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9B3A94CEF929F39B91C7D69E2B19667496FD48370DF93F57793C1DB12C710704 9664571233EF8E5E4AEFC58572CC2B5FF2EC1D4F8CB0C283C8140EFC55B24626 F35E2FE535BE6C91F5E83DEF5EB2E7DF93AAE6F2896AB7E47BE4E80DB258A827 57ADF1BC4AEBD88010A57183B72A23C2D3AC8982A53D38A13EDE0AE2B2560374 4B620E7B894B2DF0473A76EE0F8E5EF8874BD007908FA07DF3312DA191D19700 001656D2D9E5E36AA359E612183557E5259876F49C2D4D784518775EC3426845 53A02CECC08DD71A1FDC63F71547E5D3B1E832EE04F857ED8D9A3A3F910AB6B7 1F301DC4231BA0D8D7D5670472CE97159803F3E28411144BD70948F51E87C077 651E29A001035CEECC3A269C38B78D7E116924A43D4D1AF07BD418ADEBA9C104 DF35531BC18958730688C331674FA6F99CEA9209918E676E48AE34188E4A5981 D490FA3F4DDE8740496D0563D907CDF23AE39FC8E4868DEF01FD8027A9266F5D 3D701E3D4540D5334C888C3539B3997B88164799CBE5CC5F7F8CD4CD52A82C63 18BFF57275FAA2C373C436AB357F8F56A5E77438D7EECD73944A7BB6F92CA14C B0F224EEC1BF8BD3433F4052D6EA5167DD1E21A5F07DBF46A06A4A9B98809F3C B903BB3430FEA3B5E795F585F1EAFF8BDB985F033CF3CC65150F7C4DD6DF5AAB 661404BC1B0AA2FB7AC0FB2E7A38BB79A3D6975898B718D104BFD38C7967CE00 D65F987AEFA6DBAFA7DBCF7A654C9CA20589F6E4C1968DF79251F8CA3BA97B96 5A2FFEE023D1F0C3897F5AE038612B96AF9814B91309EF17CFAB46BBC11554C5 4AEE3A00C1CA181444D3E0C4AA60F65B8D761ED1530E52F785C8E0D2DD238260 7E5FBAFD5BF80193D1E9EA630AF00EF9E06A4DF3375716AB8107E89C646E7206 4E4D9FE7E0A4835B34579402E46C55280B4E888228A15B7A86E5547D9DD630F4 D7051F718CCAC631366608149DA72A4516577FA39A01024B1026EB3F79B0D17D 450F34970DB24FE0F27A05DE4A311A48E6BCA0B0665F625E7FDC0193D9B34252 CE4C6A2498E66CC131A211B044EDD1BD0F4F464542724A4060947C6DEB6DBE8C 1C475FE6E58134B084F9E1C0561EBF99069F50E3DCC0E90AEFF840EF59B9652E 59C7F46C80BBF9E8228510C17040BCFDAFA9E17673140DB38A7178A6001CCC0A 29C570AE0DC01097DA773D9D02BDDC791535C5358271C51C7A0361F2B5329452 7F692D8388569C1EF488A2233163154BB05FE5BC40192C342775881BAA20395F BF2BE12ECB2173E2E9BF03AFA883F225C96629DF37459DB423DEB5AA3AB0C11D D3409C58E6709E8BAC5841285276F3B42AF0128DF6E781B903AFA7FF1895018C 0A105D2EC8EDD897657D7D7F0CAA6A67EEEF13EF8757A23CC79FFCFA74332827 ECA1D8455162096457B9689CCEEC1406173F90F964D0535D7021 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 58 /period put dup 59 /comma put dup 110 /n put dup 111 /o put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 5 /Pi put dup 40 /parenleft put dup 41 /parenright put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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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. 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 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % 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 (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 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 78 /N put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSS12 %!PS-AdobeFont-1.1: CMSS12 1.0 %%CreationDate: 1991 Aug 20 17:33:47 % 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 (CMSS12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS12 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 73 /I put dup 77 /M put dup 80 /P put dup 85 /U put dup 88 /X put dup 101 /e put dup 102 /f put dup 114 /r put readonly def /FontBBox{-62 -251 978 758}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 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 /parenleftbig put dup 1 /parenrightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 77 /circleplusdisplay put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % 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 (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 105 /i put dup 107 /k put dup 110 /n put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % 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 (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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 54 /six put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % 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 (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 6 /plusminus put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 73 /I put dup 91 /union put dup 92 /intersection put dup 98 /floorleft put dup 99 /floorright put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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4CF4319EAD237F4FFDDF1094E3A8036C319532A35A2CDE622395F109E9B72BCA 7A1AF1487F6A2258001FC860B7FF87E36709B846DDC2F8B19E54ACA9A620E65F 7482707FC4CE3D5F6E80B39BCC6BD089B4BC395A224FD43DED12AA54017EDA26 0CCB0EC224B5A7CCA688A27096F93875B424A10D73357575BD95498CAD14971C 2409D2B2CDACD822E5387636A58CC4CBD40A9B136D83D17A2AD03D578D3525B8 5FE5F0B7273125EF389E17CF308F0D2892C5E6F6FF60A1 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 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 103 /g put dup 108 /l put dup 111 /o put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /alpha put dup 12 /beta put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 30 /phi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 68 /D put dup 73 /I put dup 79 /O put dup 83 /S put dup 84 /T put dup 88 /X put dup 89 /Y put dup 98 /b put dup 99 /c put dup 105 /i put dup 106 /j put dup 107 /k 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 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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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 (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 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 11 /alpha put dup 12 /beta put dup 15 /epsilon1 put dup 22 /mu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 30 /phi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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b Fu(b)-5 b(e)29 b(the)g(minimum)g(value)g(of)g(the)h(obje)-5 b(ctive)29 b(function)g(of)g(the)h(fol)5 b(lowing)257 3266 y(line)-5 b(ar)35 b(pr)-5 b(o)g(gr)g(am.)43 b(Then)34 b Ft(s)p Fr(\()p Fj(IP)1375 3281 y Fs(n)1422 3266 y Fr(\))27 b Fq(\025)i Ft(z)d Fq(\001)c Ft(s)p Fr(\()p Fj(IP)1887 3281 y Fs(n)p Fo(\000)p Fp(2)2024 3266 y Fr(\))p Fu(.)415 3903 y SDict begin H.S end 415 3903 a 415 3903 a SDict begin 14.5 H.A end 415 3903 a 415 3903 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4) cvn H.B /DEST pdfmark end 415 3903 a 457 3503 a Fu(Minimize)1146 3408 y Fl(X)957 3624 y Fs(T)10 b Fo(\022f)p Fs(x)1138 3633 y Fn(1)1172 3624 y Fs(;y)1227 3633 y Fn(1)1261 3624 y Fs(;x)1321 3633 y Fn(2)1355 3624 y Fs(;y)1410 3633 y Fn(2)1444 3624 y Fo(g)1496 3503 y Ft(q)1539 3518 y Fs(T)457 3805 y Fu(Subje)-5 b(ct)34 b(to)1146 3711 y Fl(X)1116 3922 y Fs(T)10 b Fp(:)p Fs(v)r Fo(2)p Fs(T)1338 3805 y Ft(q)1381 3820 y Fs(T)1679 3805 y Fq(\025)28 b Fr(1)83 b(\()p Ft(v)32 b Fq(2)c(f)p Ft(x)2232 3820 y Fp(1)2271 3805 y Ft(;)17 b(y)2363 3820 y Fp(1)2402 3805 y Ft(;)g(x)2501 3820 y Fp(2)2541 3805 y Ft(;)g(y)2633 3820 y Fp(2)2671 3805 y Fq(g)p Fr(\))1146 3996 y Fl(X)955 4212 y Fs(T)10 b Fp(:)p Fo(jf)p Fs(v)1115 4221 y Fn(1)1150 4212 y Fs(;v)1204 4221 y Fn(2)1238 4212 y Fo(g\\)p Fs(T)g Fo(j)p Fp(=1)1498 4091 y Ft(q)1541 4106 y Fs(T)1679 4091 y Fq(\025)28 b Fr(1)83 b(\()p Ft(v)2001 4106 y Fp(1)2068 4091 y Fq(2)28 b(f)p Ft(x)2267 4106 y Fp(1)2307 4091 y Ft(;)17 b(y)2399 4106 y Fp(1)2438 4091 y Fq(g)p Ft(;)g(v)2579 4106 y Fp(2)2646 4091 y Fq(2)28 b(f)p Ft(x)2845 4106 y Fp(2)2884 4091 y Ft(;)17 b(y)2976 4106 y Fp(2)3015 4091 y Fq(g)p Fr(\))p Ft(;)1227 4323 y(q)1270 4338 y Fs(T)1679 4323 y Fq(\025)28 b Fr(0)83 b(\()p Ft(T)42 b Fq(\022)28 b(f)p Ft(x)2263 4338 y Fp(1)2302 4323 y Ft(;)17 b(x)2401 4338 y Fp(2)2441 4323 y Ft(;)g(y)2533 4338 y Fp(1)2572 4323 y Ft(;)g(y)2664 4338 y Fp(2)2702 4323 y Fq(g)p Fr(\))p Ft(:)3371 3903 y Fz(\(4\))404 4545 y(LP)34 b(\()603 4545 y SDict begin H.S end 603 4545 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Fs(n)p Fo(\000)p Fs(i)1335 1176 y Fz(.)72 b(By)43 b(dividing)d Ft(P)55 b Fz(in)m(to)42 b Fr(2)2358 1140 y Fp(2)p Fs(k)2478 1176 y Fz(parts)g(and)g(letting)f Ft(q)3298 1191 y Fs(T)3395 1176 y Fz(b)s(e)257 1297 y Ft(])p Fr(\()p Ft(P)396 1312 y Fs(T)451 1297 y Fr(\))p Ft(=s)p Fr(\()p Fj(IP)710 1312 y Fs(n)p Fo(\000)p Fs(k)851 1297 y Fr(\))p Fz(,)32 b(w)m(e)i(can)f(sho)m(w)g(that:)257 1321 y SDict begin H.S end 257 1321 a 257 1321 a SDict begin 14.5 H.A end 257 1321 a 257 1321 a SDict begin [ /View [/XYZ H.V] /Dest (fact.3) cvn H.B /DEST pdfmark end 257 1321 a 186 x Fv(F)-9 b(act)46 b(3.)f Fu(Supp)-5 b(ose)41 b(that)g Ft(k)i Fq(\025)d Fr(3)p Fu(.)64 b(L)-5 b(et)42 b Ft(z)1801 1522 y Fs(k)r Fo(\000)p Fp(1)1976 1507 y Fu(and)e Ft(z)2216 1522 y Fs(k)r Fo(\000)p Fp(2)2391 1507 y Fu(b)-5 b(e)41 b(r)-5 b(e)g(al)41 b(numb)-5 b(ers)41 b(such)g(that)257 1627 y Ft(s)p Fr(\()p Fj(IP)430 1642 y Fs(n)477 1627 y Fr(\))30 b Fq(\025)h Ft(z)698 1642 y Fs(k)r Fo(\000)p Fp(1)854 1627 y Fq(\001)23 b Ft(s)p Fr(\()p Fj(IP)1078 1643 y Fs(n)p Fo(\000)p Fp(\()p Fs(k)r Fo(\000)p Fp(1\))1363 1627 y Fr(\))36 b Fu(and)g Ft(s)p Fr(\()p Fj(IP)1801 1642 y Fs(n)1848 1627 y Fr(\))30 b Fq(\025)h Ft(z)2069 1642 