XM +(q)SH +8 SS +15977 53270 MT +(i)SH +10 /Times-Roman AF +16449 52925 MT +(must be at least as large as the average value of the)SH +/Times-Italic SF +37110 XM +(q)SH +8 SS +37610 53270 MT +(j)SH +10 /Times-Roman AF +37832 52925 MT +('s:)SH +/Times-Italic SF +9200 55588 MT +(q)SH +8 SS +9700 55933 MT +(i)SH +10 /Symbol AF +10758 55588 MT +(\263 \327) +2885 W( \263)1736 W +/Times-Roman SF +18563 XM +(.)SH +12917 54924 MT +(1)SH +2049 50 12143 55343 UL +12143 56248 MT +(2)SH +/Times-Italic SF +(n)SH +/Symbol SF +(-)SH +/Times-Roman SF +(1)SH +/Times-Italic SF +14942 54924 MT +(n)SH +8 /Times-Roman AF +15442 54579 MT +(2)SH +900 50 14942 55343 UL +10 SS +15142 56248 MT +(4)SH +/Times-Italic SF +18063 54924 MT +(n)SH +500 50 18063 55343 UL +/Times-Roman SF +18063 56248 MT +(8)SH +8200 58727 MT +(This theorem shows that for) +118 W( any ordering, some multiplier output will have a graph of exponential size. This)119 W +7200 60104 MT +(leaves open the possibility) +14 W( that for each output, there could be some ordering giving a polynomial size graph for this)13 W +7200 61481 MT +(output. We) +372 W( conjecture, however, that this is not the case, namely that for certain outputs \050e.g.) +61 W( output)62 W +/Times-Italic SF +48794 XM +(n)SH +/Times-Roman SF +(\051, the graph)62 W +7200 62858 MT +(for this function is of exponential) +191 W( size regardless of the ordering.)190 W +8 SS +35017 62513 MT +(12)SH +10 SS +36697 62858 MT +(A proof of this conjecture would require)190 W +7200 64235 MT +(something stronger than our simple shifter argument. Such a proof would also) +9 W( lead to an interesting area-time lower)10 W +7200 65612 MT +(bound on circuits computing single bits in the product of two binary numbers.)SH +10800 50 7200 68504 UL +6 SS +8000 69843 MT +(12)SH +8 /Times-Bold AF +8600 70152 MT +(Update:)SH +/Times-Roman SF +11889 XM +(This conjecture was subsequently proved by the author \050R. E. Bryant, "On the Complexity of) +67 W( VLSI Implementations and Graph)66 W +7200 71076 MT +(Representations of Boolean Functions with Application) +91 W( to Integer Multiplication,")92 W +/Times-Italic SF +34603 XM +(IEEE Transactions on Computers)92 W +/Times-Roman SF +45992 XM +(40-2, pp. 205-213, Feb.,)92 W +7200 72000 MT +(1991.\051)SH +ES +%%Page: 25 25 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30100 4286 MT +(25)SH +12 /Times-Bold AF +7200 8004 MT +(References)SH +10 /Times-Roman AF +7200 9795 MT +(1.)SH +10200 XM +(C.Y. Lee, ``Representation of Switching Circuits by Binary-Decision Programs'',)182 W +/Times-Italic SF +44806 XM +(Bell System Technical)181 W +10200 10900 MT +(Journal)SH +/Times-Roman SF +(, Vol. 38, July 1959, pp. 985-999.)SH +7200 12691 MT +(2.)SH +10200 XM +(S.B. Akers,) +85 W( ``Binary Decision Diagrams'',)86 W +/Times-Italic SF +27987 XM +(IEEE Transactions on Computers)86 W +/Times-Roman SF +(, Vol.) +SH( C-27,) +86 W( No.) +SH( 6,) +86 W( June 1978,)SH +10200 13796 MT +(pp. 509-516.)SH +7200 15587 MT +(3.)SH +10200 XM +(M.R. Garey and D.S. Johnson,)121 W +/Times-Italic SF +23304 XM +(Computers and Intractibility: A Guide to the) +121 W( Theory of NP-Completeness,)120 W +/Times-Roman SF +10200 16692 MT +(Freeman, New York, 1979.)SH +7200 18483 MT +(4.)SH +10200 XM +(F.J. Hill and G.R. Peterson,)SH +/Times-Italic SF +21478 XM +(Introduction to Switching Theory and Logical Design,)SH +/Times-Roman SF +43423 XM +(Wiley, New