y Fs(k)r Fo(\000)p Fp(2)2225 1627 y Fq(\001)23 b Ft(s)p Fr(\()p Fj(IP)2449 1643 y Fs(n)p Fo(\000)p Fp(\()p Fs(k)r Fo(\000)p Fp(2\))2734 1627 y Fr(\))36 b Fu(for)g(every)h Ft(n)p Fu(.)49 b(L)-5 b(et)257 1748 y Ft(z)302 1763 y Fs(k)393 1748 y Fu(b)g(e)48 b(the)f(minimum)g(value)h(of)f(the)h(obje)-5 b(ctive)47 b(function)g(of)h(the)g(fol)5 b(lowing)46 b(line)-5 b(ar)257 1868 y(pr)g(o)g(gr)g(am.)44 b(Then)34 b Ft(s)p Fr(\()p Fj(IP)1100 1883 y Fs(n)1147 1868 y Fr(\))27 b Fq(\025)h Ft(z)1362 1883 y Fs(k)1427 1868 y Fq(\001)22 b Ft(s)p Fr(\()p Fj(IP)1650 1883 y Fs(n)p Fo(\000)p Fs(k)1790 1868 y Fr(\))p Fu(.)260 2531 y SDict begin H.S end 260 2531 a 260 2531 a SDict begin 14.5 H.A end 260 2531 a 260 2531 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5) cvn H.B /DEST pdfmark end 260 2531 a 302 2110 a Fu(Minimize)1033 2015 y Fl(X)800 2231 y Fs(T)10 b Fo(\022f)p Fs(x)981 2240 y Fn(1)1016 2231 y Fs(;y)1071 2240 y Fn(1)1105 2231 y Fs(;:::)n(;x)1243 2243 y Fm(k)1281 2231 y Fs(;y)1336 2243 y Fm(k)1373 2231 y Fo(g)1425 2110 y Ft(q)1468 2125 y Fs(T)302 2412 y Fu(Subje)-5 b(ct)34 b(to)1033 2317 y Fl(X)1002 2529 y Fs(T)10 b Fp(:)p Fs(v)r Fo(2)p Fs(T)1224 2412 y Ft(q)1267 2427 y Fs(T)1607 2412 y Fq(\025)28 b Ft(z)1757 2427 y Fs(k)r Fo(\000)p Fp(1)1973 2412 y Fr(\()p Ft(v)j Fq(2)d(f)p Ft(x)2288 2427 y Fp(1)2328 2412 y Ft(;)17 b(y)2420 2427 y Fp(1)2459 2412 y Ft(;)g(:)g(:)g(:)e(;)i(x)2732 2427 y Fs(k)2775 2412 y Ft(;)g(y)2867 2427 y Fs(k)2909 2412 y Fq(g)p Fr(\))1033 2643 y Fl(X)842 2859 y Fs(T)10 b Fp(:)p Fo(jf)p Fs(v)1002 2868 y Fn(1)1036 2859 y Fs(;v)1090 2868 y Fn(2)1124 2859 y Fo(g\\)p Fs(T)g Fo(j)p Fp(=1)1384 2738 y Ft(q)1427 2753 y Fs(T)1607 2738 y Fq(\025)28 b Ft(z)1757 2753 y Fs(k)r Fo(\000)p Fp(2)1973 2598 y Fl(\022)2088 2677 y Ft(i;)17 b(j)33 b Fq(2)28 b(f)p Fr(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(k)s Fq(g)p Ft(;)g(i)27 b Fq(6)p Fr(=)h Ft(j)2088 2797 y(v)2135 2812 y Fp(1)2202 2797 y Fq(2)g(f)p Ft(x)2401 2812 y Fs(i)2429 2797 y Ft(;)17 b(y)2521 2812 y Fs(i)2549 2797 y Fq(g)p Ft(;)g(v)2690 2812 y Fp(2)2757 2797 y Fq(2)28 b(f)p Ft(x)2956 2812 y Fs(j)2993 2797 y Ft(;)17 b(y)3085 2812 y Fs(j)3120 2797 y Fq(g)3212 2598 y Fl(\023)1113 2970 y Ft(q)1156 2985 y Fs(T)1607 2970 y Fq(\025)28 b Fr(0)212 b(\()p Ft(T)41 b Fq(\022)28 b(f)p Ft(x)2319 2985 y Fp(1)2359 2970 y Ft(;)17 b(y)2451 2985 y Fp(1)2490 2970 y Ft(;)g(:)g(:)g(:)f(;)h(x)2764 2985 y Fs(k)2806 2970 y Ft(;)g(y)2898 2985 y Fs(k)2940 2970 y Fq(g)p Fr(\))p Ft(:)3371 2531 y Fz(\(5\))404 3210 y(Note)37 b(that)g(the)g(constrain)m(t)g(matrix)e(of)i(LP)g(\()2134 3210 y SDict begin H.S end 2134 3210 a Fz(5)2183 3146 y SDict begin H.R end 2183 3146 a 2183 3210 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 2183 3210 a Fz(\))g(is)g(a)f Fr(\(2)p Ft(k)28 b Fr(+)d(4)2762 3129 y Fl(\000)2808 3166 y Fs(k)2809 3244 y Fp(2)2846 3129 y Fl(\001)2892 3210 y Fr(\))g Fq(\002)h Fr(2)3107 3174 y Fp(2)p Fs(k)3222 3210 y Fz(binary)257 3331 y(matrix)34 b(and)h(easy)h(to)e(generate)i(b)m(y)g(a)f(simple)e (computer)i(program.)49 b(In)36 b(addition,)d(if)257 3451 y(the)g(v)-5 b(alue)32 b(of)g Ft(k)k Fz(is)c(relativ)m(ely)f (small,)g(then)i(w)m(e)g(can)g(solv)m(e)g(this)f(b)m(y)i(an)e(LP)h (solv)m(er.)404 3573 y(Solving)j(LP)i(\()956 3573 y SDict begin H.S end 956 3573 a Fz(5)1004 3508 y SDict begin H.R end 1004 3508 a 1004 3573 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 1004 3573 a Fz(\))g(for)f Ft(k)i Fr(=)d(3)h Fz(with)g Ft(z)1794 3588 y Fp(1)1870 3573 y Fr(=)g(1)g Fz(and)g Ft(z)2308 3588 y Fp(2)2384 3573 y Fr(=)f(1)p Ft(:)p Fr(5)i Fz(yields)f Ft(z)2983 3588 y Fp(3)3059 3573 y Fr(=)f(2)p Fz(.)58 b(This)257 3693 y(implies)45 b Ft(s)p Fr(\()p Fj(IP)775 3708 y Fs(n)822 3693 y Fr(\))52 b Fq(\025)g Fr(2)1090 3657 y Fs(n=)p Fp(3)1259 3693 y Fq(\030)g Fr(1)p Ft(:)p Fr(2599)1660 3657 y Fs(n)1706 3693 y Fz(,)f(whic)m(h)c(is)f(sligh)m (tly)f(b)s(etter)i(than)g(the)g(lo)m(w)m(er)257 3813 y(b)s(ound)d(obtained)f(b)m(y)h(solving)f(LP)h(\()1676 3813 y SDict begin H.S end 1676 3813 a Fz(4)1724 3749 y SDict begin H.R end 1724 3749 a 1724 3813 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 1724 3813 a Fz(\).)77 b(Solving)42 b(LP)h(\()2429 3813 y SDict begin H.S end 2429 3813 a Fz(5)2478 3749 y SDict begin H.R end 2478 3749 a 2478 3813 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 2478 3813 a Fz(\))h(again)e(for)h Ft(k)49 b Fr(=)d(4)e Fz(with)257 3934 y Ft(z)302 3949 y Fp(2)372 3934 y Fr(=)29 b(1)p Ft(:)p Fr(5)k Fz(and)h Ft(z)871 3949 y Fp(3)940 3934 y Fr(=)29 b(2)34 b Fz(yields)f Ft(z)1448 3949 y Fp(4)1517 3934 y Fq(\030)d Fr(2)p Ft(:)p Fr(8333)p Fz(,)j(whic)m(h)h (implies)d(b)s(etter)j(lo)m(w)m(er)g(b)s(ound)g(of)257 4054 y Ft(s)p Fr(\()p Fj(IP)430 4069 y Fs(n)477 4054 y Fr(\))j Fq(\025)h Fr(2)p Ft(:)p Fr(8333)939 4018 y Fs(n=)p Fp(4)1093 4054 y Fq(\030)g Fr(1)p Ft(:)p Fr(2974)1480 4018 y Fs(n)1526 4054 y Fz(.)60 b(By)39 b(rep)s(eating)e(this)h(pro)s (cedure,)j(w)m(e)e(can)f(obtain)257 4175 y Ft(z)302 4190 y Fp(5)370 4175 y Fq(\030)28 b Fr(4)p Ft(:)p Fr(0277)p Fz(,)g Ft(z)847 4190 y Fp(6)914 4175 y Fq(\030)g Fr(5)p Ft(:)p Fr(7500)p Fz(,)g Ft(z)1391 4190 y Fp(7)1458 4175 y Fq(\030)g Fr(8)p Ft(:)p Fr(2541)p Fz(,)g Ft(z)1935 4190 y Fp(8)2002 4175 y Fq(\030)g Fr(11)p Ft(:)p Fr(9700)f Fz(and)g Ft(z)2684 4190 y Fp(9)2752 4175 y Fq(\030)h Fr(17)p Ft(:)p Fr(3350)p Fz(.)40 b(These)257 4295 y(imply)33 b(the)i(lo)m(w)m(er)g(b)s(ounds)h(on)e Ft(s)p Fr(\()p Fj(IP)1612 4310 y Fs(n)1659 4295 y Fr(\))g Fz(of)h Fr(1)p Ft(:)p Fr(3213)2117 4259 y Fs(n)2163 4295 y Fz(,)g Fr(1)p Ft(:)p Fr(3384)2497 4259 y Fs(n)2543 4295 y Fz(,)g Fr(1)p Ft(:)p Fr(3519)2877 4259 y Fs(n)2923 4295 y Fz(,)g Fr(1)p Ft(:)p Fr(3638)3257 4259 y Fs(n)3338 4295 y Fz(and)257 4415 y Fr(1)p Ft(:)p Fr(3729)529 4379 y Fs(n)575 4415 y Fz(,)e(resp)s(ectiv)m(ely)-8 b(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(not) e(succeeded)k(to)c(compute)g(the)h(v)-5 b(alue)32 b(of)g Ft(z)3303 4430 y Fs(k)3379 4415 y Fz(for)257 4536 y Ft(k)f Fq(\025)d Fr(10)i Fz(at)h(the)g(time)e(of)i(writing)e(this)h(article)f (\(b)s(ecause)j(GLPK)e(solv)m(er)h([)3013 4536 y SDict begin H.S end 3013 4536 a Fz(35)3111 4472 y SDict begin H.R end 3111 4472 a 3111 4536 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Mak) cvn H.B /ANN pdfmark end 3111 4536 a Fz(])g(is)f(killed)257 4656 y(b)m(y)k(the)f(out)f(of)g (memory\).)404 4778 y(Note)22 b(that)f(the)h(b)s(est)g(kno)m(wn)h(lo)m (w)m(er)f(b)s(ound)g(on)g Ft(s)p Fr(\()p Fj(IP)2313 4793 y Fs(n)2360 4778 y Fr(\))g Fz(is)f Fr(\012\(2)2664 4742 y Fs(n=)p Fp(2)2782 4778 y Ft(=n)p Fr(\))27 b(=)h(\012\(1)p Ft(:)p Fr(4142)3438 4742 y Fs(n)3484 4778 y Fr(\))257 4898 y Fz(b)m(y)34 b(F)-8 b(orster)33 b(et)h(al.)43 b([)1015 4898 y SDict begin H.S end 1015 4898 a Fz(15)1113 4834 y SDict begin H.R end 1113 4834 a 1113 4898 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.For01) cvn H.B /ANN pdfmark end 1113 4898 a Fz(])33 b(and)g(the)h(upp)s(er)f(b)s(ound)h(is)e Ft(s)p Fr(\()p Fj(IP)2384 4913 y Fs(n)2431 4898 y Fr(\))c Fq(\024)h Fr(2)2652 4862 y Fs(n)2699 4898 y Fz(.)45 b(The)34 b(lo)m(w)m(er)f(b)s(ound)257 5019 y(is)49 b(pro)m(v)m(ed)i(b)m(y)f (considering)e(the)i(rank)f(of)g(a)g(comm)m(unication)e(matrix)g(that)i (sign-)257 5139 y(represen)m(ts)36 b Fj(IP)805 5154 y Fs(n)885 5139 y Fz([)912 5139 y SDict begin H.S end 912 5139 a Fz(15)1009 5075 y SDict begin H.R end 1009 5075 a 1009 5139 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.For01) cvn H.B /ANN pdfmark end 1009 5139 a Fz(])d(\(see)h(also)e([)1488 5139 y SDict begin H.S end 1488 5139 a Fz(49)1586 5075 y SDict begin H.R end 1586 5075 a 1586 5139 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Shr08) cvn H.B /ANN pdfmark end 1586 5139 a Fz(])h(for)f(a)g(generalization\).)42 b(The)34 b(upp)s(er)f(b)s(ound)g(fol-)257 5259 y(lo)m(ws)39 b(from)e(the)i(construction)f Fj(IP)1541 5274 y Fs(n)1588 5259 y Fr(\()p Ft(X)r(;)17 b(Y)k Fr(\))38 b(=)f Fz(sign)p Fr(\()2227 5185 y Fl(P)2332 5289 y Fs(S)t Fo(\022)p Fp([)p Fs(n)p Fp(])2520 5259 y Fr(\()p Fq(\000)p Fr(2\))2722 5223 y Fo(j)p Fs(S)t Fo(j)p Fp(+1)2902 5259 y Ft(X)2983 5274 y Fs(S)3034 5259 y Ft(Y)3091 5274 y Fs(S)3141 5259 y Fr(\))p Fz(,)j(where)257 5388 y Ft(X)338 5403 y Fs(S)422 5388 y Fz(and)32 b Ft(Y)668 5403 y Fs(S)751 5388 y Fz(denote)1065 5313 y Fl(Q)1159 5417 y Fs(i)p Fo(2)p Fs(S)1298 5388 y Ft(x)1353 5403 y Fs(i)1414 5388 y Fz(and)1604 5313 y Fl(Q)1698 5417 y Fs(i)p Fo(2)p Fs(S)1836 5388 y Ft(y)1884 5403 y Fs(i)1912 5388 y Fz(,)h(resp)s(ectiv)m(ely)-8 b(.