York, 1974.)SH +7200 20274 MT +(5.)SH +10200 XM +(J.P. Roth,)SH +/Times-Italic SF +14340 XM +(Computer Logic, Testing, and Verification,)SH +/Times-Roman SF +31841 XM +(Computer Science Press, Potomac, MD., 1980.)SH +7200 22065 MT +(6.)SH +10200 XM +(R. Brayton,)89 W +/Times-Italic SF +15268 XM +(et al)90 W +/Times-Roman SF +(, ``Fast Recursive Boolean Function Manipulation'',)90 W +/Times-Italic SF +38924 XM +(International Symposium on Circuits)90 W +10200 23170 MT +(and Systems)SH +/Times-Roman SF +(, IEEE, Rome, Italy, May 1982, pp. 58-62.)SH +7200 24961 MT +(7.)SH +10200 XM +(B.M.E. Moret, ``Decision) +51 W( Trees and Diagrams'',)50 W +/Times-Italic SF +30303 XM +(ACM Computing Surveys)50 W +/Times-Roman SF +(, Vol.) +SH( 14,) +50 W( No.) +SH( 4,) +50 W( December 1982,)SH +10200 26066 MT +(pp. 593-623.)SH +7200 27857 MT +(8.)SH +10200 XM +(S. Fortune, J. Hopcroft, and E.M. Schmidt,) +166 W( ``The Complexity of Equivalence and Containment for Free)167 W +10200 28962 MT +(Single Variable Program) +320 W( Schemes'', in)319 W +/Times-Italic SF +27739 XM +(Automata, Languages, and Programming,)319 W +/Times-Roman SF +46154 XM +(Goos, Hartmannis,)319 W +10200 30067 MT +(Ausiello, and Boehm, eds., Springer-Verlag, Lecture Notes in Computer Science,) +286 W( Vol. 62, 1978, pp.)287 W +10200 31172 MT +(227-240.)SH +7200 32963 MT +(9.)SH +10200 XM +(R.W. Payne, ``Reticulation and Other Methods of) +70 W( Reducing the Size of Printed Diagnostic Keys'',)69 W +/Times-Italic SF +50889 XM +(Journal)SH +10200 34068 MT +(of General Microbiology)SH +/Times-Roman SF +(, Vol. 98, 1977, pp. 595-597.)SH +7200 35859 MT +(10.)SH +10200 XM +(R. Brayton,)SH +/Times-Italic SF +15089 XM +(et al)SH +/Times-Roman SF +(,)SH +/Times-Italic SF +17339 XM +(Logic Minimization Algorithms for VLSI Synthesis,)SH +/Times-Roman SF +38035 XM +(Kluwer Academic Publishers, 1984.)SH +7200 37650 MT +(11.)SH +10200 XM +(C.E. Shannon, ``A Symbolic Analysis of Relay and Switching Circuits'',)147 W +/Times-Italic SF +41169 XM +(Transactions of the) +147 W( AIEE)148 W +/Times-Roman SF +(, Vol.)SH +10200 38755 MT +(57, 1938, pp. 713-723.)SH +7200 40546 MT +(12.)SH +10200 XM +(C.S. Wallace, ``A Suggestion for a Fast Multiplier'',)185 W +/Times-Italic SF +33011 XM +(IEEE Transactions on Electronic) +185 W( Computing)184 W +/Times-Roman SF +(, Vol.)SH +10200 41651 MT +(EC-13, No. 1, January 1964, pp. 14-17.)SH +7200 43442 MT +(13.)SH +10200 XM +(A.V. Aho, J.E. Hopcroft, and J.D. Ullman,)164 W +/Times-Italic SF +28679 XM +(The Design) +164 W( and Analysis of Computer Algorithms,)165 W +/Times-Roman SF +50278 XM +(Addison-)SH +10200 44547 MT +(Wesley, Reading, MA., 1974.)SH +7200 46338 MT +(14.)SH +10200 XM +(J.S. Jephson, R.P. McQuarrie, and R.E. Vogelsberg, ``A Three-Level) +151 W( Design Verification System'',)150 W +/Times-Italic SF +52223 XM +(IBM)SH +10200 47443 MT +(Systems Journal)SH +/Times-Roman SF +(, Vol. 8, No. 3, 1969, pp. 178-188.)SH +7200 49234 MT +(15.)SH +10450 XM +(Texas Instruments,)SH +/Times-Italic SF +18310 XM +(TTL Data Book,)SH +/Times-Roman SF +25033 XM +(1976.)SH +7200 51025 MT +(16.)SH +10200 XM +(M. Rowan-Robinson,)SH +/Times-Italic SF +19089 XM +(Cosmology,)SH +/Times-Roman SF +24089 XM +(Oxford Univ. Press, 1977.)SH +7200 52816 MT +(17.)SH +10200 XM +(S.B. Akers, ``Functional Testing with Binary Decision Diagrams'',)164 W +/Times-Italic SF +38676 XM +(Eighth Annual Conference) +164 W( on Fault-)165 W +10200 53921 MT +(Tolerant Computing)SH +/Times-Roman SF +(, IEEE, 1978, pp. 75-82.)SH +7200 55712 MT +(18.)SH +10200 XM +(R.P. Brent, and H.T. Kung, ``The) +306 W( Area-Time Complexity of Binary Multiplication'',)305 W +/Times-Italic SF +47779 XM +(Journal of the)305 W +10200 56817 MT +(ACM)SH +/Times-Roman SF +(, Vol. 28, No. 3, July 1981, pp. 521-534.)SH +7200 58608 MT +(19.)SH +10200 XM +(H. Abelson, and P. Andreae, ``Information Transfer and Area-Time) +102 W( Trade-Offs for VLSI Multiplication'',)103 W +/Times-Italic SF +10200 59713 MT +(Communications of the ACM)SH +/Times-Roman SF +(, Vol. 23, No. 1, January 1980, pp. 20-23.)SH +ES +%%Page: i 26 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30461 4286 MT +(i)SH +12 /Times-Bold AF +26033 8004 MT +(Table of Contents)SH +11 SS +8850 9172 MT +(1. Introduction)SH +53450 XM +(1)SH +10 SS +10700 10252 MT +(1.1. Notation)SH +53500 XM +(3)SH +11 SS +8850 11420 MT +(2. Representation)SH +53450 XM +(4)SH +8850 12588 MT +(3. Properties)SH +53450 XM +(6)SH +10 SS +10700 13668 MT +(3.1. Example Functions)SH +53500 XM +(7)SH +10700 14748 MT +(3.2. Ordering Dependency)SH +53500 XM +(7)SH +10700 15828 MT +(3.3. Inherently Complex Functions)SH +53500 XM +(8)SH +11 SS +8850 16996 MT +(4. Operations)SH +53450 XM +(9)SH +10 SS +10700 18076 MT +(4.1. Data Structures)SH +53500 XM +(9)SH +10700 19156 MT +(4.2. Reduction)SH +53000 XM +(10)SH +10700 20236 MT +(4.3. Apply)SH +53000 XM +(12)SH +10700 21316 MT +(4.4. Restriction)SH +53000 XM +(15)SH +10700 22396 MT +(4.5. Composition)SH +53000 XM +(16)SH +10700 23476 MT +(4.6. Satisfy)SH +53000 XM +(16)SH +11 SS +8850 24644 MT +(5. Experimental Results)SH +52900 XM +(19)SH +8850 25812 MT +(6. Conclusion)SH +52900 XM +(21)SH +8850 26980 MT +(Appendix: The Complexity of Integer Multiplication)SH +52900 XM +(23)SH +8850 28148 MT +(References)SH +52900 XM +(25)SH +ES +%%Page: ii 27 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30322 4286 MT +(ii)SH +12 /Times-Bold AF +26866 8004 MT +(List of Figures)SH +11 SS +8850 9172 MT +(Figure 1. Example Function Graphs)SH +53450 XM +(6)SH +8850 10340 MT +(Figure 2. Example of Argument Ordering Dependency)SH +53450 XM +(7)SH +8850 11508 MT +(Figure 3. Implementation of Ordered Traversal)SH +52900 XM +(10)SH +8850 12676 MT +(Figure 4. Implementation of)SH +/Times-BoldItalic SF +22323 XM +(Reduce)SH +/Times-Bold SF +52900 XM +(11)SH +8850 13844 MT +(Figure 5. Reduction Algorithm Example)SH +52900 XM +(12)SH +8850 15012 MT +(Figure 6. Implementation of)SH +/Times-BoldItalic SF +22323 XM +(Apply)SH +/Times-Bold SF +52900 XM +(14)SH +8850 16180 MT +(Figure 7. Example of)SH +/Times-BoldItalic SF +18963 XM +(Apply)SH +/Times-Bold SF +52900 XM +(15)SH +8850 17348 MT +(Figure 8. Implementation of)SH +/Times-BoldItalic SF +22323 XM +(Compose)SH +/Times-Bold SF +52900 XM +(17)SH +8850 18516 MT +(Figure 9. Implementation of)SH +/Times-BoldItalic SF +22323 XM +(Satisfy-one)SH +/Times-Bold SF +52900 XM +(18)SH +8850 19684 MT +(Figure 10. Implementation of)SH +/Times-BoldItalic SF +22873 XM +(Satisfy-all)SH +/Times-Bold SF +52900 XM +(19)SH +8850 20852 MT +(Figure 11. ALU Output Graph Sizes for Different Input Orderings)SH +52900 XM +(21)SH +ES +%%Page: iii 28 + +61200 79200 BS +0 SI +10 /Times-Roman AF +30183 4286 MT +(iii)SH +12 /Times-Bold AF +27099 8004 MT +(List of Tables)SH +11 SS +8850 9172 MT +(Table 