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 257 266 a 404 573 a Fz(A)m(t)24 b(this)g(momen)m(t,)i(w)m(e)f(don't)g(kno)m(w)g(whether)h(our)e(metho)s (d)g(can)h(b)s(eat)f Fr(\012\(1)p Ft(:)p Fr(4142)3384 537 y Fs(n)3430 573 y Fr(\))p Fz(.)257 693 y(Ho)m(w)m(ev)m(er,)37 b(w)m(e)e(think)f(that)g(there)g(is)g(a)g(c)m(hance.)49 b(By)35 b(examining)d(LP)i(\()2912 693 y SDict begin H.S end 2912 693 a Fz(5)2961 629 y SDict begin H.R end 2961 629 a 2961 693 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.5) cvn H.B /ANN pdfmark end 2961 693 a Fz(\))g(with)f Ft(k)g Fr(=)d(8)257 814 y Fz(under)d(the)f(assumption)e Ft(s)p Fr(\()p Fj(IP)1369 829 y Fs(n)1416 814 y Fr(\))j Fq(\025)1587 731 y(p)p 1670 731 49 4 v 83 x Fr(2)7 b Fq(\001)h Ft(s)p Fr(\()p Fj(IP)1934 829 y Fs(n)p Fo(\000)p Fp(1)2071 814 y Fr(\))26 b Fz(for)f(ev)m(ery)i Ft(n)p Fz(,)g(w)m(e)g(obtain)d(somewhat)257 934 y(curious)36 b(fact)f(sa)m(ying)g(that)g(the)h(lo)m(w)m(er)f(b)s(ound)g(w)m(ould)h (b)s(e)f Fu(enhanc)-5 b(e)g(d)34 b Fz(b)m(y)i(considering)257 1054 y(a)d(large)e(LP)-8 b(.)257 1074 y SDict begin H.S end 257 1074 a 257 1074 a SDict begin 14.5 H.A end 257 1074 a 257 1074 a SDict begin [ /View [/XYZ H.V] /Dest (fact.4) cvn H.B /DEST pdfmark end 257 1074 a 207 x Fv(F)f(act)43 b(4.)h Fu(Supp)-5 b(ose)38 b(that,)i(for)e(every)h(su\036ciently)g(lar) -5 b(ge)38 b Ft(n)p Fu(,)i Ft(s)p Fr(\()p Fj(IP)2705 1296 y Fs(n)2752 1281 y Fr(\))35 b Fq(\025)2938 1198 y(p)p 3021 1198 49 4 v 83 x Fr(2)25 b Fq(\001)g Ft(s)p Fr(\()p Fj(IP)3320 1296 y Fs(n)p Fo(\000)p Fp(1)3457 1281 y Fr(\))257 1401 y Fu(holds.)44 b(Then)34 b Ft(s)p Fr(\()p Fj(IP)973 1416 y Fs(n)1020 1401 y Fr(\))27 b(=)h(\012\(1)p Ft(:)p Fr(4198)1569 1365 y Fs(n)1615 1401 y Fr(\))p Fu(.)257 1589 y SDict begin H.S end 257 1589 a 257 1589 a SDict begin 14.5 H.A end 257 1589 a 257 1589 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.3.2) cvn H.B /DEST pdfmark end 257 1589 a 130 x Fk(3.2)131 b(Lo)l(w)l(er)44 b(Bounds)g(on)g(F)-11 b(orm)l(ula)44 b(Size)g(via)h(SDP)257 1913 y Fz(A)33 b Fu(Bo)-5 b(ole)g(an)34 b(formula)f Fz(is)f(a)h(binary)g(tree)g(where) h(eac)m(h)g(in)m(ternal)e(no)s(de)h(is)g(lab)s(eled)f(with)257 2033 y Fq(^)g Fz(or)f Fq(_)p Fz(,)h(and)g(eac)m(h)g(leaf)f(is)g(lab)s (eled)f(with)h(a)g(literal,)e(i.e.,)i(a)g(v)-5 b(ariable)30 b(or)h(its)g(negation.)257 2153 y(A)d(Bo)s(olean)e(form)m(ula)g (computes)i(a)f(Bo)s(olean)f(function)h(in)g(an)g(ob)m(vious)h(w)m(a)m (y)-8 b(.)42 b(The)29 b Fu(size)257 2274 y Fz(of)36 b(a)g(form)m(ula)e (is)i(the)h(n)m(um)m(b)s(er)f(of)g(lea)m(v)m(es)h(in)f(the)g(tree.)56 b(F)-8 b(or)35 b(a)h(Bo)s(olean)f(function)g Ft(f)11 b Fz(,)257 2394 y(the)33 b Fu(formula)h(c)-5 b(omplexity)p Fz(,)32 b(denoted)i(b)m(y)f Ft(L)p Fr(\()p Ft(f)11 b Fr(\))p Fz(,)33 b(is)f(de\034ned)h(as)g(the)g(size)g(of)f(a)g(smallest) 257 2515 y(form)m(ula)h(that)h(computes)h Ft(f)11 b Fz(.)48 b(The)35 b(famous)f(result)g(of)g(Khrap)s(c)m(henk)m(o)i([)2949 2515 y SDict begin H.S end 2949 2515 a Fz(27)3046 2450 y SDict begin H.R end 3046 2450 a 3046 2515 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Khr71) cvn H.B /ANN pdfmark end 3046 2515 a Fz(])f(sa)m(ys)g(that)257 2635 y(the)28 b(form)m(ula)d(complexit)m(y)h(of)g(the)h(parit)m(y)g(of)f Ft(n)h Fz(v)-5 b(ariables)26 b(is)g(at)h(least)f Ft(n)2886 2599 y Fp(2)2926 2635 y Fz(.)41 b(The)28 b(curren)m(t)257 2755 y(b)s(est)37 b(lo)m(w)m(er)e(b)s(ound)h(for)f(an)g(explicitly)f (de\034ned)j(function)e(is)g Ft(n)2630 2719 y Fp(3)p Fo(\000)p Fs(o)p Fp(\(1\))2885 2755 y Fz(due)h(to)f(H\345stad)257 2876 y([)284 2876 y SDict begin H.S end 284 2876 a Fz(21)382 2811 y SDict begin H.R end 382 2811 a 382 2876 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Has98) cvn H.B /ANN pdfmark end 382 2876 a Fz(].)404 3001 y(The)g(quan)m(tum)h(adv)m (ersary)g(metho)s(d)e(\([)1887 3001 y SDict begin H.S end 1887 3001 a Fz(5)1936 2936 y SDict begin H.R end 1936 2936 a 1936 3001 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Amb03) cvn H.B /ANN pdfmark end 1936 3001 a Fz(],[)2017 3001 y SDict begin H.S end 2017 3001 a Fz(8)2066 2936 y SDict begin H.R end 2066 2936 a 2066 3001 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.BBS03) cvn H.B /ANN pdfmark end 2066 3001 a Fz(],[)2147 3001 y SDict begin H.S end 2147 3001 a Fz(33)2245 2936 y SDict begin H.R end 2245 2936 a 2245 3001 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.LM04) cvn H.B /ANN pdfmark end 2245 3001 a Fz(],[)2326 3001 y SDict begin H.S end 2326 3001 a Fz(56)2424 2936 y SDict begin H.R end 2424 2936 a 2424 3001 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Zha05) cvn H.B /ANN pdfmark end 2424 3001 a Fz(]\))g(has)i(originally)31 b(b)s(een)k(de-)257 3121 y(v)m(elop)s(ed)i(for)f(pro)m(ving)h(lo)m(w)m(er)f(b)s(ounds)i(on) e(quan)m(tum)h(query)h(complexit)m(y)-8 b(.)55 b(Laplan)m(te,)257 3241 y(Lee)35 b(and)f(Szegedy)i([)1030 3241 y SDict begin H.S end 1030 3241 a Fz(32)1127 3177 y SDict begin H.R end 1127 3177 a 1127 3241 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.LLS06) cvn H.B /ANN pdfmark end 1127 3241 a Fz(])e(rev)m(ealed)h(that)f(this)f(metho)s(d)h(is)f(also)g(v)m(ery)j (useful)e(for)f(lo)m(w)m(er)257 3362 y(b)s(ounding)d(the)h(form)m(ula)d (size.)43 b(In)31 b(this)e(framew)m(ork,)i(a)f(lo)m(w)m(er)h(b)s(ound)f (on)g(the)h(form)m(ula)257 3482 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Ft(;)17 b Fr(1)p Fq(g)p Fz(,)32 b(the)h(adv)m(ersary)h(b)s(ound)f(for)f Ft(f)43 b Fz(is)32 b(de\034ned)i(as)1102 4378 y Fr(AD)m(V)q(\()p Ft(f)11 b Fr(\))83 b(=)136 b(max)1698 4440 y Fp(\000)p Fo(\025)p Fp(0)p Fs(;)p Fp(\000)p Fo(6)p Fp(=0)2227 4310 y Fq(jj)p Fr(\000)p Fq(jj)p 2012 4355 602 4 v 2012 4446 a Fr(max)2194 4461 y Fs(i)2239 4446 y Fq(jj)p Fr(\000)21 b Fq(\016)h Ft(D)2530 4461 y Fs(i)2558 4446 y Fq(jj)2624 4378 y Ft(;)257 4661 y Fz(where)42 b(the)e(maxim)m(um)d(is)j(tak)m(en)h (o)m(v)m(er)g(nonnegativ)m(e)f(symmetric)f(matrices)g Fr(\000)p Fz(,)j(and)257 4782 y Ft(D)338 4797 y Fs(i)406 4782 y Fz(is)d(a)g(zero-one)g(matrix)f(where)j Ft(D)1684 4797 y Fs(i)1712 4782 y Fr([)p Ft(x;)17 b(y)t Fr(])39 b(=)g(1)g Fz(i\033)f Ft(x)2337 4797 y Fs(i)2405 4782 y Fq(6)p Fr(=)i Ft(y)2569 4797 y Fs(i)2596 4782 y Fz(.)64 b Fr(\000)27 b Fq(\016)g Ft(D)2933 4797 y Fs(i)3000 4782 y Fz(denotes)41 b(the)257 4902 y(en)m(try-wise)34 b(pro)s(duct)f(of)f Fr(\000)g Fz(and)h Ft(D)1565 4917 y Fs(i)1593 4902 y Fz(.)404 5027 y(Laplan)m(te,)40 b(Lee)g(and)g(Szegedy)h([)1633 5027 y SDict begin H.S end 1633 5027 a Fz(32)1731 4963 y SDict begin H.R end 1731 4963 a 1731 5027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.LLS06) cvn H.B /ANN pdfmark end 1731 5027 a Fz(])e(pro)m(v)m(ed)i(that)e Fr(AD)m(V)r(\()p Ft(f)11 b Fr(\))2694 4991 y Fp(2)2772 5027 y Fz(is)39 b(a)g(lo)m(w)m(er)h(b)s(ound)257 5147 y(on)35 b(the)g(form)m(ula)e (complexit)m(y)i(of)f Ft(f)11 b Fz(.)50 b(This)35 b(parameter)f(can)i (b)s(e)f(form)m(ulated)e(as)i(SDP)257 5268 y([)284 5268 y SDict begin H.S end 284 5268 a Fz(52)382 5204 y SDict begin H.R end 382 5204 a 382 5268 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.SS05) cvn H.B /ANN pdfmark end 382 5268 a Fz(]:)43 b(Let)31 b Ft(F)44 b Fz(b)s(e)31 b(a)g Fr(2)1019 5232 y Fs(n)1084 5268 y Fq(\002)19 b Fr(2)1229 5232 y Fs(n)1307 5268 y Fz(binary)30 b(matrix)g(suc)m(h)i(that)e Ft(F)14 b Fr([)p Ft(x;)j(y)t Fr(])27 b(=)h(1)i Fz(i\033)g Ft(f)11 b Fr(\()p Ft(x)p Fr(\))28 b Fq(6)p Fr(=)f Ft(f)11 b Fr(\()p Ft(y)t Fr(\))p Fz(,)257 5388 y(and)33 b(let)g Ft(D)670 5403 y Fs(i)731 5388 y Fz(b)s(e)g(de\034ned)h(as)f(ab)s(o)m(v) m(e.)45 b(The)34 b(parameter)e Fr(AD)m(V)r(\()p Ft(f)11 b Fr(\))32 b Fz(is)h(giv)m(en)g(b)m(y)g Fr(1)p Ft(=\026)3335 5403 y Fs(min)3468 5388 y Fz(,)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 257 266 a 307 x Fz(where)34 b Ft(\026)598 588 y Fs(min)764 573 y Fz(is)e(the)h(minimal)28 b(solution)j(of)h(the)h(follo)m(wing)d(semide\034nite)i(program:)850 1028 y SDict begin H.S end 850 1028 a 850 1028 a SDict begin 14.5 H.A end 850 1028 a 850 1028 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 850 1028 a 891 786 a Fz(Minimize)434 b Ft(\026)27 b Fr(=)h Fz(tr)p Fr(\001)891 907 y Fz(Sub)5 b(ject)34 b(to)653 b Fr(\001)89 b Fz(is)g(diagonal)m Ft(;)1993 1027 y(Z)h Fq(\025)84 b Fr(0)p Ft(;)1845 1147 y(Z)29 b Fq(\001)21 b Ft(F)98 b Fr(=)84 b(1)p Ft(;)1416 1268 y Fq(8)p Ft(i)28 b Fr(:)g(\001)22 b Fq(\000)h Ft(Z)29 b Fq(\016)22 b Ft(D)2039 1283 y Fs(i)2150 1268 y Fq(\027)84 b Fr(0)p Ft(;)3371 1028 y Fz(\(6\))404 1483 y(Here)33 b Fr(tr\001)g Fz(denotes)g(the)g (trace)g(of)f(a)g(matrix)f Fr(\001)p Fz(.)404 1604 y(This)46 b(parameter)g(enjo)m(ys)h(a)f(nice)g(comp)s(osition)e(prop)s(ert)m(y)-8 b(.)85 b(F)-8 b(or)46 b(t)m(w)m(o)h(Bo)s(olean)257 1724 y(functions)28 b Ft(f)38 b Fz(on)28 b Ft(n)g Fz(v)-5 b(ariables)26 b(and)i Ft(g)j Fz(on)d Ft(m)f Fz(v)-5 b(ariables,)28 b(let)f Ft(f)c Fq(\012)12 b Ft(g)32 b Fz(denote)c(a)f(comp)s(osite)257 1844 y(function)35 b(on)g Ft(nm)g Fz(v)-5 b(ariables:)47 b Fr(\()p Ft(f)34 b Fq(\012)25 b Ft(g)t Fr(\)\()p Ft(x)1809 1859 y Fp(1)1848 1844 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2122 1859 y Fs(mn)2231 1844 y Fr(\))32 b(=)f Ft(f)11 b Fr(\()p Ft(g)t Fr(\()22 b(~)-71 b Ft(x)2649 1859 y Fp(1)2688 1844 y Fr(\))p Ft(;)17 b(:)g(:)g(:)f(;)h(g)t Fr(\()26 b(~)-75 b Ft(x)3089 1859 y Fs(n)3135 1844 y Fr(\)\))35 b Fz(where)275 1965 y Fr(~)-67 b Ft(x)312 1980 y Fs(i)369 1965 y Fr(=)27 b(\()p Ft(x)565 1980 y Fp(\()p Fs(i)p Fo(\000)p Fp(1\))p Fs(m)p Fp(+1)891 1965 y Ft(;)17 b(:)g(:)g(:)f(;)h(x) 1165 1980 y Fs(im)1256 1965 y Fr(\))p Fz(.)257 1997 y SDict begin H.S end 257 1997 a 257 