1. Summary of Basic Operations)SH +53450 XM +(9)SH +8850 10340 MT +(Table 2. ALU Verification Examples)SH +52900 XM +(20)SH +ES +%%Trailer +%%Pages: 28 +%%DocumentFonts: Times-Roman Times-Bold Times-BoldItalic Times-Italic Symbol diff --git a/binary_decision_trees/bdd_datalog.pdf b/binary_decision_trees/bdd_datalog.pdf new file mode 100644 index 0000000..b5038fb Binary files /dev/null and b/binary_decision_trees/bdd_datalog.pdf differ diff --git a/binary_decision_trees/bdd_ordering_heuristics_survey.pdf b/binary_decision_trees/bdd_ordering_heuristics_survey.pdf new file mode 100644 index 0000000..7f50116 Binary files /dev/null and b/binary_decision_trees/bdd_ordering_heuristics_survey.pdf differ diff --git a/binary_decision_trees/bdd_prolog.pdf b/binary_decision_trees/bdd_prolog.pdf new file mode 100644 index 0000000..d474405 Binary files /dev/null and b/binary_decision_trees/bdd_prolog.pdf differ diff --git a/binary_decision_trees/bdd_variable_order.pdf b/binary_decision_trees/bdd_variable_order.pdf new file mode 100644 index 0000000..1547214 Binary files /dev/null and b/binary_decision_trees/bdd_variable_order.pdf differ diff --git a/binary_decision_trees/dynamic_var_order_bdd.pdf b/binary_decision_trees/dynamic_var_order_bdd.pdf new file mode 100644 index 0000000..135fbf8 Binary files /dev/null and b/binary_decision_trees/dynamic_var_order_bdd.pdf differ diff --git a/binary_decision_trees/efficient_bdd_package.pdf b/binary_decision_trees/efficient_bdd_package.pdf new file mode 100644 index 0000000..a6492e4 Binary files /dev/null and b/binary_decision_trees/efficient_bdd_package.pdf differ diff --git a/binary_decision_trees/l13-Datalog-and-BDD.pdf b/binary_decision_trees/l13-Datalog-and-BDD.pdf new file mode 100644 index 0000000..b411647 Binary files /dev/null and b/binary_decision_trees/l13-Datalog-and-BDD.pdf differ diff --git a/binary_decision_trees/tech01_zdd.pdf b/binary_decision_trees/tech01_zdd.pdf new file mode 100644 index 0000000..7be60fa Binary files /dev/null and b/binary_decision_trees/tech01_zdd.pdf differ diff --git a/constraint_handling_rules/02-Csorba-chr12.pdf b/constraint_handling_rules/02-Csorba-chr12.pdf new file mode 100644 index 0000000..8c039fe Binary files /dev/null and b/constraint_handling_rules/02-Csorba-chr12.pdf differ diff --git a/constraint_handling_rules/chr-stm.ps b/constraint_handling_rules/chr-stm.ps new file mode 100644 index 0000000..9782d0f --- /dev/null +++ b/constraint_handling_rules/chr-stm.ps @@ -0,0 +1,3494 @@ +%!PS-Adobe-2.0 +%%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software +%%Title: CCHR07.dvi +%%Pages: 5 +%%PageOrder: Ascend +%%BoundingBox: 0 0 595 842 +%%DocumentFonts: Times-Bold Times-Roman CMTT8 CMSY8 CMMI9 Times-Italic +%%+ CMTT9 CMR9 MSAM10 CMSY6 CMSY9 CMR6 CMMI6 CMMI7 CMR5 CMR8 CMMI8 MSAM7 +%%+ CMSY5 +%%DocumentPaperSizes: a4 +%%EndComments +%DVIPSWebPage: (www.radicaleye.com) +%DVIPSCommandLine: dvips -oo.ps CCHR07 +%DVIPSParameters: dpi=600 +%DVIPSSource: TeX output 2007.01.09:1705 +%%BeginProcSet: tex.pro 0 0 +%! 