1997 a SDict begin 14.5 H.A end 257 1997 a 257 1997 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.1) cvn H.B /DEST pdfmark end 257 1997 a 174 x Fv(Theorem)36 b(1.)41 b Fu(\([)929 2171 y SDict begin H.S end 929 2171 a Fu(5)979 2107 y SDict begin H.R end 979 2107 a 979 2171 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Amb03) cvn H.B /ANN pdfmark end 979 2171 a Fu(,)1043 2171 y SDict begin H.S end 1043 2171 a Fu(22)1142 2107 y SDict begin H.R end 1142 2107 a 1142 2171 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.HLS05) cvn H.B /ANN pdfmark end 1142 2171 a Fu(]\))34 b(F)-7 b(or)33 b(every)h(Bo)-5 b(ole)g(an)33 b(functions)g Ft(f)45 b Fu(and)34 b Ft(g)t Fu(,)f Fr(AD)m(V)q(\()p Ft(f)f Fq(\012)20 b Ft(g)t Fr(\))27 b(=)257 2291 y(AD)m(V)r(\()p Ft(f)11 b Fr(\))22 b Fq(\001)f Fr(AD)m(V\(g\))p Fu(.)404 2498 y Fz(This)37 b(theorem)h(sa)m(ys)h(that)e(if)g Ft(L)p Fr(\()p Ft(f)11 b Fr(\))36 b(=)g(AD)m(V)q(\()p Ft(f)11 b Fr(\))2242 2462 y Fp(2)2319 2498 y Fz(and)38 b Ft(L)p Fr(\()p Ft(g)t Fr(\))e(=)g(AD)m(V)q(\()p Ft(g)t Fr(\))3201 2462 y Fp(2)3240 2498 y Fz(,)j(then)257 2618 y Ft(L)p Fr(\()p Ft(f)j Fq(\012)30 b Ft(g)t Fr(\))47 b(=)h Ft(L)p Fr(\()p Ft(f)11 b Fr(\))30 b Fq(\001)g Ft(L)p Fr(\()p Ft(g)t Fr(\))p Fz(.)78 b(This)45 b(giv)m(es)g(a)f(nice)g (generalization)e(of)i(the)g(result)h(of)257 2739 y(Khrap)s(c)m(henk)m (o)30 b([)876 2739 y SDict begin H.S end 876 2739 a Fz(27)974 2674 y SDict begin H.R end 974 2674 a 974 2739 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Khr71) cvn H.B /ANN pdfmark end 974 2739 a Fz(].)42 b(The)29 b(parit)m(y)f(function)g (on)g Ft(n)g Fr(=)g(2)2297 2702 y Fs(k)2368 2739 y Fz(v)-5 b(ariables)27 b(can)h(b)s(e)h(written)f(as)257 2859 y Fr(\()p Ft(x)350 2874 y Fp(1)393 2859 y Fq(\010)s Ft(x)528 2874 y Fp(2)569 2859 y Fr(\))s Fq(\012)s(\001)17 b(\001)g(\001)r(\012)s Fr(\()p Ft(x)983 2874 y Fp(1)1026 2859 y Fq(\010)s Ft(x)1161 2874 y Fp(2)1201 2859 y Fr(\))p Fz(.)41 b(The)24 b(optimal)c(form)m (ula)h(for)i Ft(x)2389 2874 y Fp(1)2432 2859 y Fq(\010)s Ft(x)2567 2874 y Fp(2)2631 2859 y Fz(is)f Fr(\()p Ft(x)2812 2874 y Fp(1)2855 2859 y Fq(^)p 2924 2804 95 4 v 3 w Ft(x)2979 2874 y Fp(2)3019 2859 y Fr(\))s Fq(_)s Fr(\()p 3167 2804 V Ft(x)3222 2874 y Fp(1)3266 2859 y Fq(^)s Ft(x)3390 2874 y Fp(2)3430 2859 y Fr(\))p Fz(,)257 2979 y(whic)m(h)37 b(has)g(size)f(4.)55 b(By)37 b(using)f(this)g(recursiv)m(ely)-8 b(,)38 b(w)m(e)f(get)g(a)f(form)m(ula)e(for)i(the)h(parit)m(y)257 3100 y(on)31 b Fr(2)440 3064 y Fs(k)513 3100 y Fz(v)-5 b(ariables)30 b(whose)i(size)f(is)g Fr(4)1531 3064 y Fs(k)1601 3100 y Fr(=)c Ft(n)1762 3064 y Fp(2)1802 3100 y Fz(.)43 b(Khrap)s(c)m(henk)m(o's)33 b Ft(n)2590 3064 y Fp(2)2660 3100 y Fz(lo)m(w)m(er)e(b)s(ound)g(for)g(the)257 3220 y(parit)m(y)e(function)g(guaran)m(tees)h(the)f(exact)h(optimalit)m (y)d(of)h(suc)m(h)j(a)e(naiv)m(e)g(construction.)257 3340 y(Note)36 b(that,)h(recen)m(tly)-8 b(,)38 b(T)-8 b(arui)35 b([)1424 3340 y SDict begin H.S end 1424 3340 a Fz(53)1522 3276 y SDict begin H.R end 1522 3276 a 1522 3340 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Tar10) cvn H.B /ANN pdfmark end 1522 3340 a Fz(])h(pro)m(v)m(ed)h(that)f(the) g(form)m(ulas)f(constructed)i(in)f(this)257 3461 y(w)m(a)m(y)e(are)f (essen)m(tially)f(unique)h(smallest)e(ones.)404 3582 y(Theorem)826 3582 y SDict begin H.S end 826 3582 a Fz(1)875 3518 y SDict begin H.R end 875 3518 a 875 3582 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 875 3582 a 43 w Fz(sho)m(ws)45 b(a)d(similar)e(optimalit)m (y)g(can)k(b)s(e)f(established)g(for)g(an)m(y)g(base)257 3702 y(functions)f Ft(f)53 b Fz(with)41 b(the)i(prop)s(ert)m(y)f Ft(L)p Fr(\()p Ft(f)11 b Fr(\))44 b(=)f(AD)m(V)r(\()p Ft(f)11 b Fr(\))2323 3666 y Fp(2)2362 3702 y Fz(.)71 b(W)-8 b(e)43 b(call)d(suc)m(h)j(a)f(function)257 3823 y Fu(tight)p Fz(.)89 b(By)48 b(using)f(an)g(SDP)h(solv)m(er,)k(w)m(e)c (can)g(en)m(umerate)g(all)d(tigh)m(t)i(functions)g(on)257 3943 y(up)e(to)f(\034v)m(e)h(v)-5 b(ariables.)77 b(W)-8 b(e)44 b(sa)m(y)h(that)f(t)m(w)m(o)h(Bo)s(olean)d(functions)i(are)h(in) e(the)h(same)257 4063 y(NPN-equiv)-5 b(alence)37 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begin [ /View [/XYZ H.V] /Dest (definition.2) cvn H.B /DEST pdfmark end 257 1320 a 176 x Fv(De\034nition)26 b(2.)35 b Fu(A)27 b(Bo)-5 b(ole)g(an)26 b(function)g Ft(f)38 b Fu(over)26 b Ft(X)2100 1511 y Fs(n)2175 1496 y Fr(=)h Fq(f)p Ft(x)2383 1511 y Fp(1)2423 1496 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)2697 1511 y Fs(n)2744 1496 y Fq(g)27 b Fu(is)f Ft(k)s Fz(-mixed)g Fu(if)g(for)257 1617 y(every)31 b Ft(V)49 b Fq(\022)28 b Ft(X)803 1632 y Fs(n)880 1617 y Fu(such)j(that)f Fq(j)p Ft(V)22 b Fq(j)27 b Fr(=)g Ft(k)34 b Fu(and)c(for)g(any)g(two)h (distinct)f(p)-5 b(artial)30 b(assignments)257 1737 y Ft(\013)36 b Fu(and)e Ft(\014)41 b Fu(with)34 b(supp)-5 b(ort)35 b Ft(V)22 b Fu(,)35 b(the)f(functions)h(obtaine)-5 b(d)34 b(fr)-5 b(om)34 b Ft(f)46 b Fu(by)35 b(applying)f Ft(\013)h Fu(and)g Ft(\014)257 1858 y Fu(ar)-5 b(e)35 b(distinct,)f(i.e.,)h Ft(f)11 b Fq(j)1088 1873 y Fs(\013)1164 1858 y Fq(6)p Fr(=)28 b Ft(f)11 b Fq(j)1355 1873 y Fs(\014)1401 1858 y Fu(.)404 2059 y Fz(Note)24 b(that)g(they)i(originally)20 b(used)26 b(the)f(prop)s(ert)m(y)g(called)e Fu(str)-5 b(ongly-two-dep)g(endent)p Fz(;)257 2179 y(but)33 b(the)g(prop)s(ert)m (y)g(of)f Ft(k)s Fz(-mixed)g(is)g(stronger)h(than)f(this.)404 2300 y(The)h(main)e(p)s(ortion)g(of)h(their)g(pro)s(of)g(is)g(the)h (follo)m(wing)d(lemma.)257 2319 y SDict begin H.S end 257 2319 a 257 2319 a SDict begin 14.5 H.A end 257 2319 a 257 2319 a SDict begin [ /View [/XYZ H.V] /Dest (lemma.1) cvn H.B /DEST pdfmark end 257 2319 a 182 x Fv(Lemma)41 b(1.)i Fu(L)-5 b(et)38 b Ft(f)49 b Fu(b)-5 b(e)37 b(a)h Ft(t)p Fu(-mixe)-5 b(d)37 b Ft(n)p Fu(-variable)g(Bo)-5 b(ole)g(an)36 b(function)h(with)h Ft(n)24 b Fq(\000)h Ft(t)33 b Fr(=)g Ft(k)s Fu(.)257 2621 y(Supp)-5 b(ose)38 b(that)i Ft(n)35 b Fq(\025)h Fr(2)p Ft(k)28 b Fr(+)d(4)39 b Fu(and)f(a)h(cir)-5 b(cuit)39 b Ft(C)46 b Fu(c)-5 b(omputes)38 b Ft(f)11 b Fu(.)57 b(Then,)39 b(ther)-5 b(e)39 b(exists)f(a)257 2742 y(p)-5 b(artial)40 b(assignment)f Ft(\033)44 b Fu(that)c (satis\034es)f(the)h(fol)5 b(lowing:)53 b Fq(j)p Ft(\033)t Fq(j)37 b(\024)h Fr(2)p Fu(,)i(and)g(ther)-5 b(e)40 b(exist)f(a)257 2862 y(Bo)-5 b(ole)g(an)29 b(cir)-5 b(cuit)31 b Ft(C)998 2826 y Fo(0)1049 2862 y Fq(\021)d Ft(C)7 b Fq(j)1259 2877 y Fs(\033)1336 2862 y Fu(such)30 b(that)h Fr(SD\()p Ft(C)7 b Fr(\))27 b Fq(\025)i Fr(SD)o(\()p Ft(C)2406 2826 y Fo(0)2429 2862 y Fr(\))12 b(+)g(5)p Fq(j)p Ft(\033)t Fq(j)p Fu(.)44 b(Her)-5 b(e)30 b Fr(SD\()p Ft(C)7 b Fr(\))30 b Fu(is)g(a)257 2982 y(me)-5 b(asur)g(e)29 b(of)f(a)h(cir)-5 b(cuit)29 b(size)g(which)f(is)g(a)h(slight)g(mo)-5 b(di\034c)g(ation)27 b(of)i(the)g(normal)f("numb)-5 b(er)257 3103 y(of)35 b(gates")f(me)-5 b(asur)g(e)34 b(\(se)-5 b(e)35 b([)1275 3103 y SDict begin H.S end 1275 3103 a Fu(26)1374 3039 y SDict begin H.R end 1374 3039 a 1374 3103 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.ILMR05) cvn H.B /ANN pdfmark end 1374 3103 a Fu(])g(for)g(the)f(de\034nition\).)404 3304 y Fz(By)48 b(using)f(ab)s(o)m(v)m(e)h(lemma)d(inductiv)m(ely)i(w)m (e)i(can)e(obtain)g(the)h(lo)m(w)m(er)f(b)s(ound)h(of)257 3424 y Fr(5)p Ft(n)23 b Fq(\000)f Ft(o)p Fr(\()p Ft(n)p Fr(\))33 b Fz(for)f(ev)m(ery)i Ft(k)s Fz(-mixed)d(function)h(with)h Ft(k)d Fr(=)e Ft(n)22 b Fq(\000)h Ft(o)p Fr(\()p Ft(n)p Fr(\))p Fz(.)404 3545 y(Suc)m(h)36 b(a)e(metho)s(d)g(can)h(b)s(e)g (reform)m(ulated)e(as)i(follo)m(ws:)46 b(In)35 b(order)g(to)f(sho)m(w)i (the)f Ft(\013)q(n)257 3665 y Fz(lo)m(w)m(er)41 b(b)s(ound)g(on)g(a)f (circuit)f(complexit)m(y)h(of)h Ft(f)11 b Fz(,)42 b(it)e(is)g (su\036cien)m(t)i(to)e(giv)m(e)h(a)f(class)h(of)257 3786 y Fu(lo)-5 b(c)g(al)34 b(cir)-5 b(cuit)36 b(p)-5 b(atterns)32 b Fq(D)j Fz(that)d(satis\034es)i(the)f(follo)m(wing.)257 3987 y(\(C1\))49 b(F)-8 b(or)43 b(ev)m(ery)j(optimal)40 b(circuit)j Ft(C)50 b Fz(that)44 b(computes)g Ft(f)11 b Fz(,)46 b(some)d(pattern)h Ft(D)50 b Fq(2)d(D)501 4107 y Fz(app)s(ears)33 b(in)f Ft(C)7 b Fz(.)257 4310 y(\(C2\))49 b(F)-8 b(or)38 b(ev)m(ery)i(circuit)e Ft(C)45 b Fz(and)39 b(ev)m(ery)h Ft(D)h Fq(2)d(D)s Fz(,)i(if)d Ft(D)42 b Fz(app)s(ears)c(in)g Ft(C)7 b Fz(,)40 b(then)f(there)501 4430 y(exists)34 b(a)e(partial)e(assignmen)m(t)i Ft(\033)37 b Fz(and)32 b(a)h(circuit)e Ft(C)2411 4394 y Fo(0)2462 4430 y Fq(\021)d Ft(C)7 b Fq(j)2672 4445 y Fs(\033)2751 4430 y Fz(suc)m(h)34 b(that)1079 4648 y Fr([)p Fz(the)f(size)g(of)f Ft(C)7 b Fr(])22 b Fq(\000)h Fr([)p Fz(the)33 b(size)g(of)f Ft(C)2364 4607 y Fo(0)2387 4648 y Fr(])c Fq(\025)g Ft(\013)q Fq(j)p Ft(\033)t Fq(j)p Ft(:)404 4907 y Fz(In)m(terestingly)-8 b(,)51 b(this)c(form)m(ulation)e(of)i(the)h(gate-elimination)42 b(metho)s(d)47 b(is)g(quite)257 5027 y(resem)m(ble)e(to)e(the)i(famous) e(pro)s(of)g(of)g(the)i(F)-8 b(our)43 b(Color)g(Theorem)h([)2858 5027 y SDict begin H.S end 2858 5027 a Fz(6)2907 4963 y SDict begin H.R end 2907 4963 a 2907 5027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.AH77) cvn H.B /ANN pdfmark end 2907 5027 a Fz(,)2978 5027 y SDict begin H.S end 2978 5027 a Fz(7)3027 4963 y SDict begin H.R end 3027 4963 a 3027 5027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.AHK77) cvn H.B /ANN pdfmark end 3027 5027 a Fz(,)3098 5027 y SDict begin H.S end 3098 5027 a Fz(45)3195 4963 y SDict begin H.R end 3195 4963 a 3195 5027 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.RSST97) cvn H.B /ANN pdfmark end 3195 5027 a Fz(].)78 b(The)257 5147 y(computer)29 b(assisted)g(pro)s(of)f(of)g(this)g(theorem)g(has)h(giv)m(en)g(b)m(y)g (App)s(el)f(and)h(Hak)m(en)g(three)257 5268 y(decades)41 b(ago.)62 b(A)39 b(simpli\034ed)e(but)i(still)e(computer)h(assisted)i (pro)s(of)e(has)i(then)f(giv)m(en)257 5388 y(b)m(y)34 b(Rob)s(ertson)e(et)h(al.)42 b([)1150 5388 y SDict begin H.S end 1150 5388 a Fz(45)1248 5324 y SDict begin H.R end 1248 5324 a 1248 5388 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.RSST97) cvn H.B /ANN pdfmark end 1248 5388 a Fz(])33 b(whic)m(h)g(pro)s(ceeds)g(as)g(follo)m(ws.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 257 266 a 404 573 a Fz(Supp)s(ose)34 b(for)f(the)g(con)m(trary)h(that)f(there)h(exists)g(a)f(coun)m (terexample)h(to)f(the)h(F)-8 b(our)257 693 y(Color)35 b(Theorem.)54 b(Let)36 b Ft(P)49 b Fz(b)s(e)36 b(some)g(easily)f (describable)h(graph)g(theoretic)f(prop)s(ert)m(y)257 814 y(that)d(a)g(minimal)27 b(coun)m(terexample)32 b(to)g(the)g(F)-8 b(our)32 b(Color)e(Theorem)i(has.)44 b(Then,)33 b(they)257 934 y(exhibited)e(a)g(set)h Fq(D)h Fz(of)e(633)f("con\034gurations",)h (that)f(consists)i(of)f(small)d(graphs)k(with)257 1054 y(some)h(additional)c(information,)h(satisfying)h(the)i(follo)m(wing:) 264 1285 y(\(F1\))48 b(F)-8 b(or)32 b(ev)m(ery)i(planar)d(graph)h Ft(G)g Fz(with)f(prop)s(ert)m(y)i Ft(P)14 b Fz(,)32 b(some)g (con\034guration)f Ft(D)g Fq(2)d(D)501 1406 y Fz(app)s(ears)33 b(in)f Ft(G)p Fz(.)264 1637 y(\(F2\))48 b(If)33 b(a)f(con\034guration)f Ft(D)g Fq(2)d(D)35 b Fz(app)s(ears)e(in)e Ft(G)p Fz(,)i(then)g Ft(G)f Fz(is)g(not)h(a)f(minimal)c(coun-)501 1757 y(terexample)k(to)h (F)-8 b(our)32 b(Color)f(Theorem.)404 1988 y(By)26 b(v)m(erifying)e(ab) s(o)m(v)m(e)i(t)m(w)m(o)g(statemen)m(ts)h(using)e(computers,)i(w)m(e)f (can)g(conclude)f(that)257 2108 y(no)33 b(minimal)28 b(coun)m(terexample)33 b(exists,)g(and)g(hence)h(F)-8 b(our)32 b(Color)f(Theorem)i(is)f(true.)404 2234 y(W)-8 b(e)28 b(no)m(w)g(review)g(the)g(prop)s(ositions)e Fr(\()p Ft(C)7 b Fr(1\))27 b Fz(and)h Fr(\()p Ft(C)7 b Fr(2\))p Fz(.)41 b(Supp)s(ose)29 b(that)e(a)h(circuit)e Ft(C)257 2355 y Fz(is)36 b(a)f(minimal)c(coun)m(terexample)36 b(to)g(a)f(lo)m(w)m(er)h(b)s(ound,)g(i.e.,)g Ft(C)43 b Fz(is)35 b(an)h(optimal)c(circuit)257 2475 y(that)38 b(computes)h Ft(f)48 b Fz(and)38 b(the)h(size)f(of)f(it)g(is)h(less)g (than)g Ft(\013)q(n)p Fz(.)60 b(Recall)36 b(that)i(a)g(circuit)f(is)257 2596 y(de\034ned)32 b(as)e(a)g(graph)g(in)g(whic)m(h)h(no)s(des)f (represen)m(t)j(gates)d(and)g(edges)i(represen)m(t)g(wires.)257 2716 y(W)-8 b(e)30 b(can)f(put)g(some)g(graph)g(theoretic)g(prop)s (erties)g(for)f(a)h(circuit)f Ft(C)36 b Fz(b)m(y)30 b(its)e(optimalit)m (y)-8 b(,)257 2836 y(whic)m(h)46 b(w)m(e)g(consider)g(as)f(the)g(prop)s (ert)m(y)h Ft(P)59 b Fz(in)44 b(\(F1\).)80 b(Then)47 b(\(C2\))e(sa)m(ys)h(that)f(if)f Ft(C)257 2957 y Fz(con)m(tains)c Ft(D)k Fq(2)d(D)h Fz(then,)h Ft(C)k Fz(is)39 b(not)h(a)g(minimal)c (coun)m(terexample)41 b(\(since)f(if)f Ft(C)47 b Fz(is)39 b(a)257 3077 y(coun)m(terexample,)33 b(then)h(a)e(smaller)e(circuit)i Ft(C)7 b Fq(j)2032 3092 y Fs(\033)2111 3077 y Fz(is)32 b(also\).)404 3203 y(It)41 b(w)m(ould)g(b)s(e)g(in)m(teresting)g(to)f (giv)m(e)h(a)g(\(minimal\))c(set)k(of)g(circuit)f(patterns)h(that)257 3323 y(yield)c(a)g(lo)m(w)m(er)h(b)s(ound)f(of)g Fr(5)p Ft(n)g Fz(or)h(ev)m(en)h(higher.)57 b(W)-8 b(e)38 b(ha)m(v)m(e)h(in)d (fact)h(tried)g(to)g(pursuit)257 3444 y(this)c(approac)m(h)h(for)f (getting)f(a)h(higher)g(lo)m(w)m(er)h(b)s(ound,)f(but)h(failed.)44 b(Ho)m(w)m(ev)m(er,)36 b(during)257 3564 y(the)46 b(exp)s(erimen)m(ts,) j(the)d(author)f(and)h(T)-8 b(arui)44 b([)2039 3564 y SDict begin H.S end 2039 3564 a Fz(4)2088 3500 y SDict begin H.R end 2088 3500 a 2088 3564 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.AT08) cvn H.B /ANN pdfmark end 2088 3564 a Fz(])i(ha)m(v)m(e)g(succeeded\(?\))85 b(to)45 b(\034nd)h(the)257 3684 y(reason)33 b(wh)m(y)h(w)m(e)g(ha)m(v)m(e)g (failed.)404 3810 y(In)h(fact,)h(there)g Fu(exists)f Fz(a)g Ft(k)s Fz(-mixed)g(Bo)s(olean)f(function)g(with)h Ft(k)h Fr(=)c Ft(n)25 b Fq(\000)f Ft(o)p Fr(\()p Ft(n)p Fr(\))35 b Fz(that)257 3931 y(can)e(b)s(e)f(computed)g(b)m(y)i(a)d Ft(U)1299 3946 y Fp(2)1339 3931 y Fz(-circuit)g(of)g(size)i Fr(5)p Ft(n)21 b Fr(+)g Ft(o)p Fr(\()p Ft(n)p Fr(\))p Fz(.)43 b(The)34 b Fr(5)p Ft(n)e Fz(lo)m(w)m(er)g(b)s(ounds)h(for)257 4051 y(this)g(class)f(of)g(functions)h(ha)m(v)m(e)h(already)e(b)s(een)h (tigh)m(t.)404 4177 y(Belo)m(w)c(w)m(e)h(sk)m(etc)m(h)h(the)f (construction)f(of)f(the)i(function,)f(whic)m(h)h(is)e(a)h(mo)s (di\034cation)257 4297 y(of)j(a)h(function)f(in)m(tro)s(duced)g(b)m(y)i (Sa)m(vic)m(k\375)g(and)e(\232\341k)h([)2219 4297 y SDict begin H.S end 2219 4297 a Fz(48)2316 4233 y SDict begin H.R end 2316 4233 a 2316 4297 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.SZ96) cvn H.B /ANN pdfmark end 2316 4297 a Fz(].)404 4423 y(Let)47 b Ft(p)g Fz(b)s(e)h(a)f(prime)f(suc)m(h) j(that)e Ft(n)53 b Fq(\024)g Ft(p)f(<)h Fr(2)p Ft(n)p Fz(.)88 b(De\034ne)48 b Ft(w)2764 4438 y Fs(n)2864 4423 y Fr(:)53 b Fq(f)p Fr(0)p Ft(;)17 b Fr(1)p Ft(;)g(:)g(:)g(:)n Fq(g)53 b(!)257 4544 y(f)p Fr(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(g)45 b Fz(so)h(that)f Ft(w)1155 4559 y Fs(n)1202 4544 y Fr(\()p Ft(s)p Fr(\))h Fz(is)f(the)h(residue)g(of)f Ft(s)h Fz(mo)s(dulo)d Ft(p)p Fz(,)49 b(if)c(this)g(residue)h(lies)257 4664 y(in)e Fq(f)p Fr(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(g)p Fz(,)48 b(and)c(is)h Fr(1)f Fz(otherwise.)81 b(Put)45 b Ft(b)50 b Fr(=)e Fq(d)2370 4625 y Fs(n)p 2270 4641 244 4 v 2270 4698 a Fo(d)p Fp(log)14 b Fs(n)2448 4680 y Fn(2)2482 4698 y Fo(e)2523 4664 y Fq(e)p Fz(.)81 b(W)-8 b(e)45 b(split)f(the)h(in)m(ter-)257 4798 y(v)-5 b(al)48 b Fq(f)p Fr(1)p Ft(;)17 b(:)g(:)g(:)f(;)h(n)p Fq(g)48 b Fz(in)m(to)g Ft(b)i Fz(blo)s(c)m(ks)f Ft(D)1600 4813 y Fp(1)1639 4798 y Ft(;)17 b(:)g(:)g(:)f(;)h(D)1939 4813 y Fs(b)2022 4798 y Fz(of)49 b(equal)g(size.)92 b(F)-8 b(or)49 b(ev)m(ery)h Ft(n)56 b Fq(2)g Fi(N)p Fz(,)257 4918 y Ft(f)305 4933 y Fs(n)352 4918 y Fr(\()p Ft(x)445 4933 y Fp(1)485 4918 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)759 4933 y Fs(n)806 4918 y Fr(\))33 b Fz(outputs)g(one)g(of)f(its)g(input)g Ft(x)1976 4933 y Fs(z)2048 4918 y Fz(where)i(an)f(index)f(is)g(giv)m (en)h(b)m(y)1215 5248 y Ft(z)88 b Fr(=)83 b Ft(w)1577 5263 y Fs(n)1640 5078 y Fl( )1776 5124 y Fs(b)1719 5154 y Fl(X)1734 5364 y Fs(i)p Fp(=1)1880 5078 y Fl( )1958 5248 y Ft(i)23 b Fq(\001)2070 5154 y Fl(M)2064 5365 y Fs(j)t Fo(2)p Fs(D)2202 5375 y Fm(i)2244 5248 y Ft(x)2299 5263 y Fs(j)2336 5078 y Fl(!!)2510 5248 y Ft(:)257 5504 y SDict begin H.S end 257 5504 a 257 5504 a SDict begin 14.5 H.A end 257 5504 a 257 5504 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2) cvn H.B /DEST pdfmark end 257 5504 a eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont 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[ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Oz09) cvn H.B /ANN pdfmark end 967 4477 a Fz(].)404 4598 y(F)-8 b(or)31 b Ft(n)d Fq(\025)g Fr(1)p Fz(,)33 b(let)f Fg(D)1105 4562 y Fs(n)1184 4598 y Fz(b)s(e)h(a)f(Hadamard)g(matrix)f(of)h(order)g Fr(2)2609 4562 y Fs(n)2689 4598 y Fz(de\034ned)h(as)538 4868 y Fg(D)624 4827 y Fp(1)691 4868 y Fr(=)794 4727 y Fl(\022)909 4807 y Fr(1)122 b(1)909 4927 y(1)83 b Fq(\000)p Fr(1)1209 4727 y Fl(\023)1299 4868 y Ft(;)114 b Fg(D)1526 4827 y Fs(n)1600 4868 y Fr(=)1704 4727 y Fl(\022)1819 4807 y Fg(D)1905 4771 y Fs(n)p Fo(\000)p Fp(1)2164 4807 y Fg(D)2250 4771 y Fs(n)p Fo(\000)p Fp(1)1819 4927 y Fg(D)1905 4891 y Fs(n)p Fo(\000)p Fp(1)2125 4927 y Fq(\000)p Fg(D)2288 4891 y Fs(n)p Fo(\000)p Fp(1)2467 4727 y Fl(\023)2557 4868 y Fr(\()p Fz(for)32 b Ft(n)c Fq(\025)g Fr(2\))p Ft(:)404 5147 y Fz(The)41 b(w)m(ell)e(kno)m(wn)j(iden)m(tities)d Fg(D)1648 5111 y Fs(n)1695 5147 y Fg(D)1781 5111 y Fs(n)1868 5147 y Fr(=)i(2)2034 5111 y Fs(n)2080 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Fp(1)2136 4658 y Fr(\))83 b Ft(>)g Fg(0)p Ft(;)1237 4803 y Fz(diag)o Fr(\()p Fg(f)1488 4818 y Fp(1)1527 4803 y Fr(\))p Fg(D)1651 4762 y Fs(n)p Fo(\000)p Fp(1)1788 4803 y Fr(\()p Fg(a)1880 4818 y Fp(0)1942 4803 y Fq(\000)23 b Fg(a)2096 4818 y Fp(1)2136 4803 y Fr(\))83 b Ft(>)g Fg(0)p Ft(:)823 b Fz(\(10\))257 5023 y(By)33 b(F)-8 b(act)628 5023 y SDict begin H.S end 628 5023 a Fz(7)677 4959 y SDict begin H.R end 677 4959 a 677 5023 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (fact.7) cvn H.B /ANN pdfmark end 677 5023 a Fz(,)33 b(this)f(is)g(equiv)-5 b(alen)m(t)32 b(to)1379 5243 y SDict begin H.S