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All Rights Reserved) readonly def +/FullName (CMSY5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 48 /prime put +readonly def +/FontBBox{21 -944 1448 791}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMTT8 +%!PS-AdobeFont-1.1: CMTT8 1.0 +%%CreationDate: 1991 Aug 20 16:46:05 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMTT8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch true def +end readonly def +/FontName /CMTT8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 44 /comma put +dup 46 /period put +dup 48 /zero put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 54 /six put +dup 58 /colon put +dup 64 /at put +dup 71 /G put +dup 97 /a put +dup 99 /c put +dup 100 /d put +dup 101 /e put +dup 103 /g put +dup 108 /l put +dup 109 /m put +dup 110 /n put +dup 111 /o put +dup 112 /p put +dup 114 /r put +dup 115 /s put +dup 117 /u put +dup 122 /z put +readonly def +/FontBBox{-5 -232 545 699}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSAM7) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSAM7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 26 /arrowtailright put +readonly def +/FontBBox{0 -576 1569 1076}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI8 +%!PS-AdobeFont-1.1: CMMI8 1.100 +%%CreationDate: 1996 Jul 23 07:53:54 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +readonly def +/FontBBox{-24 -250 1110 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE +D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 +5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC +4391C9DF440285B8FC159D0E98D4258FC57892DDF753642CD526A96ACEDA4120 +788F22B1D09F149794E66DD1AC2C2B3BC6FEC59D626F427CD5AE9C54C7F78F62 +C36F49B3C2E5E62AFB56DCEE87445A12A942C14AE618D1FE1B11A9CF9FAA1F32 +617B598CE5058715EF3051E228F72F651040AD99A741F247C68007E68C84E9D1 +D0BF99AA5D777D88A7D3CED2EA67F4AE61E8BC0495E7DA382E82DDB2B009DD63 +532C74E3BE5EC555A014BCBB6AB31B8286D7712E0E926F8696830672B8214E9B +5D0740C16ADF0AFD47C4938F373575C6CA91E46D88DE24E682DEC44B57EA8AF8 +4E57D45646073250D82C4B50CBBB0B369932618301F3D4186277103B53B3C9E6 +DB42D6B30115F67B9D078220D5752644930643BDF9FACF684EBE13E39B65055E +B1BD054C324962025EC79E1D155936FE32D9F2224353F2A46C3558EF216F6BB2 +A304BAF752BEEC36C4440B556AEFECF454BA7CBBA7537BCB10EBC21047333A89 +8936419D857CD9F59EBA20B0A3D9BA4A0D3395336B4CDA4BA6451B6E4D1370FA +D9BDABB7F271BC1C6C48D9DF1E5A6FAE788F5609DE3C48D47A67097C547D9817 +AD3A7CCE2B771843D69F860DA4059A71494281C0AD8D4BAB3F67BB6739723C04 +AE05F9E35B2B2CB9C7874C114F57A185C8563C0DCCA93F8096384D71A2994748 +A3C7C8B8AF54961A8838AD279441D9A5EB6C1FE26C98BD025F353124DA68A827 +AE2AF8D25CA48031C242AA433EEEBB8ABA4B96821786C38BACB5F58C3D5DA011 +85B385124C2B6534F3CD1866AF92009D93B97F763AA3C28584857D64C6AD5C24 +68B3330EC1F0EB0EBBB855EE71262417C4C15AFBAFAC26FDEFD63C49486338FE +64211DEF7AFFBC82242BF22DD3AAD6FB1356B1FB6C464FFCBD266112D0F6D9EA +7688B88321AC6512E7BA5425531791CB08FE6ADEEBA6A503974714F7EEE8420C +9374BE70F06289C3D329DAC3A0D9696E451431465340E48CB3FC0DDEBB369028 +8DDA8A5D594537EE051D51CADBB0BAFC654AA9861BF12B3BDC5EEB6C04DA99CE +CA42B984B7D75C783CCFCFBDF5DE550E10C82CFA1582DBDDE6C3AFD64496CC8D +88C037E851FA71EC285D00266AF95D70F6E7 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndFont +%%BeginFont: CMR8 +%!PS-AdobeFont-1.1: CMR8 1.0 +%%CreationDate: 1991 Aug 20 16:39:40 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR8) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR8 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +readonly def +/FontBBox{-36 -250 1070 750}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMR5 +%!PS-AdobeFont-1.1: CMR5 1.00B +%%CreationDate: 1992 Feb 19 19:55:02 