end 1379 5243 a 1379 5243 a SDict begin 14.5 H.A end 1379 5243 a 1379 5243 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark end 1379 5243 a Fr(\()p Fg(a)1471 5258 y Fp(0)1533 5243 y Fr(+)22 b Fg(a)1685 5258 y Fp(1)1725 5243 y Fr(\))1763 5202 y Ff(T)1901 5243 y Fr(=)83 b(2)p Fg(k)2168 5258 y Fp(0)2207 5243 y Fg(Y)2292 5258 y Fp(0)2331 5243 y Ft(;)1377 5388 y Fr(\()p Fg(a)1469 5403 y Fp(0)1531 5388 y Fq(\000)23 b Fg(a)1685 5403 y Fp(1)1725 5388 y Fr(\))1763 5347 y Ff(T)1901 5388 y Fr(=)83 b(2)p Fg(k)2168 5403 y Fp(1)2207 5388 y Fg(Y)2292 5403 y Fp(1)2331 5388 y Ft(:)964 b Fz(\(11\))p eop end %%Page: 23 23 TeXDict begin 23 22 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.23) cvn H.B /DEST pdfmark end 257 266 a 307 x Fz(for)48 b(some)f(ro)m(w)h(v)m(ectors)i Fg(k)1289 588 y Fp(0)1382 573 y Ft(>)j Fg(0)48 b Fz(and)g Fg(k)1879 588 y Fp(1)1972 573 y Ft(>)54 b Fg(0)p Fz(,)d(where)e Fg(Y)2618 588 y Fs(i)2694 573 y Fz(\()p Ft(i)54 b Fr(=)f(0)p Ft(;)17 b Fr(1)p Fz(\))47 b(denotes)257 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Fz(can)h(b)s(e)g(stated)g(as)g(follo)m(ws)f (\(see)i([)3102 2017 y SDict begin H.S end 3102 2017 a Fz(1)3151 1953 y SDict begin H.R end 3151 1953 a 3151 2017 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.A10) cvn H.B /ANN pdfmark end 3151 2017 a Fz(])e(for)h(the)257 2138 y(pro)s(of)7 b(\).)257 2163 y SDict begin H.S end 257 2163 a 257 2163 a SDict begin 14.5 H.A end 257 2163 a 257 2163 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark end 257 2163 a 145 x Fv(Theorem)44 b(5.)g Fu([)900 2308 y SDict begin H.S end 900 2308 a Fu(1)950 2244 y SDict begin H.R end 950 2244 a 950 2308 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.A10) cvn H.B /ANN pdfmark end 950 2308 a Fu(])c(L)-5 b(et)40 b Ft(\017)d(>)g Fr(0)j Fu(an)f(arbitr)-5 b(ary)41 b(c)-5 b(onstant.)59 b(L)-5 b(et)40 b Ft(k)g Fq(\025)e Fr(1)i Fu(b)-5 b(e)39 b(an)h(inte)-5 b(ger)257 2428 y(and)35 b Ft(\031)j Fu(b)-5 b(e)35 b(an)f(or)-5 b(dering)34 b(of)h Fq(M)1420 2392 y Fs(k)1462 2428 y Fu(.)45 b(Then,)33 b(ther)-5 b(e)35 b(is)g(a)f(c)-5 b(onstant)35 b Ft(c)27 b(>)h Fr(0)35 b Fu(\(dep)-5 b(ending)33 b(on)257 2549 y Ft(\017)43 b Fu(and)e Ft(k)s Fu(\))i(such)e(that)i(al)5 b(l)42 b(but)g(a)g Fr(2)1566 2513 y Fo(\000)p Fs(c)p Fp(2)1687 2489 y Fm(n)1776 2549 y Fu(fr)-5 b(action)41 b(of)h Ft(n)p Fu(-variable)f(Bo)-5 b(ole)g(an)41 b(functions)257 2669 y(have)34 b(PTF)h(density)f(at)h (most)g Fr(\(1)22 b Fq(\000)g Ft(d)p Fr(\()p Ft(k)s(;)17 b(\031)t Fr(\))22 b(+)g Ft(\017)p Fr(\)2)2141 2633 y Fs(n)2188 2669 y Fu(.)404 2839 y Fz(It)39 b(is)g(a)h(bit)e(tedious)i (but)g(easy)g(to)g(v)m(erify)g(\(b)m(y)g(hand\))g(that)f Ft(d)p Fr(\(2)p Ft(;)17 b(\031)t Fr(\))39 b(=)g(5)p Ft(=)p Fr(16)g Fz(for)257 2960 y(ev)m(ery)27 b Ft(\031)h Fz(of)c Fq(M)813 2924 y Fp(2)852 2960 y Fz(;)j(this)d(giv)m(es)g Fr(\(0)p Ft(:)p Fr(688\)2)1666 2924 y Fs(n)1737 2960 y Fz(upp)s(er)g(b)s(ounds)h(on)g(the)f(a)m(v)m(erage)i(PTF)e(densit)m (y)257 3080 y(of)42 b Ft(n)p Fz(-v)-5 b(ariable)41 b(functions.)73 b(The)44 b(computation)d(of)h Ft(d)p Fr(\()p Ft(k)s(;)17 b(\031)t Fr(\))42 b Fz(for)g Ft(k)48 b Fq(\025)d Fr(3)d Fz(is)g(done)h(b)m(y)257 3200 y(a)d(computer)f(\(it)g(tak)m(es)i (double)f(exp)s(onen)m(tial)f(time)f(in)h Ft(k)s Fz(\).)65 b(Note)40 b(that)f(w)m(e)i(can)f(see)257 3321 y(whether)24 b(a)f(giv)m(en)g Ft(k)s Fz(-v)-5 b(ariable)20 b(function)j Ft(f)33 b Fz(has)23 b(a)g(sign-represen)m(ting)f(p)s(olynomial)e(with) 257 3441 y(supp)s(ort)40 b Ft(S)46 b Fz(b)m(y)41 b(c)m(hec)m(king)g (the)f(feasibilit)m(y)e(of)h(the)h(linear)e(system)j(diag)o Fr(\()p Fg(f)10 b Fr(\))p Fg(D)3186 3405 y Fs(k)3229 3441 y Fg(a)40 b Ft(>)g Fg(0)257 3562 y Fz(where)24 b Ft(a)580 3577 y Fs(i)636 3562 y Fr(=)j(0)22 b Fz(for)g(ev)m(ery)i Ft(i)k Fq(62)g Ft(S)6 b Fz(,)24 b(whic)m(h)f(is)f(an)g(easy)h(task)g (for)f(an)m(y)h(linear)d(programming)257 3682 y(solv)m(er)33 b(when)h Ft(k)h Fz(is)e(small.)404 3802 y(The)47 b(computation)d(of)i Ft(d)p Fr(\(3)p Ft(;)17 b(\031)t Fr(\))45 b Fz(for)g(ev)m(ery)j(p)s (ossible)e Ft(\031)k Fz(is)45 b(quite)h(feasible)f(b)m(y)i(a)257 3923 y(standard)33 b(PC.)h(By)f(using)f(the)h(GLPK)f(pac)m(k)-5 b(age)33 b([)2139 3923 y SDict begin H.S end 2139 3923 a Fz(35)2237 3859 y SDict begin H.R end 2237 3859 a 2237 3923 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.Mak) cvn H.B /ANN pdfmark end 2237 3923 a Fz(],)f(w)m(e)i(found)f(that)279 4090 y Ft(d)p Fr(\(3)p Ft(;)17 b Fq(f)p Fr(1)p Ft(;)g(x)659 4105 y Fp(1)697 4090 y Ft(;)g(x)796 4105 y Fp(2)835 4090 y Ft(;)g(x)934 4105 y Fp(1)974 4090 y Ft(x)1029 4105 y Fp(2)1069 4090 y Ft(;)g(x)1168 4105 y Fp(3)1207 4090 y Ft(;)g(x)1306 4105 y Fp(1)1346 4090 y Ft(x)1401 4105 y Fp(3)1441 4090 y Ft(;)g(x)1540 4105 y Fp(2)1579 4090 y Ft(x)1634 4105 y Fp(3)1674 4090 y Ft(;)g(x)1773 4105 y Fp(1)1813 4090 y Ft(x)1868 4105 y Fp(2)1908 4090 y Ft(x)1963 4105 y Fp(3)2003 4090 y Fq(g)p Fr(\))82 b(=)h(316)p Ft(=)p Fr(\(2)2615 4049 y Fp(7)2676 4090 y Fq(\002)22 b Fr(8\))28 b(=)f(0)p Ft(:)p Fr(3085)17 b Fq(\001)g(\001)g(\001)30 b Ft(;)279 4236 y(d)p Fr(\(3)p Ft(;)17 b Fq(f)p Fr(1)p Ft(;)g(x)659 4251 y Fp(1)697 4236 y Ft(;)g(x)796 4251 y Fp(2)835 4236 y Ft(;)g(x)934 4251 y Fp(3)974 4236 y Ft(;)g(x)1073 4251 y Fp(1)1113 4236 y Ft(x)1168 4251 y Fp(2)1207 4236 y Ft(;)g(x)1306 4251 y Fp(1)1346 4236 y Ft(x)1401 4251 y Fp(3)1441 4236 y Ft(;)g(x)1540 4251 y Fp(2)1579 4236 y Ft(x)1634 4251 y Fp(3)1674 4236 y Ft(;)g(x)1773 4251 y Fp(1)1813 4236 y Ft(x)1868 4251 y Fp(2)1908 4236 y Ft(x)1963 4251 y Fp(3)2003 4236 y Fq(g)p Fr(\))82 b(=)h(360)p Ft(=)p Fr(\(2)2615 4195 y Fp(7)2676 4236 y Fq(\002)22 b Fr(8\))28 b(=)f(0)p Ft(:)p Fr(3515)17 b Fq(\001)g(\001)g(\001)30 b Ft(;)257 4403 y Fz(and)g(that)f(all)f(orderings)h(of)g Fq(M)1433 4367 y Fp(3)1501 4403 y Fz(are)h(categorized)f(in)m(to)g(one)h(of)f(the)h (ab)s(o)m(v)m(e)g(t)m(w)m(o.)43 b(This)257 4524 y(giv)m(es)33 b(the)g(upp)s(er)g(b)s(ound)g(on)f(the)h(a)m(v)m(erage)h(PTF)f(densit)m (y)g(of)f 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257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.25) cvn H.B /DEST pdfmark end 257 266 a 307 x Fz(whic)m(h)33 b(giv)m(es)g(the)g(upp)s(er)g(b)s(ound)g(of)f Fr(\(0)p Ft(:)p Fr(627\)2)1982 537 y Fs(n)2028 573 y Fz(.)43 b(Ho)m(w)m(ev)m (er,)35 b(this)d(is)g(not)h(the)g(b)s(est.)404 693 y(W)-8 b(e)35 b(then)g(computed)g(the)g(a)m(v)m(erage)h(freedom)e(of)g(2000)g (randomly)f(c)m(hosen)k(order-)257 814 y(ings)28 b(of)f Fq(M)680 777 y Fp(4)720 814 y Fz(,)i(whic)m(h)f(to)s(ok)g(ab)s(out)f(t) m(w)m(o)i(da)m(ys)g(on)f(a)g(standard)g(PC.)h(The)g(b)s(est)g(ordering) 257 934 y(w)m(e)34 b(ha)m(v)m(e)g(found)e(is)561 1144 y Ft(\031)620 1103 y Fo(0)616 1169 y Fp(4)739 1144 y Fr(=)82 b Fq(f)p Fr(1)p Ft(;)17 b(x)1095 1159 y Fp(1)1135 1144 y Ft(x)1190 1159 y Fp(2)1229 1144 y Ft(x)1284 1159 y Fp(3)1324 1144 y Ft(;)g(x)1423 1159 y Fp(1)1463 1144 y Ft(x)1518 1159 y Fp(4)1558 1144 y Ft(;)g(x)1657 1159 y Fp(1)1696 1144 y Ft(x)1751 1159 y Fp(3)1791 1144 y 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/View [/XYZ H.V] /Dest (equation.14) cvn H.B /DEST pdfmark end 921 1710 a Ft(d)p Fr(\(4)p Ft(;)17 b(\031)1162 1668 y Fo(0)1158 1734 y Fp(4)1197 1710 y Fr(\))83 b(=)f(200964)p Ft(=)p Fr(\(2)1906 1668 y Fp(15)2001 1710 y Fq(\002)23 b Fr(16\))k(=)h(0)p Ft(:)p Fr(3833)17 b Fq(\001)g(\001)g(\001)30 b Ft(:)490 b Fz(\(14\))257 1920 y(Note)35 b(that)f(8)g(\(out)g(of)g(2000\))g (orderings)g(ha)m(v)m(e)h(the)g(same)f(v)-5 b(alue.)49 b(This)34 b(immediately)257 2040 y(giv)m(es)f(the)g(follo)m(wing)d(upp) s(er)j(b)s(ound,)g(whic)m(h)g(is)f(the)h(b)s(est)g(w)m(e)h(ha)m(v)m(e)g (obtained)d(so)i(far.)257 2059 y SDict begin H.S end 257 2059 a 257 2059 a SDict begin 14.5 H.A end 257 2059 a 257 2059 a SDict begin [ /View [/XYZ H.V] /Dest (corollary.1) cvn H.B /DEST pdfmark end 257 2059 a 151 x Fv(Corollary)54 b(1.)49 b Fu([)939 2210 y SDict begin H.S end 939 2210 a Fu(1)989 2146 y SDict begin H.R end 989 2146 a 989 2210 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.A10) cvn H.B /ANN pdfmark end 989 2210 a Fu(])f(A)n(lmost)g(al)5 b(l)47 b(Bo)-5 b(ole)g(an)47 b(functions)h(on)f Ft(n)i Fu(variables)d(have)i(PTF)257 2331 y(density)35 b(at)g(most)g Fr(\(0)p Ft(:)p Fr(617\)2)1293 2294 y Fs(n)1338 2331 y Fu(.)404 2501 y Fz(W)-8 b(e)39 b(pro)m(vide)h(v)m(eri\034able)f(data) g(for)g(Eqs.)65 b(\()2026 2501 y SDict begin H.S end 2026 2501 a Fz(13)2123 2437 y SDict begin H.R end 2123 2437 a 2123 2501 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 2123 2501 a Fz(\))39 b(and)h(\()2435 2501 y SDict begin H.S end 2435 2501 a Fz(14)2532 2437 y SDict begin H.R end 2532 2437 a 2532 2501 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.14) cvn H.B /ANN pdfmark end 2532 2501 a Fz(\))f(on)h(the)f(w)m(eb)i(page)e([)3392 2501 y SDict begin H.S end 3392 2501 a Fz(1)3441 2437 y SDict begin H.R end 3441 2437 a 3441 2501 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.A10) cvn H.B /ANN pdfmark end 3441 2501 a Fz(].)257 2621 y(These)h(are)d(the)i(lists)d(of)i(p) s(olynomial)c(represen)m(tations)k(of)g(all)d Fr(4)p Fz(-v)-5 b(ariable)35 b(functions)257 2742 y(with)k(maxim)m(um)e (freedom)h(with)h(resp)s(ect)h(to)f(the)g(designated)h(ordering.)62 b(The)40 b(cor-)257 2862 y(rectness)34 b(of)e(Eqs.)44 b(\()1001 2862 y SDict begin H.S end 1001 2862 a Fz(13)1099 2798 y SDict begin H.R end 1099 2798 a 1099 2862 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 1099 2862 a Fz(\))31 b(and)h(\()1395 2862 y SDict begin H.S end 1395 2862 a Fz(14)1493 2798 y SDict begin H.R end 1493 2798 a 1493 2862 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (equation.14) cvn H.B /ANN pdfmark end 1493 2862 a Fz(\))g(can)g(b)s(e)g(v)m(eri\034ed)h(b)m(y)g(hand)f(in)f (a)h(sev)m(eral)g(w)m(eeks,)j(or)257 2982 y(b)m(y)c(a)f(computer)g(in)g 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begin 14.5 H.A end 257 3705 a 257 3705 a SDict begin [ /View [/XYZ H.V] /Dest (open.6) cvn H.B /DEST pdfmark end 257 3705 a 170 x Fv(Problem)h(6.)42 b Fu(What)35 b(is)f(the)h(b)-5 b(est)35 b(achievable)e(b)-5 b(ound)34 b(obtaine)-5 b(d)34 b(by)h(this)g(metho)-5 b(d?)257 4043 y SDict begin H.S end 257 4043 a 257 4043 a SDict begin 14.5 H.A end 257 4043 a 257 4043 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 257 4043 a 163 x FA(5)156 b(Concluding)53 b(Remarks)257 4425 y Fz(In)29 b(this)g(article,)f(w)m(e)h(review)g(sev)m(eral)h (recen)m(t)g(attempts)e(to)g(use)i(computers)f(in)f(v)-5 b(arious)257 4545 y(w)m(a)m(ys)25 b(to)f(obtain)e(concrete)i(results)g (for)f(the)h(problems)f(in)f(computational)f(complexit)m(y)-8 b(.)257 4666 y(Some)37 b(of)g(them)g(are)h(\(at)f(least)g(partially\))e (succeeded)40 b(and)d(some)h(of)f(them)g(are)g(not.)257 4786 y(W)-8 b(e)31 b(strongly)f(hop)s(e)g(that)g(these)h(approac)m(hes) 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SDict begin H.S end 257 4491 a 257 4491 a SDict begin 13.6 H.A end 257 4491 a 257 4491 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Blu84) cvn H.B /DEST pdfmark end 257 4491 a 131 x FB([10])50 b(N.)30 b(Blum,)40 b(A)30 b(Bo)s(olean)h(F)-8 b(unction)32 b(Requiring)e Fc(3)p Fb(n)g FB(Net)m(w)m(ork)i(Size,)40 b Fe(The)-5 b(or)g(et.)34 b(Comput.)447 4735 y(Sci.)p FB(,)c(28,)g(337\025345,)i (1984.)257 4756 y SDict begin H.S end 257 4756 a 257 4756 a SDict begin 13.6 H.A end 257 4756 a 257 4756 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Bol09) cvn H.B /DEST pdfmark end 257 4756 a 136 x FB([11])50 b(B.)29 b(Bollig,)40 b(In)m(teger)30 b(Multiplication)i(and)d(the)h(Complexit)m(y)h(of)e (Binary)h(Decision)g(Dia-)447 5005 y(grams,)41 b Fe(Bul)5 b(letin)32 b(of)g(the)h(EA)-7 b(TCS)p FB(,)29 b(98,)i(78\025106,)g (2009.)257 5026 y SDict begin H.S end 257 5026 a 257 5026 a SDict begin 13.6 H.A end 257 5026 a 257 5026 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Bry91) cvn H.B /DEST pdfmark end 257 5026 a 136 x FB([12])50 b(R.E.)27 b(Bry)m(an)m(t,)36 b(On)27 b(the)g(Complexit)m(y)h(of)f(VLSI)f(Implemen)m(tations)j(and)f (Graph)f(Repre-)447 5275 y(sen)m(tations)34 b(of)g(Bo)s(olean)g(F)-8 b(unctions)34 b(with)g(Applications)h(to)f(In)m(teger)g (Multiplication,)447 5388 y Fe(IEEE)e(T)-7 b(r)i(ans.)32 b(on)g(Computers)p FB(,)f(40,)f(205\025213,)i(1991.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.27) cvn H.B /DEST pdfmark end 257 266 a 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 13.6 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (cite.BW02) cvn H.B /DEST pdfmark end 257 473 a 100 x FB([13])50 b(H.)32 b(Buhrman)h(and)f(R.)g(de)g(W)-8 b(olf,)46 b(Complexit)m(y)33 b(Measures)f(and)h(Decision)f(T)-8 b(ree)33 b(Com-)447 686 y(plexit)m(y:)41 b(A)30 b(Surv)m(ey)-8 b(,)41 b Fe(The)-5 b(or)g(et.)33 b(Comput.)f(Sci.)p FB(,)e(288\(1\),)h(21\02543,)h(2002.) 257 710 y SDict begin H.S end 257 710 a 257 710 a SDict begin 13.6 H.A end 257 710 a 257 710 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Cov65) cvn H.B /DEST pdfmark end 257 710 a 129 x FB([14])50 b(T.)41 b(Co)m(v)m(er,)78 b(Geometrical)43 b(and)f(Statistical)g(Prop)s(erties)g(of)g(Systems)f(of)g(Linear)h(In-) 447 951 y(equalities)33 b(with)g(Applications)g(in)g(P)m(attern)h (Recognition,)48 b Fe(IEEE)35 b(T)-7 b(r)i(ans.)33 b(Ele)-5 b(ctr)g(onic)447 1064 y(Computers)p FB(,)30 b(EC-14\(3\),)i (326\025334,)g(1965.)257 1088 y SDict begin H.S end 257 1088 a 257 1088 a SDict begin 13.6 H.A end 257 1088 a 257 1088 a SDict begin [ /View [/XYZ H.V] /Dest (cite.For01) cvn H.B /DEST pdfmark end 257 1088 a 129 x FB([15])50 b(J.)24 b(F)-8 b(orster,)26 b(M.)d(Krause,)j(S.)d(V.)h(Lok)-5 b(am,)26 b(R.)d(Mubarakzjano)m(v,)j(N.)d(Sc)m(hmitt,)k(H.)c(Simon,)447 1330 y(Relations)45 b(Bet)m(w)m(een)i(Comm)m(unication)h(Complexit)m(y) -8 b(,)51 b(Linear)46 b(Arrangemen)m(ts,)51 b(and)447 1443 y(Computational)32 b(Complexit)m(y)-8 b(.)43 b Fe(Pr)-5 b(o)g(c.)33 b(of)f(FSTTCS)e('01)p FB(,)g(171\025182,)i(2001.)257 1462 y SDict begin H.S end 257 1462 a 257 1462 a SDict begin 13.6 H.A end 257 1462 a 257 1462 a SDict begin [ /View [/XYZ H.V] /Dest (cite.FF84) cvn H.B /DEST pdfmark end 257 1462 a 134 x FB([16])50 b(P)-8 b(.)27 b(F)-8 b(rankl)27 b(and)f(F\374redi,)35 b(An)26 b(Exact)h(Result)f(for)f(3-Graphs,)35 b Fe(Discr)-5 b(ete)30 b(Math.)p FB(,)d(50\(2-3\),)447 1709 y(323\025328,)k(1984.)257 1728 y SDict begin H.S end 257 1728 a 257 1728 a SDict begin 13.6 H.A end 257 1728 a 257 1728 a SDict begin [ /View [/XYZ H.V] /Dest (cite.FS90) cvn H.B /DEST pdfmark end 257 1728 a 133 x FB([17])50 b(S.J.)32 b(F)-8 b(riedman)33 b(and)f(K.J.)h(Sup)s(o)m(wit,)45 b(Finding)33 b(the)f(Optimal)h(V)-8 b(ariable)32 b(Ordering)h(for)447 1974 y(Binary)d(Decision)h(Diagrams,)41 b Fe(IEEE)33 b(T)-7 b(r)i(ans.)31 b(Comput.)p FB(,)f(39,)h(710\025713,)h(1990.)257 1993 y SDict begin H.S end 257 1993 a 257 1993 a SDict begin 13.6 H.A end 257 1993 a 257 1993 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Fuk10) cvn H.B /DEST pdfmark end 257 1993 a 134 x FB([18])50 b(H.)30 b(F)-8 b(ukuhara,)41 b Fe(p)-5 b(ersonal)34 b(c)-5 b(ommunic)g(ation)p FB(,)30 b(2010.)257 2151 y SDict begin H.S end 257 2151 a 257 2151 a SDict begin 13.6 H.A end 257 2151 a 257 2151 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Gon08) cvn H.B /DEST pdfmark end 257 2151 a 129 x FB([19])50 b(G.)31 b(Gon)m(thier,)46 b(F)-8 b(ormal)33 b(Pro)s(of)e(\025)h(The)g(F)-8 b(our-Color)33 b(Theorem,)46 b Fe(Notic)-5 b(es)34 b(of)g(the)g(AMS)p FB(,)447 2393 y(55\(11\),)d(1382\0251393,)h(2008.)257 2417 y SDict begin H.S end 257 2417 a 257 2417 a SDict begin 13.6 H.A end 257 2417 a 257 2417 a SDict begin [ /View [/XYZ H.V] /Dest (cite.H10) cvn H.B /DEST pdfmark end 257 2417 a 128 x FB([20])50 b(T.C.)22 b(Hales,)h(J.)f(Harrison,)i(S.)e (McLaughlin,)j(T.)d(Nipk)m(o)m(w,)i(S.)e(Obua)g(and)g(R.)f(Zumk)m (eller,)447 2658 y(A)34 b(Revision)h(of)g(the)g(Pro)s(of)g(of)g(the)g (Kepler)h(Conjecture,)56 b Fe(Discr)-5 b(ete)39 b(Comput.)d(Ge)-5 b(om)p FB(,)447 2771 y(44,)30 b(1\02534,)h(2010.)257 2790 y SDict begin H.S end 257 2790 a 257 2790 a SDict begin 13.6 H.A end 257 2790 a 257 2790 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Has98) cvn H.B /DEST pdfmark end 257 2790 a 134 x FB([21])50 b(J.)37 b(H\345stad,)60 b(The)37 b(Shrink)-5 b(age)38 b(Exp)s(onen)m(t)f(of)f(de)h(Morgan)g(F)-8 b(orm)m(ulas)38 b(is)f(2,)60 b Fe(SIAM)37 b(J.)447 3037 y(Comput.)p FB(,)30 b(27\(1\),)h(48\02564,)g(1998.)257 3061 y SDict begin H.S end 257 3061 a 257 3061 a SDict begin 13.6 H.A end 257 3061 a 257 3061 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HLS05) cvn H.B /DEST pdfmark end 257 3061 a 129 x FB([22])50 b(P)-8 b(.)24 b(H\370y)m(er,)g(T.)g(Lee,)h(R.)d (S)s(\007)-66 b(palek)23 b(Tigh)m(t)h(A)m(dv)m(ersary)f(Bounds)g(for)g (Comp)s(osite)h(F)-8 b(unctions,)447 3303 y Fd(quant-ph/0509067)p FB(,)30 b(2005.)257 3324 y SDict begin H.S end 257 3324 a 257 3324 a SDict begin 13.6 H.A end 257 3324 a 257 3324 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HLS07) cvn H.B /DEST pdfmark end 257 3324 a 131 x FB([23])50 b(P)-8 b(.)40 b(H\370y)m(er,)i(T.)d(Lee,)j(R.)d(S)s(\007)-66 b(palek)61 b(Negativ)m(e)40 b(W)-8 b(eigh)m(ts)40 b(Mak)m(e)g(A)m(dv)m (ersaries)g(Stronger,)447 3568 y Fe(Pr)-5 b(o)g(c.)33 b(of)f(STOC)g('07)p FB(,)e(526\025535,)h(2007.)257 3587 y SDict begin H.S end 257 3587 a 257 3587 a SDict begin 13.6 H.A end 257 3587 a 257 3587 a SDict begin [ /View [/XYZ H.V] /Dest (cite.HKP10) cvn H.B /DEST pdfmark end 257 3587 a 134 x FB([24])50 b(P)-8 b(.)45 b(Hrub)s(es,)h(S.)e(Jukna,)k(A.)c (Kulik)m(o)m(v)h(and)f(P)-8 b(.)45 b(Pudlak,)85 b(On)44 b(Con)m(v)m(ex)h(Complexit)m(y)447 3834 y(Measures,)40 b Fe(The)-5 b(or)g(et.)33 b(Comput.)f(Sci.)p FB(,)e(411,)h (1842\0251854,)h(2010.)257 3853 y SDict begin H.S end 257 3853 a 257 3853 a SDict begin 13.6 H.A end 257 3853 a 257 3853 a SDict begin [ /View [/XYZ H.V] /Dest (cite.IM02) cvn H.B /DEST pdfmark end 257 3853 a 134 x FB([25])50 b(K.)23 b(Iw)m(ama)g(and)g(H.)g(Morizumi,)30 b(An)22 b(Explicit)i(Lo)m(w)m(er)h (Bound)e(of)f Fc(5)p Fb(n)5 b Fa(\000)g Fb(o)p Fc(\()p Fb(n)p Fc(\))22 b FB(for)h(Bo)s(olean)447 4100 y(Circuits,)41 b Fe(Pr)-5 b(o)g(c.)33 