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.00B) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR5) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR5 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 49 /one put +dup 50 /two put +dup 51 /three put +readonly def +/FontBBox{-341 -250 1304 965}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMMI7 +%!PS-AdobeFont-1.1: CMMI7 1.100 +%%CreationDate: 1996 Jul 23 07:53:53 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI7) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI7 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +dup 67 /C put +dup 84 /T put +dup 100 /d put +dup 101 /e put +dup 114 /r put +dup 117 /u put +readonly def +/FontBBox{0 -250 1171 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE +D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.100) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMMI6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 59 /comma put +dup 66 /B put +dup 68 /D put +dup 82 /R put +dup 105 /i put +dup 106 /j put +dup 109 /m put +dup 110 /n put +dup 114 /r put +dup 120 /x put +dup 121 /y put +readonly def +/FontBBox{11 -250 1241 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE +3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B +532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 +B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B +986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 91 /bracketleft put +dup 93 /bracketright put +readonly def +/FontBBox{-20 -250 1193 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 +469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 0 /minus put +dup 17 /equivalence put +dup 18 /reflexsubset put +dup 40 /arrowdblleft put +dup 41 /arrowdblright put +dup 44 /arrowdblboth put +dup 50 /element put +dup 56 /universal put +dup 57 /existential put +dup 58 /logicalnot put +dup 59 /emptyset put +dup 93 /unionmulti put +dup 102 /braceleft put +dup 103 /braceright put +dup 104 /angbracketleft put +dup 105 /angbracketright put +dup 106 /bar put +dup 107 /bardbl put +readonly def +/FontBBox{-30 -958 1146 777}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMSY6) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.035 def +/isFixedPitch false def +end readonly def +/FontName /CMSY6 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 3 /asteriskmath put +dup 48 /prime put +dup 106 /bar put +readonly def +/FontBBox{-4 -948 1329 786}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 +7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 +A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 +E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A +221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 +49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E +9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 +1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 +6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 +007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 +F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 +C5FD550F53CE5583743B50B0F9B7C4F836DEF1B263DB260571DCE3DFEC496E93 +CB244233DDAA5033CD8D8B204D87ABD75B439379225D1A63D5AB6E69E23855AE +F38E2BB7A6C2D5552261CE6C0009A08AD3A04A01BA0C1111B17CABAC3BDCD3DD +4E110C9CD1CD07B43C53B8EB3F5538C1E52D4C9EFBAE77F451041F09E3DD1AFD +873978F2F286853A80C6B732B7F13CB6E8C329BBB2050D2F3BCFA0DCD73B27B5 +19FC79CBF299D917171E983307B2A1FF73B066E40FD464B939570E518939446D +90CC173201C3B5D6DDDD7260B3498B701ABCFAD82CF8868B4886A86CDBABDAC7 +BEFB98292A8B053A11ADA107F07683E456C9254A2279A83E3EEC6D589CFDC9F4 +A6D9FA509A6E54E6261AE3ED3B064A81D81C993D6AB43D2E0C37C3C03DF746C7 +CB60B7949BC9A40192D7FBDB2F4578FEC7FA61C27BDDCAF30CC4813AFD8549A9 +3E4CDF913C46969D166702783D4B2D64C440B0EED7211B7FF8E28485E65D7C82 +822693CBB53775A31D20B827091843317A7705D720894D38B14F739F644D9664 +8292614955615ABFB4BFD5002EDB1BAEDF6D48E6A436FB9EFD76FDEB175A3B99 +C501E3D9687D7774B7882BCD50241FD3674D0CBFB907CFBB48DAEFF75E63C746 +46118F2C8513C97B6751404209C7AD069D9DF523049B683E16F901D5D6DADE82 +7973A1CD39085FE734A4112A8B4EBDA7884DFA233A13C601F4695159221CAC7B +F504A28AEC6C71C8F503950B7E5A04692F6D187A65681E67A32034A89591F2 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%EndFont +%%BeginFont: MSAM10 +%!PS-AdobeFont-1.1: MSAM10 2.1 +%%CreationDate: 1993 Sep 17 09:05:00 +% Math Symbol fonts were designed by the American Mathematical Society. +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (2.1) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (MSAM10) readonly def +/FamilyName (Euler) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /MSAM10 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 26 /arrowtailright put +readonly def +/FontBBox{8 -463 1331 1003}readonly def +currentdict end +currentfile eexec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cleartomark +%%EndFont +%%BeginFont: CMR9 +%!PS-AdobeFont-1.1: CMR9 1.0 +%%CreationDate: 1991 Aug 20 16:39:59 +% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def +/FullName (CMR9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle 0 def +/isFixedPitch false def +end readonly def +/FontName /CMR9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 35 /numbersign put +dup 38 /ampersand put +dup 40 /parenleft put +dup 41 /parenright put +dup 43 /plus put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 58 /colon put +dup 61 /equal put +dup 91 /bracketleft put +dup 93 /bracketright put +readonly def +/FontBBox{-39 -250 1036 750}readonly def +currentdict end +currentfile eexec +D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 +016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 +9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F +D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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All Rights Reserved. +11 dict begin +/FontInfo 7 dict dup begin +/version (1.0) readonly def +/Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def +/FullName (CMMI9) readonly def +/FamilyName (Computer Modern) readonly def +/Weight (Medium) readonly def +/ItalicAngle -14.04 def +/isFixedPitch false def +end readonly def +/FontName /CMMI9 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 15 /epsilon1 put +dup 18 /theta put +dup 30 /phi put +dup 58 /period put +dup 59 /comma put +dup 61 /slash put +dup 62 /greater put +dup 65 /A put +dup 66 /B put +dup 67 /C put +dup 68 /D put +dup 69 /E put +dup 72 /H put +dup 76 /L put +dup 84 /T put +dup 97 /a put +dup 98 /b put +dup 99 /c put +dup 100 /d put +dup 101 /e put +dup 103 /g put +dup 105 /i put +dup 106 /j put +dup 108 /l put +dup 110 /n put +dup 111 /o put +dup 114 /r put +dup 115 /s put +dup 116 /t put +dup 117 /u put +dup 120 /x put +dup 121 /y put +readonly def +/FontBBox{-29 -250 1075 750}readonly def +currentdict end +currentfile eexec 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