b(of)f(MF)n(CS)f('02)p FB(,)f(353\025364,)i (2002.)257 4119 y SDict begin H.S end 257 4119 a 257 4119 a SDict begin 13.6 H.A end 257 4119 a 257 4119 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ILMR05) cvn H.B /DEST pdfmark end 257 4119 a 133 x FB([26])50 b(K.)35 b(Iw)m(ama,)h(O.)f(Lac)m (hish,)i(H.)d(Morizumi)i(and)f(R.)f(Raz,)54 b(An)34 b(Explicit)i(Lo)m (w)m(er)g(Bound)447 4365 y(of)30 b Fc(5)p Fb(n)21 b Fa(\000)f Fb(o)p Fc(\()p Fb(n)p Fc(\))31 b FB(for)f(Bo)s(olean)h(Circuits,)42 b Fe(Manuscript)p FB(,)31 b(\(com)m(bined)i(v)m(ersion)e(of)f([)3203 4365 y SDict begin H.S end 3203 4365 a FB(25)3294 4307 y SDict begin H.R end 3294 4307 a 3294 4365 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.IM02) cvn H.B /ANN pdfmark end 3294 4365 a FB(])h(and)447 4478 y([)472 4478 y SDict begin H.S end 472 4478 a FB(34)562 4419 y SDict begin H.R end 562 4419 a 562 4478 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link /Dest (cite.LR01) cvn H.B /ANN pdfmark end 562 4478 a FB(]\),)g(2005.)257 4502 y SDict begin H.S end 257 4502 a 257 4502 a SDict begin 13.6 H.A end 257 4502 a 257 4502 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Khr71) cvn H.B /DEST pdfmark end 257 4502 a 129 x FB([27])50 b(V.M.)21 b(Khrap)s(c)m(henk)m(o,)29 b(A)21 b(Metho)s(d)g(of)g(Determining)i(Lo)m (w)m(er)h(Bounds)d(for)g(the)h(Complex-)447 4744 y(it)m(y)27 b(of)f Fc(\005)p FB(-sc)m(hemes,)36 b Fe(Mathematicheskie)30 b(Zametki)p FB(,)e(10\(1\),)g(83\02592,)g(1971)g(\(in)e(Russian\);)447 4857 y(English)31 b(translation)h(in)e Fe(Mathematic)-5 b(al)33 b(Notes)p FB(,)e(10\(1\),)g(474\025479,)h(1971.)257 4881 y SDict begin H.S end 257 4881 a 257 4881 a SDict begin 13.6 H.A end 257 4881 a 257 4881 a SDict begin [ /View [/XYZ H.V] /Dest (cite.KKY09) cvn H.B /DEST pdfmark end 257 4881 a 129 x FB([28])50 b(A.)37 b(A.)f(K)m(o)5 b(jevnik)m(o)m(v,)41 b(A.)36 b(S.)i(Kulik)m(o)m(v)g(and)f(G.)g(N.)g(Y)-8 b(arosla)m(vtsev,) 63 b(Finding)38 b(E\036cien)m(t)447 5122 y(Circuits)31 b(Using)f(SA)-8 b(T-solv)m(ers,)41 b Fe(Pr)-5 b(o)g(c.)34 b(of)e(SA)-7 b(T)31 b('09)p FB(,)f(LNCS)h(5584,)g(32\02544,)h(2009.)257 5141 y SDict begin H.S end 257 5141 a 257 5141 a SDict begin 13.6 H.A end 257 5141 a 257 5141 a SDict begin [ /View [/XYZ H.V] /Dest (cite.KW09) cvn H.B /DEST pdfmark end 257 5141 a 134 x FB([29])50 b(E.)32 b(Kushilevits)f(and)h(E.)g(W)-8 b(einreb,)44 b(On)32 b(the)f(Complexit)m(y)i(of)e(Comm)m(unication)j (Com-)447 5388 y(plexit)m(y)-8 b(,)41 b Fe(Pr)-5 b(o)g(c.)34 b(of)e(STOC)f('09)p FB(,)f(465\025474,)i(2009.)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.28) cvn H.B /DEST pdfmark end 257 266 a 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 13.6 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Knu08) cvn H.B /DEST pdfmark end 257 473 a 100 x FB([30])50 b(D.)31 b(Kn)m(uth,)47 b(The)32 b(Art)f(of)h(Computer)h(Programming:)45 b(V)-8 b(olume)33 b(4,)f(F)-8 b(ascicle)33 b(0:)43 b(In)m(tro-)447 686 y(duction)22 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end 257 2637 a 257 2637 a SDict begin 13.6 H.A end 257 2637 a 257 2637 a SDict begin [ /View [/XYZ H.V] /Dest (cite.MP88) cvn H.B /DEST pdfmark end 257 2637 a 134 x FB([37])50 b(M.)33 b(Minsky)g(and)h(S.)g(P)m(ap)s(ert,)53 b(P)m(erceptrons,)g(MIT) 33 b(Press,)i(Cam)m(bridge,)i(1988)d(\(First)447 2884 y(edition)d(app)s(eared)g(in)g(1968\))257 2908 y SDict begin H.S end 257 2908 a 257 2908 a SDict begin 13.6 H.A end 257 2908 a 257 2908 a SDict begin [ /View [/XYZ H.V] /Dest (cite.NS94) cvn H.B /DEST pdfmark end 257 2908 a 129 x FB([38])50 b(N.)29 b(Nisan)g(and)h(M.)g(Szegedy)-8 b(,)40 b(On)30 b(the)g(Degree)g(of)g(Bo)s(olean)g(F)-8 b(unctions)30 b(as)g(Real)f(P)m(oly-)447 3150 y(nomials,)41 b Fe(Computational)32 b(Complexity)p FB(,)e(4,)g(301\025313,)i(1994.) 257 3169 y SDict begin H.S end 257 3169 a 257 3169 a SDict begin 13.6 H.A end 257 3169 a 257 3169 a SDict begin [ /View [/XYZ H.V] /Dest (cite.NW95) cvn H.B /DEST pdfmark end 257 3169 a 134 x FB([39])50 b(N.)27 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b(72\(4\),)g (1239\0251282,)h(2007.)257 4615 y SDict begin H.S end 257 4615 a 257 4615 a SDict begin 13.6 H.A end 257 4615 a 257 4615 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Raz08) cvn H.B /DEST pdfmark end 257 4615 a 129 x FB([44])50 b(A.)43 b(Razb)s(oro)m(v,)83 b(On)44 b(3-Hyp)s(ergraphs)g(with)g(F)-8 b(orbidden)46 b(4-V)-8 b(ertex)44 b(Con\034gurations,)447 4857 y Fe(Manuscript)p FB(,)30 b(2008.)257 4876 y SDict begin H.S end 257 4876 a 257 4876 a SDict begin 13.6 H.A end 257 4876 a 257 4876 a SDict begin [ /View [/XYZ H.V] /Dest (cite.RSST97) cvn H.B /DEST pdfmark end 257 4876 a 134 x FB([45])50 b(R.)28 b(Rob)s(ertson,)g(D.P)-8 b(.)30 b(Sanders,)f(P)-8 b(.D.)30 b(Seymour)f(and)g(R.)f(Thomas,)38 b(The)29 b(F)-8 b(our)29 b(Colour)447 5122 y(Theorem,)41 b Fe(J.)33 b(Combin.)e(The)-5 b(ory.)p FB(,)30 b(Ser.)h(B,)f(70,)h (2\02544,)g(1997.)257 5141 y SDict begin H.S end 257 5141 a 257 5141 a SDict begin 13.6 H.A end 257 5141 a 257 5141 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Sak93) cvn H.B /DEST pdfmark end 257 5141 a 134 x FB([46])50 b(M.E.)39 b(Saks,)67 b(Slicing)40 b(the)f(Hyp)s(ercub)s(es,)67 b Fe(Surveys)41 b(in)f(Combinatorics)p FB(,)g(Cam)m(bridge)447 5388 y(Univ)m(ersit)m(y)31 b(Press,)f(211\025255,)i(1993.)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 257 266 a SDict begin H.S end 257 266 a 257 266 a SDict begin H.R end 257 266 a 257 266 a SDict begin [ /View [/XYZ H.V] /Dest (page.29) cvn H.B /DEST pdfmark end 257 266 a 257 473 a SDict begin H.S end 257 473 a 257 473 a SDict begin 13.6 H.A end 257 473 a 257 473 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Sau10) cvn H.B /DEST pdfmark end 257 473 a 100 x FB([47])50 b(M.)25 b(Sauerho\033,)34 b(An)25 b(Asymptotically)g(Optimal)i(Lo)m(w)m (er)g(Bound)f(on)f(the)g(OBDD)h(Size)f(of)447 686 y(the)34 b(Middle)h(Bit)f(of)g(Multiplication)i(for)e(the)h(P)m(airwise)h (Ascending)f(V)-8 b(ariable)35 b(Order,)447 799 y Fe(Disc.)d(Appl.)g (Math.)p FB(,)e(158\(11\),)i(1195\0251204,)g(2010.)257 821 y SDict begin H.S end 257 821 a 257 821 a SDict begin 13.6 H.A end 257 821 a 257 821 a SDict begin [ /View [/XYZ H.V] /Dest (cite.SZ96) cvn H.B /DEST pdfmark end 257 821 a 128 x FB([48])50 b(P)-8 b(.)43 b(Sa)m(vic)m(k\375)g(and)g(S.)f (\232\341k,)79 b(A)42 b(Large)h(Lo)m(w)m(er)h(Bound)f(for)f(1-Branc)m (hing)j(Programs,)447 1062 y Fe(ECCC)31 b(TR96-036)h(R)-5 b(ev.01)p FB(,)31 b(1996.)257 1080 y SDict begin H.S end 257 1080 a 257 1080 a SDict begin 13.6 H.A end 257 1080 a 257 1080 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Shr08) cvn H.B /DEST pdfmark end 257 1080 a 132 x FB([49])50 b(A.A.)37 b(Shersto)m(v,)67 b(The)38 b(Un)m(b)s(ounded-Error)j(Comm)m(unication)g (Complexit)m(y)e(of)f(Sym-)447 1325 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257 1869 a SDict begin 13.6 H.A end 257 1869 a 257 1869 a SDict begin [ /View [/XYZ H.V] /Dest (cite.SS05) cvn H.B /DEST pdfmark end 257 1869 a 133 x FB([52])50 b(R.\222palek)33 b(and)h(M.)f(Szegedy)-8 b(,)50 b(All)33 b(Quan)m(tum)i(A)m(dv)m(ersary) f(Metho)s(ds)f(are)h(Equiv)-5 b(alen)m(t,)447 2115 y Fe(Pr)g(o)g(c.)33 b(of)f(ICALP)g('05)p FB(,)e(1299-1311,)i(2005.)257 2132 y SDict begin H.S end 257 2132 a 257 2132 a SDict begin 13.6 H.A end 257 2132 a 257 2132 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Tar10) cvn H.B /DEST pdfmark end 257 2132 a 133 x FB([53])50 b(J.)42 b(T)-8 b(arui,)81 b(Smallest)43 b(F)-8 b(orm)m(ulas)44 b(for)e(the)g(P)m(arit)m(y)i(of)e Fc(2)2419 2232 y Fs(k)2504 2265 y FB(V)-8 b(ariables)43 b(are)g(Essen)m(tially)447 2378 y(Unique,)33 b Fe(The)-5 b(or)g(et.)30 b(Comput.)d(Sci.)p FB(,)f(\(in)g(press\),)h(2010.)f (\(Earlier)h(v)m(ersion)f(in)g(COCOON)447 2491 y('08\))257 2513 y SDict begin H.S end 257 2513 a 257 2513 a SDict begin 13.6 H.A end 257 2513 a 257 2513 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Wil08) cvn H.B /DEST pdfmark end 257 2513 a 128 x FB([54])50 b(R.)20 b(Williams,)28 b(Applying)22 b(Practice)g(to)g(Theory)-8 b(,)28 b Fe(A)n(CM)23 b(SIGA)n(CT)f(News)p FB(,)h(39\(4\):)37 b(37\02552,)447 2754 y(2008.)257 2754 y SDict begin H.S end 257 2754 a 257 2754 a SDict begin 13.6 H.A end 257 2754 a 257 2754 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Woe05) cvn H.B /DEST pdfmark end 257 2754 a 150 x FB([55])50 b(P)-8 b(.)30 b(W)-8 b(o)s(elfel,)38 b(Bounds)29 b(on)g(the)h(OBDD-size)f(of)f(In)m(teger)i(Multiplication)h (via)e(Univ)m(ersal)447 3017 y(Hashing.)40 b Fe(J.)33 b(Comput.)e(Sys.)h(Sci.)p FB(,)e(71\(4\),)h(520\025534,)g(2005.)257 3040 y SDict begin H.S end 257 3040 a 257 3040 a SDict begin 13.6 H.A end 257 3040 a 257 3040 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Zha05) cvn H.B /DEST pdfmark end 257 3040 a 127 x FB([56])50 b(S.)27 b(Zhang,)37 b(On)27 b(the)g(P)m(o)m(w)m (er)j(of)d(Am)m(bainis's)g(Lo)m(w)m(er)i(Bounds,)36 b Fe(The)-5 b(or)g(et.)31 b(Comput.)e(Sci.)p FB(,)447 3280 y(339\(2-3\),)i(241\025256,)h(2005.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF