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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 (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 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1C87678CE98C24B934A76220 4DD9B2FF3A49786028E35DDE10AD2C926BD30AD47015FFE9469DE1F793D1C53A 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 80 /summationtext put dup 88 /summationdisplay put dup 98 /hatwide put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (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 33 /arrowright put dup 55 /mapsto put dup 63 /perpendicular put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -958 1146 777}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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 /ff put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i 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 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-35 -250 1148 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 4E47B39E6712EB8908A3265E5FAB40567E866C244814449F1E993AAB422C3F1D DFA8C7118584F2E5197FD4BFA3A8AE9E953C6CD4672C0FF51E41C3A919749C1A F06650DF4C5E17492164BDBCDF22609A74BFA7F69960A64B9F949FFC2A807458 8579366C4F41BDE1FDFBCC4845FA19BBB6963D65EE8532549274BAEBDFF24FA6 03235D1BE37C06B1938AF369DA75BF38DDBC87A1FF445EAA16E1895ABE9506B9 211955753E447865D33CEF007391D2666A046277A30A49804FFCED3FEA5EB2C3 E52EE14A9F75241EA10C91974CDA6236EB840FD44D6DDE4D9B3266C3B99BD38B D835BCA8CB819C073480FB972CC028D218F6A1D344CE1B63F4FBF2C826F412E1 6E0B05A26125865A14FD7B7030B478BB8BC6BC395335C3BA940E1C348267F4F9 0AF97BBEE253511940F1048E175D3569F7D05A28851B6F50765FEB6C9654FEDC 1BF52F535DB5BB90C1BD5D2EBF75E0AEBE82B20507F3C28A03746781018D4EB2 298E4F2C27ACF73FA73EBE43F014BB575AAD516C0407B29E1653375135ECB74D C91372F06FA8EF37C31AF3FA48AE65318EAA6C34830A5377ABB2DFA5DA53A574 433484BA1466709A4B186761655C8E482833B697673E847C691079E7F1DCB8D6 1AD91101D757B83E2090337D525AEECB028FB3C9F6A6E6AD2F322CFDC5A833E6 1CE4EDBF41FD34FD61630581D222F854A76C2EA9FD72796A7C9CC1F6C2FCCD16 E95CA05826A4ECFADA6A5FB83C41A7131E52BA6585DD6DD78515D8F7327DFC6F 9404F891DDE3C21035A7EB1204E3403503B2F66527D9057C2688C4A066F61345 B461EE90CB0C3C8190A51DA0579E8AEE88F3F1FCD09C5306B503299815CB8756 579AA70B98DBF3DF85794ADB3B7B87B6ACE3F921BF5A4CE541181AB702F5C08D 28E9A6D5C9365CF5EC0199A3DB54EE879E5A66D94D1649E44C89A529CF1C727B 2C27AB39BA59B071C4B21A3D9750D89FD687845B6A794537BDFA73E3A0DFC248 D903D4CD3CB968852A045B2695037BABBC6E43DFBC2AC9EA8399D486CAADBF58 D443F4E26E837A30263B4DA1909E3129E0385A19376F834DC69B46FAC2F5D143 2B724D3B6FCE497138BF35E05B9F8A194ABAFD46B3448B042CA8F8F56D15C5E4 5B977BCD6781DAF4D73B44B036DDDF842B9530F16553E26FF979BEA03C2C2959 4F3A8FD6DFEBD9B06314F8913BD579CD9EC0C3510218F1DF06B082F5D8D18972 94328FD54E84B36F510B630439E92EA930B673A514253024F8497641DD361763 4C49939E93608D35C3836A79E4D3610518803DB12F824D1F9E7A46B29F9E31F8 9DCFBA0EE8B79361C7E62E968A9210070706334FD4A5DE49D9AF35BC89EB821D 86440199CA9FFBC18EF832A5429B914657E18EEDF6FC9E2B48DE85E66ABBFD8E B2A8E08496A5EE0C38ECA4E3A2E1691D2D2BB8001B03BD62ABF235484F21BC33 6FF2B4A5E5C158EA71F2249ACB393CD5952EECF14C90DD7BCC795046BFC76119 FEF79C8B6A489E85D5114EFEB21F80E29D95D0FC9FDFE6DFCF82BF37FEDEF454 ABDED861C750681A513CA57F138C749A92AE16CD83C7C2F0BC22F04788F80D29 C3A757108013A686486AA24D5BC611F779554ABCB189A5479B290C27D06E41EA 53F0D915786BFB4ABFA5FF00C640461455F6BD5764F4DD090BAFC4014A35CEFE 769EB89CC141FE39B9BBC61A1D9D80FE4CC8F2E9A1D57C78BCA726C068D7AC3F 7247202873E0A3824B130719ABC76AE82BC1A9664168B14074F8F12514293696 C85AC54FE1531351F98FA337E13B776CFC8A650FAD2BB4413AC9F314D6EE39A1 7F6ED129D8841B84884DE5969EF5D67AACEBAF0F0243FFB338DD22085577F041 712B797E57BA0A5CB6CE4F63F4851F5F978E2C807AEEDB7487EA097C87CAB85C ED0D8425884D29940B044FC24D4E00F0E9196938BEB2F5705DFA76C0B37E2942 BC4B077B3F25C6F797B28F6BC2B54F3F3236E91F0D064C925E5ABB45CAC24ECB 66767A6FAFB9E58ADA179837B915D4AD02A98C3BD6ED597CD6F129DC1BADFDCC 63F567DFCC85D3F28087AE9C5576E2DC4423F3843EAFA3018123A872333FE2C2 F5908325CC7761AEE945A5E5F0CBC4B93D1520D8FADCEBD5702DDFB69C66F436 2C39A1F3F578F4C1DEABB9822119AFA714076B49225B7895045112869E4A3B7A 7C208D1BA62392CA71FB64FC8FA92BCEBFBFB347E4462D17B3981706FE34B2C6 F2A3F595F8B15F3FD5BA1554E005EEE893F4C12FDBE32064717356FFD6E800AE 0E0895B5B087E73A0B683CCA14D0DEDC99FE33D78EB378FCDE11945BCB776A46 F12507CA7C08148EDA79F418A0CC6C9BE289DFF05CA6594A2DE2DFBAB41320F2 659891DA2EC68DF45156FF6796DD4C4E2E76148EA8E94A7B70625ED9FA3F4A06 76E29184D2428156FFE6137E04F4AD4D5E9E6781584FA7F612BFE3A8ED7627C2 FD274D1E276E45640BE38EE373EB00AB3C1C3BEA2C0B73AA2994E4BC7877ADFC D46E625796B4251EC308513298875BF4D7016D186AB32BA4019DBA93FD697C8D 7057102CBA34872C2008673BF305228DFFEA38C76983B807943BC0F88DE9B089 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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. 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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 11 /alpha put dup 105 /i put dup 106 /j put dup 109 /m put dup 117 /u 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 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 readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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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 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 103 /g put dup 108 /l put dup 110 /n put dup 111 /o put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % 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 (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 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 12 /fi put dup 45 /hyphen put dup 46 /period 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 65 /A put dup 67 /C put dup 73 /I put dup 75 /K put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R 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 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 12 /fi put dup 34 /quotedblright put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period 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 67 /C put dup 73 /I put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k 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 116 /t put dup 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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 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 54 /six put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (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 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 C5FD550F53CE5583743B50B0F9B7C4F836DEF0E4DAFB5591D919AA50F0078928 03C748D6E926C1F493B1C7F9573016B615ECF2B62C88534FCAD46221D7A20BC5 82BE6756538242A42EE1D9B0622CADABCA834E12346D0E0751A8A67B8A222235 062F00F8D0E42E76FABD297EBC03ED23083C7C680FF8E605B22A12B050DAD8F4 77F9B471556C55759FE4D1BF081888B416E071EDB49FF36DAF8592DCD7120AD2 D9861089B15CB519B7EE8EBB58B5B670DE3A997701B6BC011AB05B7FEC7D01C1 F557E95EA75A08330A512098E59395661D165BF505D3F03AA9A2BCFD9A758598 727B344A1416F61755D1D13F1B8B25588EB2F9C169BC6B8A97D08EBEDCFBF874 9E57D7E369E19AC21AC066470DD46A69717B52B9F1B33D4C503ABE454D8DE05B E1DD66C001D1BE5DEAAC5D9983F337EEE1944B5C29D32426990E1317570AF4C5 443C796994C15D23064218F5D7AC8C8C11F4DC6A5A44B0F582147CDD3A90D04C 41EE00A67F03B0ED4A032A5E201196A047E3E12619804009F599CAE0AFBAC89E 9A3F77320A6BBF099C6BE0F8BB87A423BB0D71389EA9B8CEF1EA1B35A8CEDA4C DA2A0DDC565D2331D57EC33F36A5A152BF93072252A6F89F1B99D93BA67E368F 1B411BCC8A7253902B44BBB6ED9A370921D8DD88F71D16AAC021A6F6F861EC1C 3D11A94E3AAC086C33CE769B163601F66134227D2702DB06897FAAF6FDB4AB74 DDC7723F08BF8E8C523E2294302A3155C27C4ADAC04EBC70AF6D6407CD8968CB 01C73C14CE4B5A6698F165877E6C00E1D409E6A4A5343F3DEB503966463AE803 8B5269A48D0EB004E0C9875A0DEE5AFB43627DB05039986807028C85AAF83B25 625CD6C490 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % 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 (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 1 /periodcentered put dup 25 /approxequal put dup 26 /propersubset put dup 33 /arrowright put dup 36 /arrowboth put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 59 /emptyset put dup 83 /S put dup 100 /ceilingleft put dup 101 /ceilingright put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 12 /fi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S 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 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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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 30 /phi put dup 33 /omega put dup 65 /A put dup 66 /B put dup 68 /D put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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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 11 /alpha put dup 14 /delta put dup 25 /pi put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less 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 71 /G put dup 72 /H put dup 73 /I put dup 75 /K 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 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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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 3 /Lambda put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash 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 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A 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 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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%%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 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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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. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 54 /six put dup 56 /eight put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 83 /S put dup 85 /U put dup 86 /V put dup 97 /a put dup 99 /c put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m 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 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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b(this)i(case)h(ev)m(ery)f(string)g(is)f(its)h(o)m(wn)g(description)448 1702 y(and)c Fv(C)690 1716 y Fu(D)754 1702 y Fx(\()p Fv(x)p Fx(\))c(=)f Fs(j)p Fv(x)p Fs(j)p Fx(.)589 1815 y(F)-8 b(or)29 b(an)m(y)g(giv)m(en)g(string)e Fv(x)h Fx(w)m(e)h(can)f(\014nd)f(a)h(description)g(mo)s(de)f Fv(D)k Fx(that)e(is)f(tailored)448 1928 y(to)23 b Fv(x)e Fx(and)g(with)h(resp)s(ect)g(to)g(whic)m(h)f Fv(x)h Fx(has)f(small)i (complexit)m(y)-8 b(.)40 b(Indeed,)22 b(let)h Fv(D)s Fx(\(\003\))j(=)f Fv(x)p Fx(.)448 2041 y(This)36 b(implies)h Fv(C)1042 2055 y Fu(D)1106 2041 y Fx(\()p Fv(x)p Fx(\))f(=)f(0.)60 b(It)36 b(ma)m(y)h(seem)g(that)g(the)g(dep)s(endence)f(of)g(complexit)m (y)448 2154 y(on)f(the)h(c)m(hoice)h(of)e(the)h(decompressor)f(mak)m (es)h(imp)s(ossible)f(an)m(y)g(general)h(theory)g(of)448 2267 y(complexit)m(y)-8 b(.)43 b(Ho)m(w)m(ev)m(er,)33 b(it)e(is)f(not)h(the)f(case.)448 2505 y Fm(Optimal)k(description)i(mo) s(des)448 2676 y Fx(A)h(description)f(mo)s(de)g(is)g(b)s(etter)g(when)g 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Fv(c)e Fx(can)h(b)s(e)f(v)m(ery)h(large.)39 b(Ho)m(w)m(ev)m(er,)25 b(nob)s(o)s(dy)19 b(managed)i(to)g(get)448 3851 y(an)m(y)30 b(reasonable)h(theory)f(that)g(o)m(v)m(ercomes)i(this)d(di\016cult)m(y) h(and)g(de\014nes)e(complexit)m(y)448 3963 y(with)i(b)s(etter)h (precision.)589 4076 y(The)26 b(follo)m(wing)h(theorem,)h(whic)m(h)d (states)i(that)g(there)f(exists)h(an)e(optimal)i(decom-)448 4189 y(pressor,)j(is)h(a)f(basis)h(of)f(the)h(theory)f(of)h(Kolmogoro)m (v)h(complexit)m(y)-8 b(.)448 4367 y Fm(Theorem)38 b(1)f Fx(\(Solomono\013)d([11)q(])f(and)f(Kolmogoro)m(v)i([5)q(]\))p Fm(.)44 b Ft(Ther)-5 b(e)35 b(is)g(a)g(description)448 4480 y(mo)-5 b(de)38 b Fv(D)i Ft(that)e(is)e(not)i(worse)f(than)h(any)g (other)f(one:)51 b(for)37 b(every)f(description)j(mo)-5 b(de)448 4592 y Fv(D)526 4559 y Fy(0)582 4592 y Ft(ther)g(e)33 b(is)g(a)g(c)-5 b(onstant)34 b Fv(c)f Ft(such)g(that)1544 4784 y Fv(C)1609 4798 y Fu(D)1673 4784 y Fx(\()p Fv(x)p Fx(\))26 b Fj(6)f Fv(C)1982 4800 y Fu(D)2042 4781 y Fq(0)2068 4784 y Fx(\()p Fv(x)p Fx(\))c(+)f Fv(c)448 4976 y Ft(for)34 b(every)e(string)h Fv(x)p Ft(.)1920 5225 y Fx(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 589 573 a Fx(A)31 b(description)f(mo)s(de)g Fv(D)j Fx(ha)m(ving)e(this)g(prop)s(ert)m(y)e(is)i(called)g Ft(optimal)p Fx(.)448 760 y Ft(Pr)-5 b(o)g(of.)47 b Fx(Recall)23 b(that)f(a)f(description)h(mo)s(de)e(b)m(y)i(de\014nition)e(is)i(a)f (computable)h(function.)448 873 y(Ev)m(ery)35 b(computable)h(function)f (has)f(a)i(program.)54 b(W)-8 b(e)36 b(assume)f(that)g(programs)g(are) 448 986 y(binary)j(strings.)64 b(Moreo)m(v)m(er,)43 b(w)m(e)c(assume)f (that)h(reading)g(the)f(program)g(bits)h(from)448 1099 y(left)30 b(to)g(righ)m(t)f(w)m(e)h(can)f(determine)g(uniquely)g(where) f(it)i(ends,)f(that)g(is,)h(programs)f(are)448 1212 y (\\self-delimiting".)57 b(Note)36 b(that)g(ev)m(ery)g(programming)e (language)j(can)e(b)s(e)f(mo)s(di\014ed)448 1325 y(in)25 b(suc)m(h)f(a)i(w)m(a)m(y)g(that)f(programs)g(are)g(self-delimiting.)40 b(F)-8 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b(as)f(input)g(to)h(that) g(program.)40 b(The)29 b(program)g(is)g(loaded)h(in)m(to)g(the)g(mem-) 448 4588 y(ory)h(and)e(then)i(it)f(decompresses)h(the)f(rest)h(of)f (the)h(\014le.)589 4701 y(Note)h(that)e(in)g(our)f(construction)i (optimal)g(decompressor)e(w)m(orks)h(v)m(ery)h(long)f(on)448 4814 y(some)h(inputs)f(\(some)h(programs)g(ha)m(v)m(e)g(large)h (running)d(time\),)j(and)e(is)h(unde\014ned)d(on)448 4927 y(some)j(inputs.)1920 5225 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 448 573 a Fm(Kolmogoro)m(v)37 b(complexit)m(y)448 744 y Fx(Fix)31 b(an)g(optimal)g(description)g(mo)s(de)f Fv(D)j Fx(and)d(call)i Fv(C)2299 758 y Fu(D)2363 744 y Fx(\()p Fv(x)p Fx(\))f(the)g(\(plain\))g Ft(Kolmo)-5 b(gor)g(ov)448 857 y(c)g(omplexity)41 b Fx(of)32 b(the)f(string)h Fv(x)p Fx(.)43 b(In)31 b(the)g(notation)i Fv(C)2245 871 y Fu(D)2309 857 y Fx(\()p Fv(x)p Fx(\))f(w)m(e)f(drop)g(the)g (subscript)f Fv(D)448 970 y Fx(and)g(write)h(just)e Fv(C)7 b Fx(\()p Fv(x)p Fx(\).)589 1083 y(If)37 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b(ha)m(ving)g(in)f(mind)g(a)g(sp)s(eci\014c)h (optimal)g(description)g(mo)s(de)f(used)g(in)g(the)g(de\014-)448 2056 y(nition)i(of)g Fv(C)7 b Fx(\()p Fv(x)p Fx(\)?)39 b(No,)29 b(since)e(b)m(y)f(adjusting)h(the)g(optimal)g(description)g (mo)s(de)g(w)m(e)g(can)448 2169 y(mak)m(e)e(the)f(complexit)m(y)i(of)e Fv(x)g Fx(arbitrarily)g(small)g(or)g(arbitrarily)g(large.)40 b(Similarly)-8 b(,)26 b(the)448 2282 y(relation)32 b(\\string)f Fv(x)f Fx(is)h(simpler)f(than)g Fv(y)s Fx(",)h(that)g(is,)g Fv(C)7 b Fx(\()p Fv(x)p Fx(\))25 b Fv(<)h(C)7 b Fx(\()p Fv(y)s Fx(\),)30 b(has)g(no)h(meaning)448 2395 y(for)d(t)m(w)m(o)g (\014xed)f(strings)h Fv(x)f Fx(and)g Fv(y)s Fx(:)39 b(b)m(y)28 b(adjusting)f(the)g(optimal)i(description)e(mo)s(de)h(w)m(e)448 2508 y(can)j(mak)m(e)g(an)m(y)g(of)g(these)f(t)m(w)m(o)i(strings)e (simpler)g(than)g(the)h(other)g(one.)589 2621 y(One)i(ma)m(y)g(w)m (onder)f(whether)g(Kolmogoro)m(v)i(complexit)m(y)h(has)d(an)m(y)h (sense)g(at)g(all.)448 2734 y(Let)i(us)e(recall)j(the)e(construction)h (of)f(the)h(optimal)g(description)f(mo)s(de)g(used)f(in)h(the)448 2847 y(pro)s(of)29 b(of)g(the)g(Solomono\013{Kolmogoro)m(v)j(theorem.) 41 b(This)28 b(construction)i(uses)f(some)448 2960 y(programming)38 b(language,)k(and)37 b(t)m(w)m(o)i(di\013eren)m(t)g(c)m(hoices)g(of)g (this)e(language)j(lead)e(to)448 3072 y(t)m(w)m(o)33 b(complexities)f(that)g(di\013er)e(at)i(most)f(b)m(y)g(a)h(constan)m (t.)43 b(This)30 b(constan)m(t)j(is)e(in)f(fact)448 3185 y(the)g(length)f(of)g(the)g(program)g(that)h(is)f(written)g(in)g(one)g (of)g(these)h(t)m(w)m(o)g(languages)h(and)448 3298 y(in)m(terprets)40 b(the)f(other)g(one.)67 b(If)39 b(b)s(oth)f(languages)j(are)e (\\natural",)k(w)m(e)c(can)h(exp)s(ect)448 3411 y(this)g(constan)m(t)h (to)g(b)s(e)e(not)h(that)g(h)m(uge,)j(just)c(sev)m(eral)j(thousands)c (or)i(ev)m(en)h(sev)m(eral)448 3524 y(h)m(undreds.)71 b(Therefore)41 b(if)h(w)m(e)f(sp)s(eak)g(ab)s(out)g(strings)g(whose)g (complexit)m(y)i(is,)i(sa)m(y)-8 b(,)448 3637 y(ab)s(out)28 b(10)796 3604 y Fl(5)863 3637 y Fx(\(i.e.,)i(a)e(text)h(of)e(a)h(no)m (v)m(el\),)i(or)e(10)1993 3604 y Fl(6)2061 3637 y Fx(\(DNA)g (sequence\))h(then)e(the)h(c)m(hoice)h(of)448 3750 y(the)i(programming) f(language)i(is)e(not)h(that)g(imp)s(ortan)m(t.)589 3863 y(Nev)m(ertheless)50 b(one)f(should)e(alw)m(a)m(ys)i(remem)m(b)s(er)f (that)h(all)g(statemen)m(ts)g(ab)s(out)448 3976 y(Kolmogoro)m(v)33 b(complexit)m(y)g(are)f(inheren)m(tly)f(asymptotic:)44 b(they)31 b(in)m(v)m(olv)m(e)j(in\014nite)d(se-)448 4089 y(quences)c(of)g(strings.)39 b(This)26 b(situation)h(is)g(t)m(ypical)h (also)f(for)g(computational)h(complex-)448 4202 y(it)m(y:)44 b(usually)31 b(upp)s(er)f(and)h(lo)m(w)m(er)h(b)s(ounds)e(for)h (complexit)m(y)i(of)f(some)f(computational)448 4314 y(problem)f(are)h (asymptotic)h(b)s(ounds.)448 4558 y Fo(The)38 b(n)m(um)m(b)s(er)g(of)g (strings)g(of)f(small)i(complexit)m(y)448 4729 y Fx(Let)21 b Fv(n)f Fx(b)s(e)g(an)g(in)m(teger.)39 b(Then)19 b(there)i(are)g(less) f(than)h(2)2236 4696 y Fu(n)2303 4729 y Fx(strings)g Fv(x)f Fx(suc)m(h)g(that)h Fv(C)7 b Fx(\()p Fv(x)p Fx(\))25 b Fv(<)g(n)p Fx(.)448 4842 y(Indeed,)46 b(let)e Fv(D)i Fx(b)s(e)d(the)g(optimal)h(description)g(mo)s(de)e(used)h(in)g(the)g (de\014nition)g(of)448 4955 y(Kolmogoro)m(v)33 b(complexit)m(y)-8 b(.)45 b(Then)30 b(only)i(strings)e Fv(D)s Fx(\()p Fv(y)s Fx(\))i(for)f(all)h Fv(y)h Fx(suc)m(h)e(that)h Fs(j)p Fv(y)s Fs(j)26 b Fv(<)h(n)1920 5225 y Fx(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 448 573 a Fx(ha)m(v)m(e)37 b(complexit)m(y)g (less)f(than)f Fv(n)p Fx(.)55 b(The)35 b(n)m(um)m(b)s(er)f(of)h(suc)m (h)g(strings)g(do)s(es)g(not)h(exceed)448 686 y(the)31 b(n)m(um)m(b)s(er)e(of)h(strings)h Fv(y)i Fx(suc)m(h)d(that)h Fs(j)p Fv(y)s Fs(j)25 b Fv(<)g(n)p Fx(,)31 b(i.e.,)g(the)g(sum)1245 870 y(1)20 b(+)g(2)h(+)f(4)g(+)g(8)h(+)f Fv(:)15 b(:)g(:)21 b Fx(+)f(2)2133 833 y Fu(n)p Fy(\000)p Fl(1)2296 870 y Fx(=)25 b(2)2437 833 y Fu(n)2504 870 y Fs(\000)20 b Fx(1)448 1055 y(\(there)31 b(are)g(2)913 1022 y Fu(k)986 1055 y Fx(strings)g(for)f(eac)m(h)h(length)g Fv(k)e(<)24 b(n)p Fx(\).)448 1292 y Fm(Complexit)m(y)35 b(and)g(information)448 1464 y Fx(One)e(can)g(consider)g(the)h(Kolmogoro)m(v)h(complexit)m(y)f (of)g Fv(x)f Fx(as)g(the)g Ft(amount)k(of)e(infor-)448 1576 y(mation)43 b Fx(in)34 b Fv(x)h Fx(\(w)m(e)g(do)g(not)f(try)h(to)g (distinguish)f(b)s(et)m(w)m(een)h(useful)f(and)f(meaningless)448 1689 y(information,)25 b(a)e(string)f(has)g(m)m(uc)m(h)g(information)h (unless)f(it)h(has)f(a)g(short)g(description\).)589 1802 y(If)31 b(the)g(complexit)m(y)h(of)f(a)g(string)g Fv(x)g Fx(is)f(equal)i(to)f Fv(k)s Fx(,)g(w)m(e)g(sa)m(y)h(that)f Fv(x)g Fx(has)f Fv(k)k Fx(bits)d(of)448 1915 y(information.)39 b(One)23 b(can)g(exp)s(ect)h(that)g(the)f(amoun)m(t)h(of)f(information) h(in)f(a)g(string)g(do)s(es)448 2028 y(not)36 b(exceed)h(its)e(length,) j(that)e(is,)h Fv(C)7 b Fx(\()p Fv(x)p Fx(\))34 b Fj(6)g Fs(j)p Fv(x)p Fs(j)p Fx(.)56 b(This)35 b(is)g(true)h(\(up)e(to)j(an)e (additiv)m(e)448 2141 y(constan)m(t,)49 b(as)c(w)m(e)f(ha)m(v)m(e)i (already)e(said\):)69 b Fv(C)7 b Fx(\()p Fv(x)p Fx(\))48 b Fj(6)g Fs(j)p Fv(x)p Fs(j)30 b Fx(+)f Fv(O)s Fx(\(1\))p Fv(:)45 b Fx(This)f(inequalit)m(y)448 2254 y(implies,)32 b(in)g(particular,)g(that)g(Kolmogoro)m(v)h(complexit)m(y)g(is)f(alw)m (a)m(ys)h(\014nite,)f(that)g(is,)448 2367 y(ev)m(ery)f(string)g(has)f (a)h(description.)589 2480 y(Here)22 b(is)g(another)f(prop)s(ert)m(y)g (of)h(\\amoun)m(t)g(of)g(information")g(that)g(one)g(can)f(exp)s(ect:) 448 2593 y(the)32 b(amoun)m(t)h(of)f(information)g(do)s(es)g(not)g (increase)g(when)f(algorithmic)j(transforma-)448 2706 y(tion)h(is)f(p)s(erformed.)49 b(\(More)35 b(precisely)-8 b(,)36 b(the)e(increase)h(is)f(b)s(ounded)d(b)m(y)j(an)g(additiv)m(e) 448 2818 y(constan)m(t)e(dep)s(ending)d(on)h(the)h(transformation)f (algorithm.\))448 2990 y Fm(Theorem)35 b(2.)42 b Ft(F)-7 b(or)34 b(every)e(algorithm)j Fv(A)e Ft(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fv(c)d Ft(such)h(that)1543 3174 y Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x)p Fx(\)\))27 b Fj(6)e Fv(C)7 b Fx(\()p Fv(x)p Fx(\))20 b(+)g Fv(c)448 3359 y Ft(for)34 b(al)5 b(l)33 b Fv(x)f Ft(such)h(that)h Fv(A)p Fx(\()p Fv(x)p Fx(\))f Ft(is)g(de\014ne)-5 b(d.)448 3530 y(Pr)g(o)g(of.)47 b Fx(Let)35 b Fv(D)j Fx(b)s(e)c(an)g(optimal)i (decompressor)e(that)i(is)e(used)g(in)h(the)g(de\014nition)f(of)448 3643 y(the)d(Kolmogoro)m(v)h(complexit)m(y)-8 b(.)43 b(Consider)30 b(another)g(decompressor)g Fv(D)2958 3610 y Fy(0)2981 3643 y Fx(:)1594 3828 y Fv(D)1672 3790 y Fy(0)1695 3828 y Fx(\()p Fv(p)p Fx(\))c(=)f Fv(A)p Fx(\()p Fv(D)s Fx(\()p Fv(p)p Fx(\)\))p Fv(:)448 4012 y Fx(\(W)-8 b(e)26 b(apply)d(\014rst)g Fv(D)j Fx(and)d(then)h Fv(A)p Fx(.\))39 b(If)23 b Fv(p)g Fx(is)h(a)g(description)g(of)f(a)h(string)g Fv(x)g Fx(with)f(resp)s(ect)448 4125 y(to)28 b Fv(D)i Fx(and)c Fv(A)p Fx(\()p Fv(x)p Fx(\))i(is)f(de\014ned,)f(then)h Fv(p)g Fx(is)f(a)i(description)f(of)g Fv(A)p Fx(\()p Fv(x)p Fx(\))h(with)e(resp)s(ect)h(to)h Fv(D)3389 4092 y Fy(0)3412 4125 y Fx(.)448 4238 y(Let)j Fv(p)f Fx(b)s(e)g(a)h (shortest)f(description)h(of)f Fv(x)h Fx(with)f(resp)s(ect)g(to)h Fv(D)s Fx(.)41 b(Then)29 b(w)m(e)i(ha)m(v)m(e)1271 4423 y Fv(C)1336 4438 y Fu(D)1396 4419 y Fq(0)1422 4423 y Fx(\()p Fv(A)p Fx(\()p Fv(x)p Fx(\)\))c Fj(6)e Fs(j)p Fv(p)p Fs(j)g Fx(=)g Fv(C)2087 4437 y Fu(D)2151 4423 y Fx(\()p Fv(x)p Fx(\))h(=)f Fv(C)7 b Fx(\()p Fv(x)p Fx(\))p Fv(:)448 4607 y Fx(By)31 b(optimalit)m(y)h(w)m(e)f(obtain)1201 4792 y Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x)p Fx(\)\))26 b Fj(6)f Fv(C)1720 4807 y Fu(D)1780 4788 y Fq(0)1807 4792 y Fx(\()p Fv(A)p Fx(\()p Fv(x)p Fx(\)\))c(+)f Fv(c)26 b Fj(6)f Fv(C)7 b Fx(\()p Fv(x)p Fx(\))20 b(+)g Fv(c)448 4976 y Fx(for)30 b(some)h Fv(c)g Fx(and)f(all)h Fv(x)p Fx(.)p 3371 4976 4 62 v 3375 4918 55 4 v 3375 4976 V 3429 4976 4 62 v 1920 5225 a(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 589 573 a Fx(This)30 b(theorem)h(implies)f(that)h (the)g(amoun)m(t)g(of)f(information)h(\\do)s(es)f(not)h(dep)s(end)448 686 y(on)g(the)f(sp)s(eci\014c)h(enco)s(ding".)41 b(F)-8 b(or)31 b(instance,)g(if)g(w)m(e)g(rev)m(erse)g(all)g(bits)g(of)f(some) h(string)448 799 y(\(replace)26 b(0)f(b)m(y)f(1)h(and)f(vice)h(v)m (ersa\),)i(or)e(add)f(a)g(zero)i(bit)e(after)h(eac)m(h)h(bit)e(of)h (that)g(string,)448 912 y(the)h(resulting)g(string)f(has)h(the)f(same)h (Kolmogoro)m(v)i(complexit)m(y)f(as)f(the)g(original)h(one)448 1024 y(\(up)21 b(to)h(an)f(additiv)m(e)h(constan)m(t\).)40 b(Indeed,)22 b(the)f(transformation)h(itself)g(and)f(its)g(in)m(v)m (erse)448 1137 y(can)31 b(b)s(e)f(p)s(erformed)f(b)m(y)h(an)g (algorithm.)589 1250 y(So)f(w)m(e)f(can)h(de\014ne)f(Kolmogoro)m(v)i (complexit)m(y)g(for)e(an)m(y)h(\014nite)f(ob)5 b(jects,)30 b(lik)m(e)f(nat-)448 1363 y(ural)35 b(n)m(um)m(b)s(ers,)h(pairs)f(of)g (binary)g(strings,)i(\014nite)e(sets)h(of)f(binary)g(strings,)i(etc.)57 b(T)-8 b(o)448 1476 y(de\014ne)40 b(Kolmogoro)m(v)j(complexit)m(y)f (for)f(elemen)m(ts)h(in)e(a)i(class)f Fv(L)g Fx(of)f(\014nite)h(ob)5 b(jects,)448 1589 y(\014x)42 b(a)h(computable)g(bijection)g Fv(\031)48 b Fx(:)d Fv(L)g Fs(!)g(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)2195 1556 y Fy(\003)2279 1589 y Fx(and)42 b(let)h Fv(C)7 b Fx(\()p Fv(x)p Fx(\))45 b(=)g Fv(C)7 b Fx(\()p Fv(\031)s Fx(\()p Fv(x)p Fx(\)\))43 b(for)448 1702 y(all)37 b Fv(x)d Fs(2)g Fv(L)p Fx(.)57 b(Theorem)35 b(2)i(guaran)m(tees)g(that)f(for)g(an)m(y)g(other)g(computable)g (bijection)448 1815 y Fv(\034)42 b Fx(:)32 b Fv(L)g Fs(!)f(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)1023 1782 y Fy(\003)1099 1815 y Fx(w)m(e)35 b(ha)m(v)m(e)h Fv(C)7 b Fx(\()p Fv(\031)s Fx(\()p Fv(x)p Fx(\)\))32 b(=)g Fv(C)7 b Fx(\()p Fv(\034)j Fx(\()p Fv(x)p Fx(\)\))23 b(+)g Fv(O)s Fx(\(1\))35 b(for)f(all)h Fv(x)d Fs(2)f Fv(L)p Fx(.)53 b(Indeed,)448 1928 y(b)s(oth)28 b(functions)g Fv(\031)s Fx(\()p Fv(\034)1191 1895 y Fy(\000)p Fl(1)1285 1928 y Fx(\()p Fv(p)p Fx(\)\))h(and)f Fv(\034)10 b Fx(\()p Fv(\031)1780 1895 y Fy(\000)p Fl(1)1875 1928 y Fx(\()p Fv(p)p Fx(\)\))29 b(are)f(computable.)41 b(Th)m(us)27 b(Kolmogoro)m(v)448 2041 y(complexit)m(y)32 b(is)f(w)m(ell)g(de\014ned) e(\(up)h(to)h(an)f(additiv)m(e)i(constan)m(t\))g(for)e(ob)5 b(jects)31 b(in)f Fv(L)p Fx(.)589 2154 y(A)23 b(theorem,)i(similar)e (to)g(Theorem)g(2,)h(holds)e(for)h(algorithmic)h(transformation)f(of) 448 2267 y(sev)m(eral)h(ob)5 b(jects)23 b(in)m(to)g(one)f(ob)5 b(ject.)39 b(Let)23 b(us)f(state)h(it)g(for)f(algorithmic)i (transformation)448 2379 y(of)31 b(2)g(strings)f(in)m(to)h(one)g (string.)448 2532 y Fm(Theorem)50 b(3.)e Ft(F)-7 b(or)46 b(every)d(algorithm)k Fv(A)d Ft(that)h(has)g(two)g(input)g(strings)g (and)g(one)448 2645 y(output)34 b(string)f(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fv(c)e Ft(such)f(that)1099 2807 y Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\)\))27 b Fj(6)e Fv(C)7 b Fx(\()p Fv(x)p Fx(\))20 b(+)g Fv(C)7 b Fx(\()p Fv(y)s Fx(\))20 b(+)g(2)15 b(log)j Fv(C)7 b Fx(\()p Fv(x)p Fx(\))21 b(+)f Fv(c)448 2968 y Ft(for)34 b(al)5 b(l)33 b Fv(x;)15 b(y)35 b Ft(such)e(that)h Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\))34 b Ft(is)e(de\014ne)-5 b(d.)448 3121 y(Pr)g(o)g(of.)47 b Fx(W)-8 b(e)31 b(can)g(\014nd)d Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\))31 b(giv)m(en)h(a)e(description)g (of)g Fv(A)h Fx(\(in)f(constan)m(t)h(n)m(um)m(b)s(er)e(of)448 3234 y(bits\))g(and)f(minimal)g(descriptions)h(of)f Fv(x;)15 b(y)32 b Fx(\(in)c Fv(C)7 b Fx(\()p Fv(x)p Fx(\))29 b(and)f Fv(C)7 b Fx(\()p Fv(y)s Fx(\))28 b(bits,)h(resp)s(ectiv)m(ely\).)448 3347 y(This)43 b(informal)g(observ)-5 b(ation)44 b(yields)g(a)g (stronger)f(b)s(ound)f Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\)\))48 b Fj(6)e Fv(C)7 b Fx(\()p Fv(x)p Fx(\))29 b(+)448 3460 y Fv(C)7 b Fx(\()p Fv(y)s Fx(\))21 b(+)e Fv(O)s Fx(\(1\),)32 b(without)e(the)h(term)f(2)15 b(log)k Fv(C)7 b Fx(\()p Fv(x)p Fx(\),)30 b(whic)m(h)g(is)h(actually)h (wrong.)589 3572 y(What)k(is)g(wrong)f(with)g(this)g(argumen)m(t?)56 b(Let)35 b Fv(D)k Fx(b)s(e)34 b(the)i(optimal)g(description)448 3685 y(mo)s(de)h(that)g(is)g(used)f(in)h(the)g(de\014nition)f(of)h (Kolmogoro)m(v)i(complexit)m(y)-8 b(.)63 b(W)-8 b(e)38 b(try)e(to)448 3798 y(de\014ne)i(the)g(follo)m(wing)i(description)e(mo) s(de)g Fv(D)2079 3765 y Fy(0)2102 3798 y Fx(.)65 b(If)38 b Fv(D)s Fx(\()p Fv(p)p Fx(\))g(=)h Fv(x)f Fx(and)g Fv(D)s Fx(\()p Fv(q)s Fx(\))g(=)h Fv(y)i Fx(w)m(e)448 3911 y(consider)d Fv(pq)j Fx(as)e(a)f(description)h(of)f Fv(xy)s Fx(,)i(that)f(is,)i(w)m (e)d(let)h Fv(D)2581 3878 y Fy(0)2604 3911 y Fx(\()p Fv(pq)s Fx(\))g(=)f Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\).)66 b(The)448 4024 y(problem)38 b(is)f(that)i Fv(D)1191 3991 y Fy(0)1252 4024 y Fx(is)f(not)g(w)m(ell)g(de\014ned,)h(as)f Fv(D)2256 3991 y Fy(0)2317 4024 y Fx(has)g(no)f(means)h(to)h(separate)f Fv(p)448 4137 y Fx(from)30 b Fv(q)s Fx(.)589 4250 y(T)-8 b(o)23 b(\014x)f(this)g(bug)f(let)i(us)f(prep)s(end)e(the)i(string)h Fv(pq)h Fx(b)m(y)e(the)h(length)f Fs(j)p Fv(p)p Fs(j)h Fx(of)f(string)g Fv(p)g Fx(\(in)448 4363 y(binary)31 b(notation\).)47 b(This)30 b(allo)m(ws)j(us)e(to)i(separate)g Fv(p)e Fx(and)g Fv(q)s Fx(.)44 b(Ho)m(w)m(ev)m(er,)35 b(w)m(e)d(need)f(to)448 4476 y(\014nd)26 b(where)h Fs(j)p Fv(p)p Fs(j)h Fx(ends,)f(so)h(let)g(us)f(double)g(all)h(the)g(bits)f (in)h(the)f(binary)g(represen)m(tation)448 4589 y(of)k Fs(j)p Fv(p)p Fs(j)g Fx(and)f(then)g(put)g(01)i(as)f(separator.)42 b(More)32 b(sp)s(eci\014cally)-8 b(,)32 b(let)f Ft(bin)p Fx(\()p Fv(k)s Fx(\))h(denote)f(the)448 4702 y(binary)j(represen)m (tation)h(of)f(in)m(teger)i Fv(k)h Fx(and)c(let)p 2147 4651 52 4 v 35 w Fv(x)h Fx(b)s(e)g(the)g(result)g(of)g(doubling)g(eac)m (h)448 4814 y(bit)d(in)f Fv(x)p Fx(.)40 b(\(F)-8 b(or)32 b(example,)f Ft(bin)p Fx(\(5\))26 b(=)f(101,)32 b(and)p 2120 4735 237 4 v 30 w Ft(bin)p Fx(\(5\))26 b(=)f(110011.\))44 b(Let)1251 4976 y Fv(D)1329 4939 y Fy(0)1352 4976 y Fx(\()p 1402 4897 288 4 v 15 w Ft(bin)q Fx(\()p Fs(j)p Fv(p)p Fs(j)p Fx(\))16 b(01)p Fv(pq)s Fx(\))26 b(=)f Fv(A)p Fx(\()p Fv(D)s Fx(\()p Fv(p)p Fx(\))p Fv(;)15 b(D)s Fx(\()p Fv(q)s Fx(\)\))p Fv(:)1920 5225 y Fx(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 448 573 a Fx(Th)m(us)30 b Fv(D)757 540 y Fy(0)811 573 y Fx(is)h(w)m(ell)h(de\014ned:)40 b(the)31 b(algorithm)h Fv(D)2087 540 y Fy(0)2141 573 y Fx(scans)p 2379 494 288 4 v 30 w Ft(bin)p Fx(\()p Fs(j)p Fv(p)p Fs(j)p Fx(\))g(while)f(all)g(the)g(digits)448 686 y(are)42 b(doubled.)71 b(Once)41 b(it)h(sees)f(01,)k(it)c (determines)g Fs(j)p Fv(p)p Fs(j)g Fx(and)g(then)f(scans)h Fs(j)p Fv(p)p Fs(j)g Fx(digits)448 799 y(to)35 b(\014nd)e Fv(p)p Fx(.)52 b(The)34 b(rest)h(of)f(the)h(input)e(is)i Fv(q)i Fx(and)c(the)i(algorithm)g(is)g(able)g(to)g(compute)448 912 y Fv(A)p Fx(\()p Fv(D)s Fx(\()p Fv(p)p Fx(\))p Fv(;)15 b(D)s Fx(\()p Fv(q)s Fx(\)\).)589 1024 y(No)m(w)28 b(w)m(e)g(see)g (that)g Fv(C)1327 1040 y Fu(D)1387 1021 y Fq(0)1413 1024 y Fx(\()p Fv(xy)s Fx(\))g(is)f(at)h(most)f(2)p Fs(j)p Ft(bin)q Fx(\()p Fs(j)p Fv(p)p Fs(j)p Fx(\))p Fs(j)14 b Fx(+)g(2)g(+)g Fs(j)p Fv(p)p Fs(j)g Fx(+)g Fs(j)p Fv(q)s Fs(j)p Fx(.)40 b(The)27 b(length)448 1137 y(of)k(the)g(binary)f (represen)m(tation)h(of)g Fs(j)p Fv(p)p Fs(j)g Fx(is)g(at)g(most)g(log) 2363 1159 y Fl(2)2418 1137 y Fs(j)p Fv(p)p Fs(j)20 b Fx(+)h(1.)41 b(Therefore,)31 b Fv(xy)i Fx(has)448 1250 y(a)i(description)e(of)h(length)h(at)f(most)g(2)15 b(log)1903 1272 y Fl(2)1958 1250 y Fs(j)p Fv(p)p Fs(j)23 b Fx(+)f(4)h(+)f Fs(j)p Fv(p)p Fs(j)h Fx(+)f Fs(j)p Fv(q)s Fs(j)34 b Fx(with)f(resp)s (ect)h(to)h Fv(D)3389 1217 y Fy(0)3412 1250 y Fx(.)448 1363 y(By)c(optimalit)m(y)-8 b(,)448 1534 y Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\)\))27 b Fj(6)e Fv(C)1056 1549 y Fu(D)1116 1530 y Fq(0)1142 1534 y Fx(\()p Fv(A)p Fx(\()p Fv(x;)15 b(y)s Fx(\)\))t(+)t Fv(O)s Fx(\(1\))28 b Fj(6)d Fv(C)1945 1548 y Fu(D)2009 1534 y Fx(\()p Fv(x)p Fx(\))t(+)t Fv(C)2275 1548 y Fu(D)2340 1534 y Fx(\()p Fv(y)s Fx(\))t(+)t(2)15 b(log)k Fv(C)2796 1548 y Fu(D)2860 1534 y Fx(\()p Fv(x)p Fx(\))t(+)t Fv(O)s Fx(\(1\))p Fv(:)p 3371 1534 4 62 v 3375 1476 55 4 v 3375 1534 V 3429 1534 4 62 v 589 1704 a Fx(F)-8 b(or)25 b(algorithms)g(receiving)g Fv(k)i Fx(input)c(strings,)j(w)m(e)e(obtain)h(the)f(follo)m(wing)h (inequal-)448 1817 y(it)m(y)448 1988 y Fv(C)7 b Fx(\()p Fv(A)p Fx(\()p Fv(x;)15 b(:)g(:)g(:)i(;)e(x)964 2003 y Fu(k)1007 1988 y Fx(\)\))26 b Fj(6)f Fv(C)7 b Fx(\()p Fv(x)1358 2002 y Fl(1)1397 1988 y Fx(\))j(+)g Fs(\001)15 b(\001)g(\001)c Fx(+)f Fv(C)d Fx(\()p Fv(x)1879 2003 y Fu(k)1922 1988 y Fx(\))j(+)g(2)15 b(log)j Fv(C)7 b Fx(\()p Fv(x)2400 2002 y Fl(1)2439 1988 y Fx(\))j(+)g Fs(\001)15 b(\001)g(\001)c Fx(+)f(2)15 b(log)j Fv(C)7 b Fx(\()p Fv(x)3114 2003 y Fu(k)r Fy(\000)p Fl(1)3247 1988 y Fx(\))j(+)g Fv(c:)448 2223 y Fm(Conditional)35 b(complexit)m(y)448 2394 y Fx(When)f(transmitting)g(a)f(\014le,)i(one)f (could)f(try)h(to)g(sa)m(v)m(e)h(comm)m(unication)g(c)m(harges)f(b)m(y) 448 2507 y(compressing)i(it.)56 b(The)34 b(transmission)h(could)h(b)s (e)e(made)i(ev)m(en)g(more)f(e\013ectiv)m(e)j(if)d(an)448 2620 y(old)d(v)m(ersion)f(of)g(the)h(same)f(\014le)g(already)h(exists)g (at)g(the)f(other)g(side.)43 b(In)30 b(this)h(case)i(w)m(e)448 2733 y(need)27 b(only)h(to)g(describ)s(e)f(the)g(c)m(hanges)i(made.)40 b(This)26 b(could)i(b)s(e)e(considered)h(as)h(a)g(kind)448 2846 y(of)g(motiv)-5 b(ation)29 b(for)f(the)f(de\014nition)h(of)f (conditional)i(complexit)m(y)g(of)f(a)g(giv)m(en)h(string)e Fv(x)448 2959 y Fx(relativ)m(e)33 b(to)e(\(kno)m(wn\))g(string)f Fv(y)s Fx(.)589 3072 y(A)j Ft(c)-5 b(onditional)37 b(de)-5 b(c)g(ompr)g(essor)45 b Fx(is)33 b(an)m(y)g(computable)f(function)h Fv(D)i Fx(of)d(t)m(w)m(o)i(argu-)448 3185 y(men)m(ts)22 b(\(b)s(oth)g(argumen)m(ts)g(and)f(the)g(v)-5 b(alue)23 b(of)e Fv(D)k Fx(are)d(binary)f(strings\).)38 b(If)21 b Fv(D)s Fx(\()p Fv(y)s(;)15 b(z)t Fx(\))26 b(=)f Fv(x)448 3298 y Fx(w)m(e)h(sa)m(y)g(that)g Fv(y)i Fx(is)d(a)h(\(conditional\))h Ft(description)i(of)f Fv(x)g Ft(when)h Fv(z)j Ft(is)c(known)33 b Fx(\(or)25 b Ft(r)-5 b(elative)448 3410 y(to)36 b Fv(z)t Fx(\))d(The)g(complexit)m(y)h Fv(C)1393 3424 y Fu(D)1457 3410 y Fx(\()p Fv(x)p Fs(j)p Fv(z)t Fx(\))h(is)e(then)f(de\014ned)g(as) h(the)g(length)g(of)h(the)f(shortest)448 3523 y(conditional)f (description:)1272 3694 y Fv(C)1337 3708 y Fu(D)1401 3694 y Fx(\()p Fv(x)p Fs(j)p Fv(z)t Fx(\))26 b(=)f(min)o Fs(fj)p Fv(y)s Fs(j)h Fx(:)g Fv(D)s Fx(\()p Fv(y)s(;)15 b(z)t Fx(\))26 b(=)f Fv(x)p Fs(g)p Fv(:)448 3864 y Fx(W)-8 b(e)32 b(sa)m(y)f(that)g(\(conditional\))h(decompressor)e Fv(D)2136 3878 y Fl(1)2206 3864 y Fx(is)h Ft(not)i(worse)38 b Fx(than)31 b Fv(D)3007 3878 y Fl(2)3077 3864 y Fx(if)1451 4035 y Fv(C)1516 4049 y Fu(D)1574 4058 y Fk(1)1613 4035 y Fx(\()p Fv(x)p Fs(j)p Fv(z)t Fx(\))26 b Fj(6)f Fv(C)1993 4049 y Fu(D)2051 4058 y Fk(2)2090 4035 y Fx(\()p Fv(x)p Fs(j)p Fv(z)t Fx(\))c(+)f Fv(c)448 4206 y Fx(for)43 b(some)f(constan)m (t)i Fv(c)f Fx(and)f(for)g(all)h Fv(x)g Fx(and)e Fv(z)t Fx(.)78 b(A)42 b(conditional)i(decompressor)e(is)448 4318 y Ft(optimal)g Fx(if)30 b(it)h(is)g(not)f(w)m(orse)h(than)f(an)m (y)h(other)g(conditional)g(decompressor.)448 4478 y Fm(Theorem)k(4.)42 b Ft(Ther)-5 b(e)33 b(exist)g(optimal)i(c)-5 b(onditional)35 b(de)-5 b(c)g(ompr)g(essors.)589 4638 y Fx(This)39 b(\\conditional")i (v)m(ersion)e(of)h(Kolmogoro)m(v{Solomono\013)i(theorem)d(can)h(b)s(e) 448 4751 y(pro)m(v)m(ed)29 b(in)g(the)g(same)g(w)m(a)m(y)g(as)g(the)g (unconditional)g(one)g(\(Theorem)g(1,)h(p.)e(4\).)41 b(Again,)448 4863 y(w)m(e)35 b(\014x)f(some)h(optimal)h(conditional)g (decompressor)e Fv(D)k Fx(and)c(omit)h(index)f Fv(D)k Fx(in)c(the)448 4976 y(notation.)1920 5225 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 448 573 a Fw(3)135 b(Information)64 b(distance)g(b)t(et)l(w)l(een)g(t)l(w)l(o)h(strings)e(and)650 722 y(on-line)45 b(edge)h(coloring)g(of)f(a)g(graph)448 925 y Fm(Information)35 b(distance)g(and)g(conditional)h(complexit)m(y) 448 1097 y Fx(The)d(\\information)h(distance")h(b)s(et)m(w)m(een)f (strings)f Fv(x)g Fx(and)g Fv(y)j 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Fx(\))26 b(=)f Fv(y)s Fs(g)p Fv(:)448 2070 y Fx(Similar)43 b(to)h (Solomono\013{Kolmogoro)m(v)i(theorem,)h(w)m(e)c(can)g(sho)m(w)g(that)h (there)f(are)448 2183 y(optimal)34 b(algorithms)g Fv(A;)15 b(B)5 b Fx(,)34 b(for)e(whic)m(h)h Fv(C)1947 2197 y Fu(AB)2093 2183 y Fx(is)g(minimal)g(up)f(to)i(an)e(additiv)m(e)i(con-)448 2296 y(stan)m(t.)46 b(Fixing)33 b(optimal)g(algorithms)f Fv(A;)15 b(B)37 b Fx(w)m(e)32 b(obtain)h(the)f(de\014nition)f(of)h Ft(informa-)448 2409 y(tion)k(distanc)-5 b(e)36 b Fv(C)1057 2423 y Fu(AB)1171 2409 y Fx(\()p Fv(x)30 b Fs($)g Fv(y)s Fx(\))35 b Ft(b)-5 b(etwe)g(en)36 b Fv(x)f Ft(and)h Fv(y)s Fx(.)49 b(It)33 b(is)h(de\014ned,)f(lik)m(e)h(Kolmogoro)m(v)448 2522 y(complexit)m(y)-8 b(,)33 b(up)c(to)j(an)e(additiv)m(e)h(constan)m (t.)589 2634 y(The)i(trivial)i(lo)m(w)m(er)f(b)s(ound)e(for)h Fv(C)7 b Fx(\()p Fv(x)30 b Fs($)g Fv(y)s Fx(\))j(is)h(max)p Fs(f)p Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))p Fv(;)15 b(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))p Fs(g)35 b Fx(\(up)e(to)h(a)448 2747 y(constan)m(t)e(error)e(term\).)42 b(Indeed,)30 b(if)g Fv(A)p Fx(\()p Fv(p;)15 b(y)s Fx(\))27 b(=)e Fv(x)30 b Fx(then)h(giv)m(en)g Fv(p)f Fx(and)g Fv(y)k Fx(w)m(e)d(can)g(\014nd)448 2860 y Fv(x)j Fx(th)m(us)g Fv(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))32 b Fj(6)f Fs(j)p Fv(p)p Fs(j)23 b Fx(+)f Fv(O)s Fx(\(1\).)53 b(Similarly)-8 b(,)36 b Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))32 b Fj(6)f Fs(j)p Fv(p)p Fs(j)23 b Fx(+)g Fv(O)s Fx(\(1\))35 b(if)f Fv(B)5 b Fx(\()p Fv(p;)15 b(x)p Fx(\))32 b(=)f Fv(y)s Fx(.)448 2973 y(The)f(constan)m(ts)i(in)e Fv(O)s Fx(\(1\))h(notation)h(dep)s(end)c(on)j Fv(A;)15 b(B)5 b Fx(.)589 3086 y(It)26 b(turns)e(out)i(that)g Fv(C)7 b Fx(\()p Fv(x)25 b Fs($)g Fv(y)s Fx(\))h(is)f(close)i(to)f(this)g(lo)m (w)m(er)g(b)s(ound)e(for)h(all)h Fv(x;)15 b(y)s Fx(,)27 b(whic)m(h)448 3199 y(w)m(as)k(pro)m(v)m(ed)g(in)f([1)q(].)448 3387 y Fm(Theorem)35 b(5.)42 b Ft(F)-7 b(or)34 b(al)5 b(l)33 b Fv(x;)15 b(y)36 b Ft(we)c(have)650 3591 y Fv(C)7 b Fx(\()p Fv(x)25 b Fs($)g Fv(y)s Fx(\))g(=)g(max)p Fs(f)p Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))p Fv(;)15 b(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))p Fs(g)22 b Fx(+)e Fv(O)s Fx(\(log)e(max)p Fs(f)p Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))p Fv(;)15 b(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))p Fs(g)p Fx(\))p Fv(:)448 3831 y Fm(The)35 b(edge)g(coloring)h(game)448 4003 y Fx(The)c(pro)s(of)g(of)h(Theorem)f(5)h(is)g(based)f(on)h(the)g(follo)m (wing)h(edge)f(coloring)h(game.)49 b(Let)448 4115 y Fv(G)37 b Fx(b)s(e)f(an)g(undirected)f(graph.)58 b(An)36 b(edge)h(coloring)h (of)e Fv(G)h Fx(is)f(a)h(mapping)f(assigning)448 4228 y(an)g(in)m(teger)i(n)m(um)m(b)s(er)d(to)i(eac)m(h)h(edge)e(of)h Fv(G)p Fx(.)59 b(An)36 b(edge)h(coloring)g(is)f Ft(pr)-5 b(op)g(er)49 b Fx(if)36 b(ev)m(ery)448 4341 y(t)m(w)m(o)g(adjacen)m(t)g (edges)f(\(edges)g(sharing)f(a)h(common)g(v)m(ertex\))h(ha)m(v)m(e)g (di\013eren)m(t)f(colors.)448 4454 y(W)-8 b(e)33 b(w)m(an)m(t)e(to)h (\014nd)e(a)h(prop)s(er)f(edge)h(coloring)i(of)e(a)g(giv)m(en)h(graph)f Fv(G)g Fx(using)g(minim)m(um)448 4567 y(n)m(um)m(b)s(er)g(of)h(colors.) 45 b(Let)33 b Fv(d)p Fx(\()p Fv(G)p Fx(\))g(denote)f(the)g(degree)h(of) e Fv(G)i Fx(\(the)f(maximal)g(degree)h(of)448 4680 y(a)e(no)s(de)f(in)g Fv(G)p Fx(\).)42 b(Clearly)-8 b(,)31 b(w)m(e)g(need)f(at)h(least)h Fv(d)p Fx(\()p Fv(G)p Fx(\))g(colors,)f(as)g(all)g(edges)g(inciden)m(t) g(to)448 4793 y(an)m(y)36 b(no)s(de)e(m)m(ust)i(ha)m(v)m(e)g (di\013eren)m(t)g(colors.)56 b(By)35 b(Vizing's)i(theorem)e(\(see)i ([3,)f(16)q(]\))f(for)448 4906 y(ev)m(ery)c(graph)f Fv(G)h Fx(there)g(is)f(a)h(prop)s(er)e(edge)i(coloring)g(using)f Fv(d)p Fx(\()p Fv(G)p Fx(\))22 b(+)e(1)31 b(colors.)1897 5225 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 589 573 a Fx(W)-8 b(e)33 b(are)f(in)m (terested,)h(ho)m(w)m(ev)m(er,)h(in)d(ho)m(w)h(man)m(y)g(colors)g(w)m (e)g(need)g(if)f(the)h(graph)f(is)448 686 y(not)38 b(giv)m(en)g(to)f 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y(generated)h(b)m(y)e(Alice)i(on)e(the)h(preceding)f(mo)m(v)m(e)i (using)e(one)h(of)g(a)m(v)-5 b(ailable)41 b Fv(l)f Fx(colors.)448 1702 y(Bob)31 b(wins)f(if)g(the)h(resulting)g(coloring)g(of)g(the)g (graph)f(is)g(prop)s(er)f(\(that)j(is,)e(ev)m(ery)i(t)m(w)m(o)448 1815 y(adjacen)m(t)g(edges)f(ha)m(v)m(e)g(di\013eren)m(t)g(colors\).) 448 2055 y Fm(An)k(analysis)h(of)f(the)f(game)448 2227 y Fx(It)f(is)g(not)g(hard)e(to)j(see)f(that,)h(in)e(con)m(trast)i(to)g (Vizing's)g(theorem,)g Fv(d)22 b Fx(+)f(1)33 b(colors)h(are)448 2340 y(not)25 b(enough)f(for)g(Bob)g(to)h(win)f(\(for)g(all)h Fv(d)h(>)f Fx(2\).)39 b(The)24 b(next)h(lemma)f(sho)m(ws)g(that)h(2)p Fv(d)8 b Fs(\000)g Fx(1)448 2452 y(colors)32 b(su\016ce.)448 2640 y Fm(Lemma)g(1.)40 b Ft(If)30 b Fv(l)d Fx(=)e(2)p Fv(d)16 b Fs(\000)f Fx(1)31 b Ft(then)g(Bob)g(has)g(a)g(c)-5 b(omputable)32 b(winning)e(str)-5 b(ate)g(gy)32 b(in)f(the)448 2753 y(game.)448 2941 y(Pr)-5 b(o)g(of.)47 b Fx(Apply)27 b(the)g(greedy)g(strategy:)41 b(color)28 b(eac)m(h)g(edge)g Fv(e)g Fx(in)e(the)i(\014rst)e(color)i(that)g(is)448 3054 y(di\013eren)m(t)k(from)e(all)i(colors)g(used)e(so)h(far)g(for)f (edges)i(adjacen)m(t)g(to)g Fv(e)p Fx(.)42 b(The)31 b(edge)g Fv(e)g Fx(has)448 3166 y(2)g(ends,)f(and)g(eac)m(h)i(of)f(them)f(has)g (at)i(most)f Fv(d)20 b Fs(\000)g Fx(1)31 b(inciden)m(t)g(edges)g (di\013eren)m(t)g(from)g Fv(e)p Fx(.)448 3279 y(Th)m(us)k(there)g(are)h (at)h(most)f(2\()p Fv(d)24 b Fs(\000)g Fx(1\))36 b(edges)g(adjacen)m(t) h(to)f Fv(e)g Fx(and)f(hence)g(w)m(e)h(do)g(not)448 3392 y(need)30 b(more)h(than)f(2\()p Fv(d)21 b Fs(\000)f Fx(1\))h(+)f(1)31 b(colors.)p 3371 3392 4 62 v 3375 3334 55 4 v 3375 3392 V 3429 3392 4 62 v 448 3632 a Fm(Pro)s(of)36 b(of)f(Theorem)g(5)448 3804 y Fx(Ho)m(w)k(do)f(w)m(e)g(use)f(the)h(lemma?)64 b(Let)38 b Fv(k)j Fx(=)c(max)p Fs(f)p Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))p Fv(;)15 b(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))p Fs(g)p Fx(.)65 b(Consider)37 b(the)448 3917 y(graph)24 b Fv(G)771 3932 y Fu(k)814 3917 y Fx(,)i(where)e(no)s(des)f(are)i(binary)f(strings,)h(and)f(no)s (des)g Fv(x)2584 3884 y Fy(0)2631 3917 y Fx(and)g Fv(y)2850 3884 y Fy(0)2897 3917 y Fx(are)h(connected)448 4030 y(b)m(y)e(an)g (edge)g(if)g Fv(K)7 b Fx(\()p Fv(x)1132 3997 y Fy(0)1155 4030 y Fs(j)p Fv(y)1228 3997 y Fy(0)1252 4030 y Fx(\))25 b Fj(6)g Fv(k)h Fx(and)c Fv(K)7 b Fx(\()p Fv(y)1817 3997 y Fy(0)1841 4030 y Fs(j)p Fv(x)1918 3997 y Fy(0)1941 4030 y Fx(\))26 b Fj(6)f Fv(k)s Fx(.)38 b(Clearly)23 b(the)g(degree)h(of)f(this)g(graph)448 4143 y(is)33 b(less)g(than)g Fv(d)c Fx(=)g(2)1148 4110 y Fu(k)r Fl(+1)1281 4143 y Fx(.)48 b(Let)34 b(Alice)g(generate)g(edges)f(of)g(this)g(graph)f(\(en) m(umerating)448 4256 y(the)39 b(set)g(of)g(pairs)f Fv(x)1158 4223 y Fy(0)1181 4256 y Fv(;)15 b(y)1269 4223 y Fy(0)1331 4256 y Fx(with)38 b Fv(K)7 b Fx(\()p Fv(x)1717 4223 y Fy(0)1740 4256 y Fs(j)p Fv(y)1813 4223 y Fy(0)1837 4256 y Fx(\))39 b Fj(6)f Fv(k)s Fx(\):)58 b(once)39 b(she)f(has)g(found)f(a) i(new)f(pair)448 4369 y(suc)m(h)32 b(that)g Fv(K)7 b Fx(\()p Fv(x)1024 4336 y Fy(0)1047 4369 y Fs(j)p Fv(y)1120 4336 y Fy(0)1143 4369 y Fx(\))28 b Fj(6)f Fv(k)35 b Fx(and)c Fv(K)7 b Fx(\()p Fv(y)1731 4336 y Fy(0)1754 4369 y Fs(j)p Fv(x)1831 4336 y Fy(0)1854 4369 y Fx(\))28 b Fj(6)f Fv(k)s Fx(,)32 b(she)f(mak)m(es)i(the)f(next)f(mo)m(v)m(e)i(\()p Fv(x)3241 4336 y Fy(0)3265 4369 y Fv(;)15 b(y)3353 4336 y Fy(0)3377 4369 y Fx(\).)448 4481 y(Apply)39 b(Bob's)i(computable)f (strategy)h(against)g(Alice's)g(strategy)-8 b(.)71 b(W)-8 b(e)41 b(obtain)f(an)448 4594 y(algorithm)34 b(to)g(color)g(prop)s (erly)e(the)i(graph)e Fv(G)2062 4609 y Fu(k)2138 4594 y Fx(using)h(2)p Fv(d)23 b Fs(\000)f Fx(1)30 b Fv(<)f Fx(2)2807 4561 y Fu(k)r Fl(+2)2974 4594 y Fx(colors.)49 b(Call)448 4707 y(this)33 b(algorithm)h Fv(T)13 b Fx(,)33 b(it)g(receiv)m(es)i Fv(k)h Fx(as)d(input)e(and)i(it)g(prin)m(ts)f(on)h (its)g(output)f(tap)s(e)h(all)448 4820 y(the)e(pairs)f Fs(h)p Fx(an)g(edge)i(of)e Fv(G)1371 4835 y Fu(k)1414 4820 y Fv(;)15 b Fx(its)31 b(color)r Fs(i)p Fx(.)1897 5225 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 589 573 a Fx(W)-8 b(e)32 b(will)f(no)m(w)f (design)g(particular)h(algorithms)g Fv(A;)15 b(B)36 b Fx(suc)m(h)30 b(that)1207 777 y Fv(C)1272 791 y Fu(AB)1385 777 y Fx(\()p Fv(x)c Fs($)f Fv(y)s Fx(\))g Fj(6)g Fx(max)q Fs(f)p Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))p Fv(;)15 b(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))p Fs(g)p Fv(;)448 981 y Fx(up)30 b(to)h(a)g(logarithmic)h(error)e(term)g (\(actually)i Fv(A)f Fx(will)f(coincide)i(with)e Fv(B)5 b Fx(\).)589 1094 y(Let)765 1070 y(^)763 1094 y Fv(k)44 b Fx(is)d(a)g(self-delimiting)h(description)f(of)g Fv(k)s Fx(,)j(sa)m(y)-8 b(,)2541 1070 y(^)2538 1094 y Fv(k)46 b Fx(=)p 2745 1015 242 4 v 43 w Ft(bin)o Fx(\()p Fv(k)s Fx(\))q(01)c(where)p 3391 1044 46 4 v 40 w Fv(p)448 1207 y Fx(stands)21 b(for)h(the)g(string)f Fv(p)h Fx(with)f(all)i(bits)e (doubled,)i(and)e Ft(bin)p Fx(\()p Fv(k)s Fx(\))h(is)g(the)g(binary)f (notation)448 1320 y(of)32 b Fv(k)s Fx(.)46 b(Consider)31 b(the)h(follo)m(wing)h(algorithm)g Fv(A)p Fx(:)44 b(on)32 b(inputs)2561 1296 y(^)2559 1320 y Fv(k)s(i)g Fx(and)g Fv(y)2899 1287 y Fy(0)2954 1320 y Fx(it)g(runs)e Fv(T)13 b Fx(\()p Fv(k)s Fx(\))448 1433 y(and)36 b(outputs)g Fv(x)1023 1400 y Fy(0)1083 1433 y Fx(once)h Fv(T)13 b Fx(\()p Fv(k)s Fx(\))37 b(prin)m(ts)f(the)h(pair)f Fs(h)p Fx(\()p Fv(x)2264 1400 y Fy(0)2288 1433 y Fv(;)15 b(y)2376 1400 y Fy(0)2399 1433 y Fx(\))p Fv(;)g(i)p Fs(i)p Fx(.)61 b(As)36 b(b)s(oth)g Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(y)s Fx(\))37 b(and)448 1546 y Fv(C)7 b Fx(\()p Fv(y)s Fs(j)p Fv(x)p Fx(\))28 b(do)f(not)h(exceed)g Fv(k)s Fx(,)h(the)e(edge)h(\()p Fv(x;)15 b(y)s Fx(\))29 b(b)s(elongs)e(to)h Fv(G)2514 1561 y Fu(k)2584 1546 y Fx(and)f(th)m(us)g Fv(A)p Fx(\()3060 1522 y(^)3058 1546 y Fv(k)t(i;)15 b(x)p Fx(\))26 b(=)f Fv(y)448 1659 y Fx(and)30 b Fv(A)p Fx(\()730 1635 y(^)728 1659 y Fv(k)t(i;)15 b(y)s Fx(\))26 b(=)f Fv(x)30 b Fx(for)g(some)h Fv(i)p Fx(.)41 b(Hence)1256 1863 y Fv(C)1321 1877 y Fu(AA)1431 1863 y Fx(\()p Fv(x)26 b Fs($)f Fv(y)s Fx(\))g Fj(6)g Fs(j)1891 1839 y Fx(^)1889 1863 y Fv(k)t(i)p Fs(j)g Fx(=)g Fv(k)f Fx(+)c Fv(O)s Fx(\(log)d Fv(k)s Fx(\))p Fv(:)448 2067 y Fx(By)31 b(optimalit)m(y)-8 b(,)33 b(w)m(e)e(ha)m(v)m(e)994 2271 y Fv(C)7 b Fx(\()p Fv(x)25 b Fs($)g Fv(y)s Fx(\))g Fj(6)g Fv(C)1563 2285 y Fu(AA)1673 2271 y Fx(\()p Fv(x)h Fs($)f Fv(y)s Fx(\))20 b(+)g Fv(O)s Fx(\(1\))26 b Fj(6)f Fv(k)f Fx(+)c Fv(O)s Fx(\(log)d Fv(k)s Fx(\))448 2476 y(as)31 b(w)m(ell.)448 2762 y Fw(4)135 b(Pro)t(ducer)44 b(vs.)60 b(Consumer)45 b(game)448 2965 y Fm(The)35 b(n)m(um)m(b)s(er)g (of)g(shortest)g(descriptions)448 3137 y Fx(Ho)m(w)k(man)m(y)f(minimal) h(length)f(descriptions)g(can)h(ha)m(v)m(e)g(a)g(string)f Fv(x)g Fx(\(with)g(resp)s(ect)448 3250 y(to)j(an)f(optimal)g (description)g(mo)s(de)f Fv(D)s Fx(\)?)69 b(The)39 b(n)m(um)m(b)s(er)g (of)h(suc)m(h)f(descriptions)h(is)448 3363 y(b)s(ounded)29 b(b)m(y)h(a)h(constan)m(t.)589 3475 y(Indeed,)24 b(assume)e(that)h (there)g(are)g(2)1813 3442 y Fu(m)1903 3475 y Fx(length-)p Fv(k)j Fx(descriptions)d(of)f Fv(x)p Fx(.)38 b(The)22 b(n)m(um)m(b)s(er)448 3588 y(of)30 b Fv(y)s Fx('s)e(that)i(ha)m(v)m(e)g (at)g(least)g(2)1458 3555 y Fu(m)1554 3588 y Fx(descriptions)f(of)h (length)f Fv(k)j Fx(is)d(at)h(most)g(2)2979 3555 y Fu(k)r Fy(\000)p Fu(m)3139 3588 y Fx(.)40 b(Giv)m(en)448 3701 y Fv(m)34 b Fx(and)g Fv(k)j Fx(w)m(e)e(can)f(en)m(umerate)h(all)g(suc)m (h)f Fv(y)s Fx('s.)52 b(Hence)35 b(ev)m(ery)g(suc)m(h)f Fv(y)i Fx(can)f(b)s(e)e(found)448 3814 y(from)h Fv(m)f Fx(and)h(its)g(\()p Fv(k)26 b Fs(\000)c Fv(m)p Fx(\))34 b(bit)g(index)g(in)f(that)i(en)m(umeration)f(\()p Fv(k)k Fx(can)c(b)s(e)f(retriev)m(ed)448 3927 y(from)40 b Fv(k)31 b Fs(\000)26 b Fv(m)41 b Fx(and)f Fv(m)p Fx(\).)72 b(Th)m(us)39 b(if)i Fv(x)f Fx(has)h(2)2012 3894 y Fu(m)2119 3927 y Fx(descriptions)g(of)g(length)g Fv(k)i Fx(then)e(its)448 4040 y(complexit)m(y)32 b(is)f(b)s(ounded)d(b)m(y)1419 4244 y Fv(C)7 b Fx(\()p Fv(x)p Fx(\))25 b Fj(6)g Fv(k)f Fs(\000)19 b Fv(m)h Fx(+)g Fv(O)s Fx(\(log)e Fv(m)p Fx(\))p Fv(:)448 4448 y Fx(F)-8 b(or)31 b Fv(k)e Fx(=)c Fv(C)7 b Fx(\()p Fv(x)p Fx(\))30 b(this)h(implies)f(that)h Fv(m)f Fx(is)h(b)s(ounded)d(b)m(y)i(a)h(constan)m(t.)589 4561 y(Is)37 b(an)h(analogous)g(statemen)m(t)h(true)f(for)f(appro)m(ximate)h (descriptions?)61 b(W)-8 b(e)39 b(call)448 4674 y Fv(p)c Fx(an)g(appro)m(ximate)h(description)f(of)h Fv(x)f Fx(if)g(the)g(n)m (um)m(b)s(er)f(of)h(p)s(ositions)g(where)g Fv(x)g Fx(and)448 4787 y Fv(D)s Fx(\()p Fv(p)p Fx(\))29 b(di\013er)f(\(that)i(is,)f (Hamming)g(distance)g(b)s(et)m(w)m(een)h Fv(x)e Fx(and)g Fv(D)s Fx(\()p Fv(p)p Fx(\))h(is)g(at)g(most)g Fv(d)p Fx(\))g(is)448 4900 y(small.)42 b(Here)30 b Fv(D)k Fx(is)c(an)g (optimal)i(description)e(mo)s(de)g(in)g(the)h(de\014nition)f(of)g Fv(C)7 b Fx(\()p Fv(x)p Fx(\).)1897 5225 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 589 573 a Fx(More)31 b(formally)-8 b(,)31 b(let)g Fv(d)f Fx(b)s(e)f(an)g(in)m(teger.)42 b(Let)31 b Fv(x)e Fx(b)s(e)g(a)i(binary)d(string)i(of)g(length)g Fv(n)p Fx(.)448 686 y(Let)i Fv(C)677 701 y Fu(d)717 686 y Fx(\()p Fv(x)p Fx(\))f(denote)g(the)g(minimal)g(Kolmogoro)m(v)i (complexit)m(y)f(of)f(a)g(string)g(of)g(length)448 799 y Fv(n)f Fx(suc)m(h)g(that)h(the)f(Hamming)g(distance)h(b)s(et)m(w)m (een)g Fv(x)f Fx(and)f Fv(y)k Fx(is)e(at)f(most)h Fv(d)p Fx(.)41 b(Is)30 b(it)g(true)448 912 y(that)e(the)f(n)m(um)m(b)s(er)f (of)i Fv(y)s Fx('s)f(of)g(complexit)m(y)i Fv(C)1984 927 y Fu(d)2024 912 y Fx(\()p Fv(x)p Fx(\))f(at)g(Hamming)g(distance)f Fv(d)h Fx(\(or)f(less\))448 1024 y(from)38 b Fv(x)g Fx(is)g(small?)65 b(More)39 b(sp)s(eci\014cally)-8 b(,)41 b(is)e(it)f(b)s(ounded)f(b)m(y) h(a)g(p)s(olynomial)h(in)f Fs(j)p Fv(x)p Fs(j)p Fx(?)448 1137 y(More)33 b(generally)-8 b(,)34 b(is)e(it)h(true)f(that)g Fv(C)1734 1152 y Fu(d)1775 1137 y Fx(\()p Fv(x)p Fx(\))d Fj(6)e Fv(k)e Fs(\000)c Fv(m)g Fx(+)g Fv(O)s Fx(\(log)d Fv(n)i Fx(+)h(log)d Fv(m)p Fx(\))32 b(whenev)m(er)448 1250 y(there)26 b(are)g(at)h(least)g(2)1183 1217 y Fu(m)1275 1250 y Fx(strings)f Fv(y)i Fx(of)e(complexit)m(y)i(at)e(most)g Fv(k)j Fx(at)d(Hamming)g(distance)448 1363 y Fv(d)34 b Fx(from)e Fv(x)p Fx(?)48 b(If)33 b(the)g(second)g(question)g(answ)m (ers)g(in)f(p)s(ositiv)m(e)i(then)f(so)g(do)s(es)g(the)g(\014rst)448 1476 y(one)e(\(substitute)g Fv(C)1141 1491 y Fu(d)1181 1476 y Fx(\()p Fv(x)p Fx(\))g(for)f Fv(k)s Fx(\).)41 b(W)-8 b(e)32 b(will)f(sho)m(w)f(that)h(this)f(is)h(indeed)e(the)i (case.)589 1589 y(T)-8 b(o)25 b(see)f(wh)m(y)g(this)g(is)g(not)g (straigh)m(tforw)m(ard,)i(let)f(us)e(try)h(\014rst)f(the)i(argumen)m (ts)f(used)448 1702 y(in)j(the)h(case)h(of)e(precise)h(\(normal\))g (descriptions.)40 b(The)27 b(n)m(um)m(b)s(er)f(of)h Fv(x)p Fx('s)h(that)g(ha)m(v)m(e)g(at)448 1815 y(least)j(2)705 1782 y Fu(m)801 1815 y Fx(di\013eren)m(t)f Fv(y)s Fx('s)f(of)g (complexit)m(y)i Fv(k)i Fx(at)d(Hamming)f(distance)h Fv(d)g Fx(or)f(less)g(from)g Fv(x)448 1928 y Fx(is)i(b)s(ounded)d(b)m (y)1684 2041 y(2)1729 2003 y Fu(k)r Fy(\000)p Fu(m)1890 2041 y Fv(B)5 b Fx(\()p Fv(n;)15 b(d)p Fx(\))p Fv(;)448 2208 y Fx(where)36 b Fv(B)5 b Fx(\()p Fv(n;)15 b(d)p Fx(\))36 b(stands)f(for)h(the)g(cardinalit)m(y)h(of)f(a)h(Hamming)f (ball)g(of)g(radius)f Fv(d)h Fx(in)448 2320 y Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)668 2287 y Fu(n)718 2320 y Fx(.)38 b(Suc)m(h)22 b Fv(x)p Fx('s)h(can)g(b)s(e)f(en)m(umerated)h (giv)m(en)h Fv(n;)15 b(m;)g(k)s(;)g(d)24 b Fx(and)e(th)m(us)g(the)h (complexit)m(y)448 2433 y(of)31 b(eac)m(h)g(suc)m(h)f Fv(x)h Fx(is)f(at)h(most)1162 2638 y Fv(k)24 b Fs(\000)c Fv(m)g Fx(+)f(log)f Fv(B)5 b Fx(\()p Fv(n;)15 b(d)p Fx(\))21 b(+)f Fv(O)s Fx(\(log)d Fv(n)j Fx(+)g(log)e Fv(m)p Fx(\))p Fv(:)448 2842 y Fx(W)-8 b(e)30 b(need)f(to)g(pro)m(v)m(e)h(that)f(for)f (a)h(smaller)h(quan)m(tit)m(y)g Fv(C)2313 2857 y Fu(d)2353 2842 y Fx(\()p Fv(x)p Fx(\))f(w)m(e)g(ha)m(v)m(e)h(a)f(b)s(etter)g(b)s (ound)1253 3046 y Fv(C)1318 3061 y Fu(d)1358 3046 y Fx(\()p Fv(x)p Fx(\))d Fj(6)f Fv(k)e Fs(\000)d Fv(m)g Fx(+)g Fv(O)s Fx(\(log)e Fv(n)i Fx(+)g(log)d Fv(m)p Fx(\))p Fv(:)448 3250 y Fx(T)-8 b(o)29 b(this)g(end)f(w)m(e)h(ha)m(v)m(e)h(to)f (co)m(v)m(er)h(all)g(en)m(umerated)f Fv(x)p Fx('s)f(b)m(y)h(2)2535 3217 y Fu(k)r Fy(\000)p Fu(m)p Fl(+)p Fu(O)r Fl(\(log)13 b Fu(n)p Fl(+log)f Fu(m)p Fl(\))3255 3250 y Fx(balls)448 3363 y(of)33 b(radius)f Fv(d)p Fx(.)48 b(Moreo)m(v)m(er,)36 b(w)m(e)d(need)f(to)i(pro)s(duce)d(suc)m(h)h(co)m(v)m(ering)j(on-line.) 48 b(W)-8 b(e)34 b(need)448 3476 y(to)e(co)m(v)m(er)g(ev)m(ery)f(en)m (umerated)g Fv(x)f Fx(immediately)-8 b(,)33 b(w)m(e)e(cannot)g(w)m(ait) h(un)m(til)e(all)i(suc)m(h)e Fv(x)p Fx('s)448 3589 y(ha)m(v)m(e)i(b)s (een)e(en)m(umerated,)g(as)h(w)m(e)g(do)f(not)h(kno)m(w)f(when)g(it)g (happ)s(ens.)589 3702 y(As)38 b(w)m(e)f(said)h(ab)s(o)m(v)m(e,)i(the)d (answ)m(er)g(to)h(the)g(question)f(is)g(p)s(ositiv)m(e:)56 b(if)37 b Fv(x)g Fx(b)s(elongs)448 3815 y(to)g(at)g(least)g(2)945 3782 y Fu(m)1048 3815 y Fx(di\013eren)m(t)f(Hamming)h(balls)f(of)g (radius)f Fv(d)h Fx(and)g(complexit)m(y)h Fv(k)i Fx(then)448 3928 y Fv(x)32 b Fx(b)s(elongs)g(to)h(a)g(Hamming)f(ball)h(of)f(radius) f Fv(d)i Fx(and)e(complexit)m(y)j Fv(k)25 b Fs(\000)c Fv(m)g Fx(+)g Fv(O)s Fx(\(log)d Fv(n)p Fx(\))448 4041 y(\(up)34 b(to)h(an)f(additiv)m(e)i(error)e(term)g Fv(O)s Fx(\(log)18 b Fv(n)p Fx(\),)36 b(the)e(complexit)m(y)j(of)d(a)h (Hamming)g(ball)448 4154 y(coincides)24 b(with)f(complexit)m(y)i(of)f (its)f(cen)m(ter,)j(th)m(us)d(this)g(is)h(an)f(equiv)-5 b(alen)m(t)24 b(form)m(ulation)448 4267 y(of)j(the)g(question\).)40 b(This)26 b(is)h(true)f(not)h(only)g(for)f(Hamming)h(balls)g(but)f(for) h(an)m(y)g(family)448 4379 y(of)k(subsets)f(of)g Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)1190 4346 y Fu(n)1270 4379 y Fx(of)30 b(lo)m(w)h(Kolmogoro)m(v)i(complexit)m(y)-8 b(.)589 4492 y(Let)47 b Fs(S)53 b Fx(b)s(e)46 b(an)g(arbitrary)g (family)h(of)f(subsets)g(of)g Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)2644 4459 y Fu(n)2740 4492 y Fx(and)45 b Fv(k)s(;)15 b(m)47 b Fx(natural)448 4605 y(n)m(um)m(b)s(ers.)74 b(Let)43 b Fv(C)7 b Fx(\()p Fs(S)g Fx(\))42 b(denote)g(the)g(complexit) m(y)i(of)e Fs(S)49 b Fx(regarded)41 b(as)i(a)f(\014nite)g(ob-)448 4718 y(ject.)95 b(F)-8 b(or)49 b(example,)k(if)48 b Fs(S)55 b Fx(consist)49 b(of)f(all)h(Hamming)f(balls)h(of)f(radius)f Fv(d)h Fx(then)448 4831 y Fv(C)7 b Fx(\()p Fs(S)g Fx(\))26 b(=)f Fv(O)s Fx(\(log)17 b Fv(n)j Fx(+)g(log)e Fv(d)p Fx(\))26 b(=)f Fv(O)s Fx(\(log)17 b Fv(n)p Fx(\).)1897 5225 y(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 448 573 a Fm(Theorem)25 b(6)h Fx(\([14)q(]\))p Fm(.)35 b Ft(If)25 b(a)g(string)g Fv(x)g Ft(b)-5 b(elongs)25 b(to)g(at)h(le)-5 b(ast)25 b Fx(2)2471 540 y Fu(m)2563 573 y Ft(sets)g(in)g Fs(S)31 b Ft(of)25 b(c)-5 b(omplexity)448 686 y(at)34 b(most)f Fv(k)j Ft(then)d Fv(x)g Ft(b)-5 b(elongs)33 b(to)g(a)g(set)g(in)f Fs(S)40 b Ft(of)33 b(c)-5 b(omplexity)34 b(at)f(most)1123 883 y Fv(k)23 b Fs(\000)d Fv(m)g Fx(+)g Fv(O)s Fx(\(log)e Fv(n)i Fx(+)g(log)d Fv(k)24 b Fx(+)19 b(log)f Fv(m)i Fx(+)g Fv(C)7 b Fx(\()p Fs(S)g Fx(\)\))p Fv(:)589 1081 y Fx(F)-8 b(or)35 b(instance,)g(in)f(the)f(case)i(of)f(Hamming)g(balls) g(Theorem)f(sa)m(ys:)48 b(if)34 b Fv(x)f Fx(b)s(elongs)448 1194 y(to)j(2)609 1161 y Fu(m)710 1194 y Fx(balls)f(of)f(radius)g Fv(d)h Fx(and)f(complexit)m(y)i(at)f(most)g Fv(k)j Fx(then)c Fv(x)g Fx(b)s(elongs)h(to)g(a)g(ball)448 1307 y(of)d(radius)f Fv(d)g Fx(and)g(complexit)m(y)j(at)e(most)g Fv(k)24 b Fs(\000)d Fv(m)g Fx(+)f Fv(O)s Fx(\(log)e Fv(n)p Fx(\).)44 b(\(Note)33 b(that)g(the)e(total)448 1420 y(n)m(um)m(b)s(er)e(of)h (balls)h(of)f(\014xed)f(radius)g(is)h(2)1826 1387 y Fu(n)1904 1420 y Fx(hence)g Fv(m)25 b Fj(6)g Fv(n)k Fx(and)h(the)g(term)g Fv(O)s Fx(\(log)18 b Fv(m)p Fx(\))30 b(is)448 1533 y(absorb)s(ed)f(b)m (y)i Fv(O)s Fx(\(log)18 b Fv(n)p Fx(\),)30 b(as)h(w)m(ell)g(as)g(terms) f Fv(O)s Fx(\(log)18 b Fv(k)s Fx(\))30 b(and)g Fv(O)s Fx(\()p Fv(C)7 b Fx(\()p Fs(S)g Fx(\)\).\))589 1645 y(The)33 b(reader)h(ma)m(y)g(w)m(onder)f(what)g(happ)s(ens)f(if)h(Kolmogoro)m(v) i(complexit)m(y)h(of)d(all)448 1758 y(sets)e(in)f Fs(S)36 b Fx(is)31 b(close)g(to)g Fv(k)s Fx(.)41 b(In)29 b(this)h(case)h(the)f (statemen)m(t)i(of)f(the)f(theorem)g(just)g(yields)448 1871 y(an)f(upp)s(er)e(b)s(ound)g Fv(C)7 b Fx(\()p Fs(S)g Fx(\))17 b(+)g Fv(O)s Fx(\(log)g Fv(n)g Fx(+)g(log)g Fv(k)s Fx(\))30 b(for)e Fv(m)p Fx(.)40 b(Note)31 b(that)e(in)f(this)h (case)h Fv(C)7 b Fx(\()p Fs(S)g Fx(\))448 1984 y(is)31 b(at)g(least)g Fv(k)24 b Fs(\000)c Fv(O)s Fx(\(1\))31 b(and)f(th)m(us)g(this)g(b)s(ound)e(is)j(not)f(v)m(ery)h(in)m (teresting.)589 2097 y(T)-8 b(o)30 b(pro)m(v)m(e)h(the)e(theorem)h(let) g(us)f(try)g(the)h(follo)m(wing)h(argumen)m(t.)41 b(Let)30 b Fv(P)3070 2111 y Fl(1)3109 2097 y Fv(;)15 b(:)g(:)g(:)i(;)e(P)3369 2111 y Fu(N)448 2210 y Fx(where)38 b Fv(N)47 b(<)38 b Fx(2)993 2177 y Fu(k)r Fl(+1)1164 2210 y Fx(b)s(e)f(all)i(sets)f(in)f Fs(S)45 b Fx(of)38 b(complexit)m(y)i(at)e(most)g Fv(k)s Fx(.)64 b(Let)38 b Fv(T)50 b Fx(consist)448 2323 y(of)34 b(all)h Fv(x)30 b Fs(2)g(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)1078 2290 y Fu(n)1161 2323 y Fx(of)34 b(m)m(ultiplicit)m(y)h Fj(>)c Fx(2)1905 2290 y Fu(m)2005 2323 y Fx(in)j(the)f(sets)h Fv(P)2513 2337 y Fl(1)2553 2323 y Fv(;)15 b(:)g(:)g(:)i(;)e(P)2813 2337 y Fu(N)2881 2323 y Fx(.)50 b(Let)34 b(us)f(sho)m(w)448 2436 y(\014rst)42 b(that)h(it)g(is)f(p)s(ossible)g(to)h(co)m(v)m(er)h Fv(T)55 b Fx(b)m(y)42 b(at)h(most)f Fv(N)10 b Fx(2)2518 2403 y Fy(\000)p Fu(m)2640 2436 y Ft(p)-5 b(oly)r Fx(\()p Fv(n;)15 b(k)s(;)g(m)p Fx(\))43 b(sets)g(of)448 2549 y Fv(P)506 2563 y Fl(1)546 2549 y Fv(;)15 b(:)g(:)g(:)i(;)e(P)806 2563 y Fu(N)904 2549 y Fx(\(here,)31 b Ft(p)-5 b(oly)r Fx(\()p Fv(k)s(;)15 b(m;)g(n)p Fx(\))31 b(denotes)g(a)f(p)s (olynomial\).)589 2662 y(Consider)42 b(the)g(bipartite)g(graph)g(whose) f(left)i(no)s(des)e(are)i(elemen)m(ts)g(of)f Fv(T)55 b Fx(and)448 2775 y(righ)m(t)33 b(no)s(des)e(are)i(sets)f Fv(P)1318 2789 y Fl(1)1358 2775 y Fv(;)15 b(:)g(:)g(:)h(;)f(P)1617 2789 y Fu(N)1685 2775 y Fx(.)46 b(Dra)m(w)33 b(an)f(edge)g(b)s(et)m(w)m (een)h Fv(x)f Fx(and)f Fv(P)3011 2789 y Fu(i)3072 2775 y Fx(if)h Fv(x)c Fs(2)f Fv(P)3383 2789 y Fu(i)3412 2775 y Fx(.)448 2888 y(The)38 b(degree)g(of)g(eac)m(h)h(left)g(no)s(de)e(is) h(at)g(least)i(2)2133 2855 y Fu(m)2237 2888 y Fx(hence)e(the)g(n)m(um)m (b)s(er)f(of)h(edges)g(in)448 3000 y(the)g(graph)e(is)i(at)f(least)i(2) 1357 2967 y Fu(m)1424 3000 y Fs(j)p Fv(T)13 b Fs(j)p Fx(.)61 b(Th)m(us)36 b(there)i(is)f(a)g(righ)m(t)h(no)s(de)f(of)g (degree)h(at)g(least)448 3113 y(2)493 3080 y Fu(m)560 3113 y Fs(j)p Fv(T)13 b Fs(j)p Fv(=)-5 b(N)10 b Fx(.)64 b(In)38 b(other)g(w)m(ords,)h(there)f(is)g(a)h(set)f Fv(P)2180 3127 y Fu(u)2264 3113 y Fx(that)g(co)m(v)m(ers)i(at)e(least)h (2)3130 3080 y Fu(m)3197 3113 y Fs(j)p Fv(T)13 b Fs(j)p Fv(=)-5 b(N)448 3226 y Fx(elemen)m(t)28 b(in)e Fv(T)13 b Fx(.)39 b(Include)25 b(in)h(the)h(co)m(v)m(ering)h(an)m(y)e(suc)m(h)g (set)h Fv(P)2499 3240 y Fu(u)2544 3226 y Fx(.)39 b(Then)25 b(apply)h(the)h(same)448 3339 y(argumen)m(t)g(to)h(the)e(set)h Fv(T)f Fs(n)13 b Fv(P)1444 3353 y Fu(u)1516 3339 y Fx(in)26 b(place)h(of)g Fv(T)39 b Fx(and)26 b(include)g(in)h(the)f(co)m(v)m (ering)j(a)e(set)g Fv(P)3396 3353 y Fu(v)448 3452 y Fx(that)35 b(co)m(v)m(ers)g(at)g(least)g(2)1299 3419 y Fu(m)1366 3452 y Fv(=)-5 b(N)44 b Fx(fraction)35 b(of)f(elemen)m(ts)h(in)e Fv(T)j Fs(n)23 b Fv(P)2669 3466 y Fu(u)2714 3452 y Fx(,)35 b(and)e(so)h(on.)51 b(Eac)m(h)448 3565 y(time)34 b(the)g(fraction)g(of) f(non-co)m(v)m(ered)i Fv(x)30 b Fs(2)g Fv(T)45 b Fx(decreases)35 b(b)m(y)e(a)g(factor)i(of)e(1)23 b Fs(\000)e Fx(2)3221 3532 y Fu(m)3288 3565 y Fv(=)-5 b(N)10 b Fx(.)448 3678 y(After)32 b(including)g(2)1128 3645 y Fy(\000)p Fu(m)1250 3678 y Fv(N)10 b(n)15 b Fx(ln)f(2)33 b(sets,)f(the)g(n)m(um)m(b)s(er)f (of)h(non-co)m(v)m(ered)h(strings)f(in)f Fv(T)45 b Fx(is)448 3791 y(at)31 b(most)930 3988 y Fs(j)p Fv(T)13 b Fs(j)p Fx(\(1)22 b Fs(\000)e Fx(2)1284 3951 y Fu(m)1351 3988 y Fv(=)-5 b(N)10 b Fx(\))1509 3951 y Fl(2)1544 3927 y Fq(\000)p Fi(m)1651 3951 y Fu(N)d(n)12 b Fl(ln)g(2)1904 3988 y Fv(<)25 b Fs(j)p Fv(T)13 b Fs(j)p Fv(e)2158 3951 y Fy(\000)p Fu(n)f Fl(ln)g(2)2403 3988 y Fx(=)25 b Fs(j)p Fv(T)13 b Fs(j)p Fx(2)2660 3951 y Fy(\000)p Fu(n)2788 3988 y Fj(6)25 b Fx(1)p Fv(:)448 4186 y Fx(That)20 b(is,)j(all)e Fv(x)k Fs(2)g Fv(T)33 b Fx(are)21 b(co)m(v)m(ered.)39 b(\(Alternativ)m(ely)-8 b(,)26 b(w)m(e)20 b(could)h(sho)m(w)f(that)h(2) 3038 4153 y Fy(\000)p Fu(m)3160 4186 y Fv(N)10 b(n)15 b Fx(ln)f(2)448 4299 y(randomly)34 b(c)m(hosen)h(sets)g(co)m(v)m(er)h Fv(T)47 b Fx(with)34 b(p)s(ositiv)m(e)h(probabilit)m(y)-8 b(.)53 b(W)-8 b(e)36 b(prefer)d(the)i(de-)448 4412 y(scrib)s(ed)30 b(greedy)g(co)m(v)m(ering)j(algorithm,)e(as)g(it)g(is)f(more)h (constructiv)m(e.\))589 4525 y(Wh)m(y)39 b(this)e(argumen)m(t)i(do)s (es)e(not)h(su\016ce?)63 b(The)38 b(problem)f(is)h(that)g(w)m(e)h(are)f (not)448 4638 y(giv)m(en)e(the)f(set)h Fs(f)p Fv(P)1101 4652 y Fl(1)1141 4638 y Fv(;)15 b(:)g(:)g(:)i(;)e(P)1401 4652 y Fu(N)1469 4638 y Fs(g)p Fx(.)54 b(This)35 b(set)g(migh)m(t)h(ha) m(v)m(e)g(a)f(h)m(uge)g(Kolmogoro)m(v)i(com-)448 4751 y(plexit)m(y)-8 b(.)56 b(W)-8 b(e)36 b(can)f(only)g(assume)g(that)g(a)h (pro)s(cedure)d(to)j(en)m(umerate)g Fv(P)2974 4765 y Fl(1)3014 4751 y Fv(;)15 b(:)g(:)g(:)h(;)f(P)3273 4765 y Fu(N)3376 4751 y Fx(is)448 4863 y(a)m(v)-5 b(ailable,)41 b(suc)m(h)36 b(pro)s(cedure)f(can)i(b)s(e)f(describ)s(ed)f(in)h Fv(O)s Fx(\(log)18 b Fv(k)27 b Fx(+)d Fv(C)7 b Fx(\()p Fs(S)g Fx(\)\))37 b(bits.)58 b(More)448 4976 y(sp)s(eci\014cally)-8 b(,)30 b(there)e(is)g(a)g(non-halting)g(algorithm)h(with)e(inputs)g Fv(k)s(;)15 b(C)7 b Fx(\()p Fs(S)g Fx(\))28 b(that)h(prin)m(ts)1897 5225 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 448 573 a Fx(out)27 b Fv(P)664 587 y Fl(1)704 573 y Fv(;)15 b(:)g(:)g(:)i(;)e(P)964 587 y Fu(N)1059 573 y Fx(is)26 b(some)i(order.)39 b(More)27 b(formally)-8 b(,)29 b(the)e(algorithm)h(prin)m(ts)e(the)h(co)s(des)448 686 y(of)e(sets)h Fv(P)777 700 y Fl(1)817 686 y Fv(;)15 b(:)g(:)g(:)h(;)f(P)1076 700 y Fu(N)1169 686 y Fx(with)25 b(resp)s(ect)g(to)g(a)h(\014xed)e(enco)s(ding)h(of)g(\014nite)g (subsets)f(of)h Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)3395 653 y Fy(\003)448 799 y Fx(b)m(y)39 b(binary)f(strings.)66 b(And)38 b(w)m(e)h(do)g(not)g(kno)m(w)g(at)g(whic)m(h)g(momen)m(t)g (the)g(last)h(set)f(is)448 912 y(prin)m(ted.)448 1146 y Fm(The)c(game)448 1318 y Fx(Th)m(us)30 b(w)m(e)h(naturally)g(come)g (to)g(the)g(follo)m(wing)h(game)g(of)f(t)m(w)m(o)g(pla)m(y)m(ers,)h(w)m (e)f(call)h(them)448 1431 y(Pro)s(ducer)26 b(\(P\))i(and)e(Consumer)g (\(C\).)h(Let)h Fv(m;)15 b(n;)g(k)s(;)g(L)27 b Fx(b)s(e)g(natural)g (parameters.)40 b(The)448 1544 y(game)c(consists)g(of)f(alternating)i (mo)m(v)m(es)g(b)m(y)e(the)g(pla)m(y)m(ers,)i(eac)m(h)g(making)e(2)3093 1511 y Fu(k)3171 1544 y Fx(mo)m(v)m(es,)448 1657 y(starting)44 b(with)f(P's)h(mo)m(v)m(e.)81 b(A)43 b(mo)m(v)m(e)i(of)e(P)h(consists)f (in)g(pro)s(ducing)f(a)i(subset)f(of)448 1770 y Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)668 1737 y Fu(n)718 1770 y Fx(.)40 b(A)30 b(mo)m(v)m(e)h(of)e(C)g(consists)h(in)g(marking)f (some)h(sets)g(previously)f(pro)s(duced)f(b)m(y)448 1882 y(P)i(\(the)g(n)m(um)m(b)s(er)f(of)g(mark)m(ed)h(sets)g(on)g(an)m(y)g (mo)m(v)m(e)h(can)f(b)s(e)f(0\).)42 b(The)29 b(total)i(n)m(um)m(b)s(er) e(of)448 1995 y(mark)m(ed)h(sets)f(m)m(ust)g(not)h(exceed)g Fv(L)p Fx(.)40 b(C)29 b(wins)f(if,)i(follo)m(wing)g(ev)m(ery)g(one)g (of)f(its)g(mo)m(v)m(es,)448 2108 y(ev)m(ery)43 b Fv(x)g Fs(2)h(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)1120 2075 y Fu(n)1210 2108 y Fx(that)42 b(is)g(co)m(v)m(ered)h(at)f(least)h(2)2249 2075 y Fu(m)2357 2108 y Fx(times)f(b)m(y)f(P's)h(sets)3060 2075 y Fl(1)3141 2108 y Fx(b)s(elongs)448 2221 y(to)35 b(a)g(mark)m(ed)f(set.)54 b(It)34 b(is)g(imp)s(ortan)m(t)h(that)g(this) f(condition)h(is)f(c)m(hec)m(k)m(ed)i(after)f(ev)m(ery)448 2334 y(Consumer's)29 b(mo)m(v)m(e.)43 b(Consumer)29 b(cannot)i(w)m(ait) g(un)m(til)g(all)g(2)2523 2301 y Fu(k)2597 2334 y Fx(sets)f(app)s(ear.) 448 2569 y Fm(An)35 b(analysis)h(of)f(the)f(game)448 2740 y Fx(Consumer)k(can)h(easily)h(win)e(if)h Fv(L)g Fx(=)g(2)1849 2707 y Fu(k)1892 2740 y Fx(:)57 b(she)39 b(marks)f(ev)m(ery)h(set)h(pro)s(duced)d(b)m(y)i(P)-8 b(.)448 2853 y(On)37 b(the)h(other)f(extreme,)k(Pro)s(ducer)36 b(wins)h(if)g Fv(L)g(<)g Fx(2)2353 2820 y Fu(k)r Fy(\000)p Fu(m)2550 2853 y Fx(and)g Fv(n)g Fx(is)h(large)g(enough)448 2966 y(\(sp)s(eci\014cally)-8 b(,)38 b(w)m(e)e(need)f(2)1368 2933 y Fu(n)1448 2966 y Fj(>)e Fx(\()p Fv(m)23 b Fx(+)g(1\)2)1909 2933 y Fu(k)r Fy(\000)p Fu(m)2070 2966 y Fx(\):)51 b(he)35 b(generates)h(2)2756 2933 y Fu(m)2858 2966 y Fx(groups)e(of)h(sets,)448 3079 y(eac)m(h)25 b(group)d(has)h(2)1096 3046 y Fu(k)r Fy(\000)p Fu(m)1280 3079 y Fx(sets)g(and)g(all)h(sets)f(within)g(a)h (group)e(ha)m(v)m(e)j(a)e(common)h(elemen)m(t)448 3192 y(and)34 b(sets)h(from)f(di\013eren)m(t)h(groups)f(are)g(disjoin)m(t) 2139 3159 y Fl(2)2179 3192 y Fx(.)53 b(What)35 b(happ)s(ens)e(for)h Fv(L)g Fx(b)s(et)m(w)m(een)448 3305 y(2)493 3272 y Fu(k)r Fy(\000)p Fu(m)684 3305 y Fx(and)c(2)906 3272 y Fu(k)949 3305 y Fx(?)40 b(It)31 b(turns)e(out)i(that)f(for)h(most)f(suc)m(h)g Fv(L)h Fx(Consumer)e(wins.)448 3464 y Fm(Lemma)35 b(2.)42 b Ft(Consumer)33 b(has)h(a)f(winning)g(str)-5 b(ate)g(gy)34 b(in)f(the)g(game)g(pr)-5 b(ovide)g(d)1490 3633 y Fv(L)25 b Fx(=)g(2)1718 3596 y Fu(k)r Fy(\000)p Fu(m)1878 3633 y Ft(p)-5 b(oly)r Fx(\()p Fv(k)s(;)15 b(m;)g(n)p Fx(\))p Fv(:)589 3803 y Fx(There)20 b(are)h(t)m(w)m(o)h(pro)s(ofs)d(of)h(Lemma) h(2:)36 b(a)20 b(constructiv)m(e)i(one)f(and)e(a)i(non-constructiv)m(e) 448 3916 y(one)36 b(\(the)h(latter)g(b)s(elongs)e(to)i(An.)e(Muc)m (hnik\).)57 b(Giv)m(en)37 b(the)f(ab)s(o)m(v)m(e)h(analysis)f(of)g(the) 448 4029 y(\\o\013-line")42 b(v)m(ersion)e(of)g(the)g(game,)k(is)c(is)f (easier)i(to)g(understand)d(the)i(constructiv)m(e)448 4142 y(pro)s(of.)448 4323 y Ft(The)33 b(c)-5 b(onstructive)33 b(pr)-5 b(o)g(of)35 b(of)e(the)g(lemma.)47 b Fx(Consumer)19 b(sim)m(ultaneously)i(uses)f Fv(k)j Fx(strate-)448 4436 y(gies)33 b(denoted)g(b)m(y)f Fv(j)i Fx(=)28 b(1)p Fv(;)15 b Fx(2)p Fv(;)g(:)g(:)g(:)j(;)d(k)s Fx(.)46 b(Strategy)34 b Fv(j)j Fx(w)m(orks)32 b(as)h(follo)m(ws.)47 b(Divide)33 b(the)g(se-)448 4549 y(quence)27 b(of)f(Pro)s(ducer's)f(sets)i(in)m(to) g(2)1693 4516 y Fu(k)r Fy(\000)p Fu(j)1850 4549 y Fx(segmen)m(ts)g(of)f (2)2377 4516 y Fu(j)2440 4549 y Fx(sets)h(eac)m(h.)40 b(After)27 b(receiving)p 448 4617 1196 4 v 552 4671 a Fk(1)587 4702 y Fp(This)f(means)g(that)f Fh(x)h Fp(b)r(elongs)g(to)g (at)g(least)h(2)1914 4671 y Fi(m)1998 4702 y Fg(di\013er)l(ent)34 b Fp(Pro)r(ducer's)27 b(sets.)552 4762 y Fk(2)587 4794 y Fp(T)-6 b(o)32 b(run)f(this)h(strategy)-6 b(,)33 b(w)n(e)g(need)e (enough)g(strings)i(of)g(length)e Fh(n)p Fp(.)53 b(That's)33 b(wh)n(y)e Fh(n)h Fp(should)g(b)r(e)448 4885 y(large.)j(The)24 b(union)g(of)g(sets)g(of)h(one)f(group)g(m)n(ust)f(ha)n(v)n(e)g Fh(m)13 b Fp(+)g(1)24 b(elemen)n(ts|one)g(shared)g(elemen)n(t)f(and)448 4976 y Fh(m)j Fp(other)f(elemen)n(ts)h(to)g(obtain)g(2)1427 4945 y Fi(m)1512 4976 y Fp(di\013eren)n(t)f(sets.)1897 5225 y Fx(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 448 573 a Fx(eac)m(h)38 b(segmen)m(t)g Fv(Q)1089 587 y Fl(1)1129 573 y Fv(;)15 b(:)g(:)g(:)h(;)f(Q)1402 593 y Fl(2)1437 575 y Fi(j)1475 573 y Fx(,)38 b(consider)f(the)g(set)h Fv(T)49 b Fx(consisting)38 b(of)f(all)h Fv(x)e Fs(2)g(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)3388 540 y Fu(n)448 686 y Fx(of)32 b(m)m(ultiplicit)m(y)h Fj(>)27 b Fx(2)1184 653 y Fu(m)1251 686 y Fv(=k)35 b Fx(in)c(the)h(sets)f Fv(Q)1893 700 y Fl(1)1933 686 y Fv(;)15 b(:)g(:)g(:)h(;)f(Q)2206 706 y Fl(2)2241 687 y Fi(j)2278 686 y Fx(.)44 b(Using)32 b(the)f(greedy)h(algorithm)448 799 y(describ)s(ed)25 b(ab)s(o)m(v)m(e)j(mark)e(2)1376 766 y Fu(j)t Fy(\000)p Fu(m)1530 799 y Fv(k)s(n)15 b Fx(ln)g(2)26 b(sets)h(among)g Fv(Q)2347 813 y Fl(1)2386 799 y Fv(;)15 b(:)g(:)g(:)i(;)e(Q)2660 819 y Fl(2)2695 800 y Fi(j)2758 799 y Fx(so)27 b(as)f(to)h(co)m(v)m(er) h(the)448 912 y(set)j Fv(T)43 b Fx(b)m(y)31 b(mark)m(ed)f(sets.)589 1024 y(Since)h(there)g(are)g(2)1257 992 y Fu(k)r Fy(\000)p Fu(j)1418 1024 y Fx(segmen)m(ts)h(\(for)e(\014xed)g Fv(j)5 b Fx(\),)32 b(the)f(total)i(n)m(um)m(b)s(er)c(of)i(mark)m(ed)448 1137 y(sets)26 b(C)f(needs)g(to)g(use)g(is)h(2)1344 1104 y Fu(j)t Fy(\000)p Fu(m)1498 1137 y Fx(2)1543 1104 y Fu(k)r Fy(\000)p Fu(j)1673 1137 y Fv(k)s(n)15 b Fx(ln)g(2)26 b(=)f(2)2096 1104 y Fu(k)r Fy(\000)p Fu(m)2256 1137 y Fv(k)s(n)15 b Fx(ln)g(2)26 b(\(for)f(\014xed)f Fv(j)5 b Fx(\).)40 b(Summing)448 1250 y(o)m(v)m(er)32 b(all)f Fv(j)5 b Fx(,)31 b(this)g(comes)g(to)g(2)1469 1217 y Fu(k)r Fy(\000)p Fu(m)1629 1250 y Fv(k)1679 1217 y Fl(2)1719 1250 y Fv(n)15 b Fx(ln)g(2)30 b(mark)m(ed)h(sets.)589 1363 y(W)-8 b(e)36 b(claim)g(that)f(after)h(ev)m(ery)f(mo)m(v)m(e)h Fv(t)d Fx(=)f(1)p Fv(;)15 b(:)g(:)g(:)i(;)e Fx(2)2362 1330 y Fu(k)2440 1363 y Fx(of)35 b(C,)g(eac)m(h)g Fv(x)e 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y(will)h(pro)m(v)m(e)f(that)h(P)e(has)h(no)g (winning)f(strategy)i(\(b)m(y)f(K\177)-45 b(onig's)35 b(theorem)h(one)f(of)g(the)448 2680 y(pla)m(y)m(ers)d(m)m(ust)e(ha)m(v) m(e)h(a)g(winning)f(strategy\);)i(\(2\))f(T)-8 b(o)31 b(pro)m(v)m(e)g(that)g(P)g(has)f(no)g(winning)448 2793 y(strategy)35 b(w)m(e)f(will)f(design)g(a)h(randomized)f(strategy)h (for)f(C)g(that)h(b)s(eats)f(ev)m(ery)h(\014xed)448 2906 y(P's)d(strategy)g(with)f(p)s(ositiv)m(e)i(probabilit)m(y)-8 b(.)589 3019 y(The)38 b(randomized)g(C's)f(strategy)i(is)f(v)m(ery)h (simple:)55 b(mark)38 b(eac)m(h)h(set)g(of)f(P)f(with)448 3132 y(probabilit)m(y)c Fv(p)28 b Fx(=)g(2)1133 3099 y Fy(\000)p Fu(m)1255 3132 y Fx(\()p Fv(n)21 b Fx(+)g(1\))15 b(ln)h(2.)47 b(Fix)32 b(a)h(strategy)h Fv(S)j Fx(of)32 b Fv(P)13 b Fx(.)46 b(It)32 b(su\016ces)g(to)h(pro)m(v)m(e)448 3245 y(that)38 b(\(1\))f(with)g(probabilit)m(y)g(more)g(than)1951 3209 y Fl(1)p 1951 3224 36 4 v 1951 3276 a(2)1997 3245 y Fx(,)h(follo)m(wing)g(eac)m(h)g(mo)m(v)m(e)g(of)f(Consumer,)448 3357 y(ev)m(ery)30 b(elemen)m(t)g(of)f(m)m(ultiplicit)m(y)h(2)1653 3324 y Fu(m)1749 3357 y Fx(or)e(more)h(is)g(co)m(v)m(ered)h(b)m(y)e (mark)m(ed)h(sets;)h(and)e(\(2\))448 3470 y(with)j(probabilit)m(y)g (more)g(than)1571 3435 y Fl(1)p 1571 3450 V 1571 3502 a(2)1616 3470 y Fx(,)g(Consumer)e(marks)h(at)i(most)f(2)2753 3437 y Fu(k)r Fy(\000)p Fu(m)p Fl(+1)3003 3470 y Fx(\()p Fv(n)21 b Fx(+)f(1\))15 b(ln)h(2)448 3583 y(sets.)589 3696 y(T)-8 b(o)28 b(pro)m(v)m(e)g(\(2\))g(note)g(that)g(the)g(exp)s (ected)f(n)m(um)m(b)s(er)f(of)h(mark)m(ed)h(sets)f(is)h Fv(p)p Fx(2)3130 3663 y Fu(k)3173 3696 y Fx(.)39 b(Th)m(us)448 3809 y(b)m(y)30 b(Mark)m(o)m(v's)h(inequalit)m(y)-8 b(,)32 b(the)d(probabilit)m(y)h(that)g(it)g(exceeds)h Fv(p)p Fx(2)2726 3776 y Fu(k)r Fl(+1)2888 3809 y Fx(is)f(less)f(than)3367 3773 y Fl(1)p 3367 3788 V 3367 3840 a(2)3412 3809 y Fx(.)589 3922 y(T)-8 b(o)37 b(pro)m(v)m(e)f(\(1\))h(\014x)e Fv(x)g Fs(2)f(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)1667 3889 y Fu(n)1752 3922 y Fx(and)35 b(estimate)i(the)f(probabilit)m(y)h(that)f(there)g(is) 448 4035 y(mo)m(v)m(e)g(of)f(C)f(follo)m(wing)i(whic)m(h)f Fv(x)f Fx(is)h(co)m(v)m(ered)h(2)2105 4002 y Fu(m)2207 4035 y Fx(times)f(b)m(y)f(sets)h(of)g(P)f(but)g(b)s(elongs)448 4148 y(to)44 b(no)e(mark)m(ed)h(set.)79 b(W)-8 b(e)44 b(need)e(to)i(sho)m(w)e(that)i(this)e(happ)s(ens)f(with)i(probabilit)m (y)448 4261 y(less)f(than)g(2)896 4228 y Fy(\000)p Fu(n)p Fy(\000)p Fl(1)1088 4261 y Fx(.)75 b(T)-8 b(o)42 b(this)g(end)f(denote) h(b)m(y)g Fv(R)2216 4275 y Fu(i)2286 4261 y Fx(the)g(ev)m(en)m(t)h (\\follo)m(wing)g(a)g(mo)m(v)m(e)448 4374 y(of)36 b(C,)g(string)g Fv(x)f Fx(is)h(co)m(v)m(ered)h(at)g(least)g Fv(i)f Fx(times)g(b)m(y)g (sets)g(of)g(P)g(but)f(none)g(of)h(them)g(is)448 4487 y(mark)m(ed".)62 b(Let)38 b(us)e(pro)m(v)m(e)i(b)m(y)f(induction)g (that)h(Prob)o([)p Fv(R)2453 4501 y Fu(i)2481 4487 y Fx(])f Fj(6)f Fx(\(1)26 b Fs(\000)e Fv(p)p Fx(\))2932 4454 y Fu(i)2961 4487 y Fx(.)61 b(F)-8 b(or)38 b Fv(i)e Fx(=)h(0)448 4599 y(the)c(statemen)m(t)h(is)f(trivial.)48 b(T)-8 b(o)33 b(pro)m(v)m(e)g(the)g(induction)f(step)h(w)m(e)g(need)f (to)h(sho)m(w)f(that)448 4712 y(Prob[)p Fv(R)736 4726 y Fu(i)p Fl(+1)854 4712 y Fs(j)p Fv(R)948 4726 y Fu(i)977 4712 y Fx(])d Fj(6)h Fx(1)22 b Fs(\000)f Fv(p)p Fx(.)48 b(Let)34 b Fv(z)f Fx(=)c Fv(z)1793 4726 y Fl(1)1833 4712 y Fv(;)15 b(z)1915 4726 y Fl(2)1955 4712 y Fv(;)g(:)g(:)g(:)i(;)e(z) 2199 4726 y Fu(s)2269 4712 y Fx(b)s(e)32 b(a)i(sequence)f(of)g (decisions)g(b)m(y)448 4825 y(C:)i Fv(z)616 4839 y Fu(j)685 4825 y Fx(=)d(1)j(if)f(C)h(marks)f(the)h Fv(j)5 b Fx(th)35 b(set)g(of)g(P)f(and)g Fv(z)2228 4839 y Fu(j)2297 4825 y Fx(=)e(0)j(otherwise.)54 b(Call)35 b Fv(z)k Ft(b)-5 b(ad)45 b Fx(if)448 4938 y(follo)m(wing)32 b(C's)d Fv(s)p Fx(th)g(mo)m(v)m(e)i(it)g(happ)s(ens)d(for)h(the)h(\014rst)f(time)i (that)f Fv(x)g Fx(b)s(elongs)f(to)i Fv(i)f Fx(sets)1897 5225 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 448 573 a Fx(of)26 b(P)g(but)f(none)g(of)h (them)g(is)f(mark)m(ed.)40 b(Then)24 b Fv(R)2081 587 y Fu(i)2135 573 y Fx(is)i(the)g(disjoin)m(t)g(union)f(of)h(the)g(ev)m (en)m(ts)448 686 y(\\C)37 b(has)g(made)g(the)g(decisions)g Fv(z)t Fx(")h(\(denoted)f(b)m(y)g Fv(Q)2282 700 y Fu(z)2321 686 y Fx(\))g(o)m(v)m(er)h(all)g(bad)e Fv(z)t Fx(.)61 b(Th)m(us)36 b(it)h(is)448 799 y(enough)e(to)h(pro)m(v)m(e)g(that)f (Prob[)p Fv(R)1621 813 y Fu(i)p Fl(+1)1740 799 y Fs(j)p Fv(Q)1837 813 y Fu(z)1876 799 y Fx(])f Fj(6)e Fx(1)24 b Fs(\000)f Fv(p)p Fx(.)55 b(Giv)m(en)36 b(that)f(C)g(has)g(made)g(the) 448 912 y(decisions)d Fv(z)t Fx(,)h(the)f(ev)m(en)m(t)h Fv(R)1401 926 y Fu(i)p Fl(+1)1551 912 y Fx(means)e(that)i(after)f (those)g(decisions)g(the)g(strategy)h Fv(S)448 1024 y Fx(will)28 b(ev)m(er)g(pro)s(duce)e Fv(i)14 b Fx(+)g(1st)27 b(set)g(including)g Fv(x)g Fx(but)f(C)h(will)g(not)h(mark)f(it.)40 b(C's)26 b(decision)448 1137 y(not)38 b(mark)g(that)g(set)h(do)s(es)e 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Fx(of)g(length)g Fv(n)f Fx(b)s(elongs)g(to)h(2)1412 1852 y Fu(m)1513 1885 y Fx(Pro)s(ducer's)e(set)i(but)f(do)s(es)g(not)g(b)s(elong)h(to)g(a)g (mark)m(ed)448 1998 y(set)d(is)g(at)g(most)f(1/2.)p 3371 1998 4 62 v 3375 1939 55 4 v 3375 1998 V 3429 1998 4 62 v 448 2238 a Fm(Pro)s(of)36 b(of)f(Theorem)g(6)448 2409 y Fx(The)29 b(strategy)h(of)g(Lemma)f(2)h(can)f(b)s(e)g(found)e(b) m(y)i(the)h(brute)e(force)i(searc)m(h)g(giv)m(en)g Fv(n)p Fx(,)f Fv(k)448 2522 y Fx(and)23 b Fv(m)f Fx(and)h Fs(S)7 b Fx(.)38 b(En)m(umerate)23 b(the)g(sets)h(in)e Fs(S)30 b Fx(of)23 b(complexit)m(y)i(at)f(most)f Fv(k)s Fx(,)i(and)d(consider) 448 2635 y(app)s(earing)31 b(sets)g(as)g(the)g(mo)m(v)m(es)h(of)e(the)h (Pro)s(ducer.)41 b(Use)31 b(the)g(strategy)h(of)f(Lemma)g(2)448 2748 y(against)40 b(it.)65 b(W)-8 b(e)39 b(will)g(mark)f(at)h(most)f(2) 1901 2715 y Fu(k)r Fy(\000)p Fu(m)2062 2748 y Ft(p)-5 b(oly)q Fx(\()p Fv(m;)15 b(k)s(;)g(n)p Fx(\))40 b(of)e(the)g(generated) i(sets)448 2861 y(that)33 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Fv(C)7 b Fx(\()p Fv(!)2183 4982 y Fu(n)2230 4968 y Fs(j)p Fv(n)p Fx(\))25 b(=)g Fv(O)s Fx(\(1\))p Fv(:)643 b Fx(\(1\))1897 5225 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 448 573 a Fx(This)31 b(result)g(is)h (attributed)g(in)f([8])h(to)g(A.R.)g(Mey)m(er)h(\(see)f(also)h([17)q(,) f(7]\).)45 b(Our)30 b(goal)j(is)448 686 y(to)e(in)m(v)m(estigate)j (quan)m(titativ)m(e)f(v)m(ersions)e(of)f(Mey)m(er's)i(criterion.)589 799 y(The)42 b(follo)m(wing)i(t)m(w)m(o)g(natural)e(complexit)m(y)i (measure)e(are)h(related)g(to)g(the)f(left)448 912 y(hand)34 b(side)i(of)42 b(\(1\))q(.)55 b(Fix)36 b(a)f(programming)g(language)i (and)d(let)j Fv(U)10 b Fx(\()p Fv(p;)15 b(n)p Fx(\))35 b(denote)h(the)448 1024 y(output)24 b(of)g(the)g(program)g Fv(p)g Fx(on)g(input)f Fv(n)g Fx(\(here)h Fv(p)g Fx(is)g(a)g(binary)g (string)g(and)f Fv(n)h Fx(a)g(natural)448 1137 y(n)m(um)m(b)s(er\).)40 b(De\014ne)1106 1339 y Fv(C)1171 1353 y Fu(U)1230 1339 y Fx(\()p Fv(!)s Fx(\))26 b(=)f(min)o Fs(fj)p Fv(p)p Fs(j)h 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b(Bob)c(de\014nes)f Fv(l)j Fx(functions)d Fv(h)2437 1789 y Fl(1)2492 1775 y Fv(:)15 b(:)g(:)h(;)f(h)2705 1790 y Fu(l)2754 1775 y Fx(from)21 b Ff(N)h Fx(to)g Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)3370 1742 y Fy(\003)3412 1775 y Fx(;)448 1888 y(in)32 b(his)f(turn)g(Bob)h(can)g(de\014ne)f(eac) m(h)i(of)f Fv(h)1890 1902 y Fl(1)1945 1888 y Fv(:)15 b(:)g(:)h(;)f(h)2158 1903 y Fu(l)2217 1888 y Fx(on)31 b(an)m(y)h(\014nite)g(set)h(of)e(argumen)m(ts.)448 2001 y(Once)h Fv(h)734 2015 y Fu(i)763 2001 y Fx(\()p Fv(n)p Fx(\))g(is)g(de\014ned,)f(it)h(cannot)h(b)s(e)e(c)m(hanged)h(on)g (later)h(steps.)45 b(The)31 b(game)i(lasts)448 2114 y(in\014nitely)28 b(long.)41 b(After)28 b(ha)m(ving)h(done)f(in\014nitely)g(man)m(y)h(mo) m(v)m(es)g(Alice)h(has)e(de\014ned)f(a)448 2227 y(set)k Fv(T)43 b Fx(and)30 b(Bob)h(has)f(de\014ned)f(partial)i(functions)f Fv(h)2266 2241 y Fl(1)2306 2227 y Fv(;)15 b(:)g(:)g(:)i(;)e(h)2560 2242 y Fu(l)2612 2227 y Fx(:)25 b Ff(N)g Fs(!)g(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)3089 2194 y Fy(\003)3131 2227 y Fx(.)589 2340 y(Bob)40 b(wins)f(if)g(the)h(follo)m(wing)h(holds.)68 b(Call)40 b(an)f(in\014nite)h(binary)e(sequence)i Fv(!)j Fx(a)448 2452 y Ft(br)-5 b(anch)35 b Fx(of)27 b Fv(T)40 b Fx(if)27 b(almost)g(all)h(pre\014xes)e(of)h Fv(!)j Fx(are)d(in)f Fv(T)13 b Fx(.)40 b(Bob)27 b(wins)f(if)g(for)h(ev)m(ery)h (branc)m(h)448 2565 y Fv(!)k Fx(of)d Fv(T)41 b Fx(there)29 b(is)g Fv(i)g Fx(suc)m(h)f(that)i Fv(h)1565 2579 y Fu(i)1593 2565 y Fx(\()p Fv(n)p Fx(\))f(is)g(de\014ned)e(and)h(equal)i(to)f Fv(!)2728 2579 y Fu(n)2803 2565 y Fx(for)g(almost)h(all)f Fv(n)p Fx(.)448 2678 y(\(W)-8 b(e)32 b(call)g(suc)m(h)e Fv(i)h Fx(a)f(name)h(of)f Fv(!)s Fx(.\))589 2791 y(Call)h(this)g(game)g (the)f Fv(k)s(;)15 b(l)r Fx(-game.)43 b(F)-8 b(or)31 b(eac)m(h)h Fv(k)s(;)15 b(l)32 b Fx(the)f(winning)e(rule)h(in)g Fv(k)s(;)15 b(l)r Fx(-game)448 2904 y(is)44 b(de\014ned)e(b)m(y)h(a)h (Borel)g(set.)80 b(By)44 b(Martin's)f(theorem)h([9)q(])f(ev)m(ery)h (Borel)h(game)f(is)448 3017 y(deterministic.)e(In)28 b(particular,)i(for)g(eac)m(h)g Fv(k)s(;)15 b(l)32 b Fx(either)e(Alice,)h(or)e(Bob)h(has)f(a)h(winning)448 3130 y(strategy)i(in)e Fv(k)s(;)15 b(l)r Fx(-game.)448 3370 y Fm(An)35 b(analysis)h(of)f(the)f(game)448 3542 y Fx(As)22 b Fv(T)34 b Fx(has)21 b(at)i(most)e Fv(k)k Fx(strings)c(of)h(eac)m(h)h(length,)h Fv(T)34 b Fx(has)21 b(at)h(most)g Fv(k)j Fx(branc)m(hes.)37 b(\(Indeed,)448 3655 y(if)f Fv(T)48 b Fx(had)34 b Fv(k)27 b Fx(+)c(1)36 b(branc)m(hes,)g(for)f(large)i(enough)e Fv(n)g Fx(the)g(length-)p Fv(n)h Fx(pre\014xes)e(of)i(those)448 3767 y(branc)m(hes)g(w)m(ould)g (b)s(e)f(pair-wise)i(di\013eren)m(t)f(strings)g(of)g(length)h Fv(n)f Fx(in)f Fv(T)13 b Fx(.\))58 b(Alice)38 b(can)448 3880 y(easily)g(construct)g(a)f(set)h Fv(T)49 b Fx(ha)m(ving)38 b(exactly)h Fv(k)h Fx(branc)m(hes.)60 b(As)37 b(di\013eren)m(t)g(branc) m(hes)448 3993 y(m)m(ust)27 b(ha)m(v)m(e)g(di\013eren)m(t)g(names,)g (Alice)h(wins)d(if)i Fv(l)g(<)e(k)s Fx(.)39 b(On)26 b(the)g(other)h (hand,)f(it)h(is)f(ev)m(en)448 4106 y(not)31 b(eviden)m(t)g(that)g(for) f(ev)m(ery)h Fv(k)j Fx(there)c(is)h Fv(l)h Fx(suc)m(h)e(Bob)h(wins)e (in)h Fv(k)s(;)15 b(l)r Fx(-game.)589 4219 y(It)39 b(turns)d(out)j (that)f(the)g(b)s(order)f(line)h(separating)h(\()p Fv(k)s(;)15 b(l)r Fx(\)-games)41 b(w)m(on)d(b)m(y)g(Alice)448 4332 y(and)e(Bob)h(is)f(close)i(to)f(the)f(parab)s(ola)g Fv(l)h Fx(=)e Fv(k)2030 4299 y Fl(2)2070 4332 y Fx(,)j(whic)m(h)e(implies)g (Theorem)g(7.)59 b(More)448 4445 y(sp)s(eci\014cally)-8 b(,)31 b(Theorem)e(7)h(follo)m(ws)g(more)g(or)f(less)g(directly)h(from) f(the)h(follo)m(wing)g(lem-)448 4558 y(mas.)448 4745 y Fm(Lemma)h(3.)38 b Ft(F)-7 b(or)31 b(every)e Fv(k)j Ft(Bob)d(has)h(a)g(c)-5 b(omputable)31 b(winning)e(str)-5 b(ate)g(gy)31 b(in)e(the)g Fv(k)s(;)15 b(k)3364 4712 y Fl(2)3405 4745 y Ft(-)448 4858 y(game)33 b(\(the)g(winning)g (algorithm)i(has)f Fv(k)h Ft(as)f(an)f(input\).)1897 5225 y Fx(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 448 573 a Fm(Lemma)26 b(4.)36 b Ft(A)n(lic)-5 b(e)24 b(has)j(a)f(c)-5 b(omputable)27 b(winning)e(str)-5 b(ate)g(gy)28 b(in)d(the)h Fv(k)s(;)15 b(k)2888 540 y Fl(2)2928 573 y Fv(=)p Fx(4)p Ft(-game)26 b(\(the)448 686 y(winning)39 b(algorithm)h(has)f Fv(k)j Ft(as)d(an)f(input\).)60 b(Mor)-5 b(e)g(over,)40 b(A)n(lic)-5 b(e's)38 b(winning)g(str)-5 b(ate)g(gy)448 799 y(always)48 b(c)-5 b(onstructs)48 b(a)e(tr)-5 b(e)g(e)47 b(\(if)f Fv(x)g Ft(is)g(a)h(pr)-5 b(e\014x)47 b(of)f Fv(y)53 b Fs(2)c Fv(T)59 b Ft(then)46 b Fv(x)k Fs(2)f Fv(T)13 b Ft(\).)83 b(This)448 912 y(implies)47 b(that)g(we)f(c)-5 b(an)46 b(r)-5 b(e)g(quir)g(e)47 b(the)f(ine)-5 b(quality)46 b Fv(M)10 b Fx(\()p Fv(!)s Fx(\))50 b Fj(6)f Fv(m)29 b Fx(+)h Fv(O)s Fx(\(1\))47 b Ft(inste)-5 b(ad)47 b(of)482 1002 y Fx(~)448 1024 y Fv(M)10 b Fx(\()p Fv(!)s Fx(\))26 b Fj(6)f Fv(m)20 b Fx(+)g Fv(O)s Fx(\(1\))34 b Ft(in)e(The)-5 b(or)g(em)35 b(7.)589 1212 y Fx(W)-8 b(e)25 b(will)e(presen)m(t)g(here) f(Muc)m(hnik's)h(pro)s(of)f(of)h(Lemma)g(3)h(and)e(will)h(omit)h(the)f (pro)s(of)448 1325 y(of)31 b(Lemma)f(4,)h(as)g(it)g(is)f(to)s(o)h(in)m (v)m(olv)m(ed.)448 1513 y Ft(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)34 b(3.)46 b Fx(Bob's)39 b(functions)f(will)i(b)s(e)e(indexed)g (b)m(y)h(pairs)f Fv(a;)15 b(b)p Fx(,)42 b(where)c Fv(a)448 1626 y Fx(and)e Fv(b)g Fx(are)h(natural)f(n)m(um)m(b)s(ers)f(in)g(the)i (range)f(1)p Fv(;)15 b(:)g(:)g(:)i(;)e(k)s Fx(.)59 b(Let)37 b(us)e(explain)i(ho)m(w)f(Bob)448 1738 y(de\014nes)22 b Fv(h)791 1753 y Fu(ab)863 1738 y Fx(\()p Fv(n)p Fx(\).)38 b(Observing)22 b(the)g(gro)m(wing)h(set)g Fv(T)13 b Fx(,)24 b(Bob)f(lo)s(oks)f(for)h(all)g(pairs)f(of)g(strings)448 1851 y Fv(u)31 b Fx(and)e Fv(v)34 b Fx(suc)m(h)c(that:)514 2039 y(\(a\))47 b(the)35 b(ordinal)g(n)m(um)m(b)s(er)f(of)i Fv(u)f Fx(in)g(the)g(lexicographic)j(ordering)c(of)i(all)g(\(already) 676 2152 y(app)s(eared\))30 b(strings)g(of)g(length)h Fs(j)p Fv(u)p Fs(j)g Fx(in)f Fv(T)43 b Fx(is)30 b Fv(a)p Fx(;)509 2340 y(\(b\))46 b(the)37 b(ordinal)f(n)m(um)m(b)s(er)g(of)h Fv(v)j Fx(in)d(the)g(rev)m(erse)g(lexicographic)i(ordering)3155 2307 y Fl(4)3231 2340 y Fx(of)e(all)676 2452 y(\(already)31 b(app)s(eared\))f(strings)g(of)h(length)f Fs(j)p Fv(v)s Fs(j)i Fx(in)e Fv(T)43 b Fx(is)30 b Fv(b)p Fx(;)519 2640 y(\(c\))47 b Fv(u)30 b Fx(is)g(a)h(pre\014x)e(of)i Fv(v)s Fx(.)448 2828 y(After)h(suc)m(h)g(a)g(pair)g(of)g(strings)f(is)h (found,)f(Bob)h(sets)g Fv(h)2358 2843 y Fu(ab)2431 2828 y Fx(\()p Fv(n)p Fx(\))c(=)f Fv(u)2734 2842 y Fu(n)2813 2828 y Fx(\(length-)p Fv(n)32 b Fx(pre\014x)448 2941 y(of)j Fv(u)p Fx(\))f(for)g(all)i Fv(n)31 b Fj(6)g Fs(j)p Fv(u)p Fs(j)k Fx(suc)m(h)f(that)h Fv(h)1738 2956 y Fu(ab)1810 2941 y Fx(\()p Fv(n)p Fx(\))g(has)f(not)g(b)s(een)g(de\014ned)f(y)m (et.)53 b(Then)34 b(Bob)448 3054 y(lo)s(oks)d(for)f(another)h(pair)f (of)h(strings)f Fv(u)g Fx(and)g Fv(v)j Fx(with)e(the)f(same)h(prop)s (erties,)f(etc.)589 3166 y(Let)k Fv(T)45 b Fx(b)s(e)32 b(the)h(set)h(constructed)f(b)m(y)f(Alice)i(after)g(in\014nite)e(n)m (um)m(b)s(er)g(mo)m(v)m(es.)49 b(W)-8 b(e)448 3279 y(need)28 b(to)h(pro)m(v)m(e)g(that)f(this)g(strategy)h(guaran)m(tees)h(that)e (at)h(the)f(end)f(of)i(the)f(game)h(an)m(y)448 3392 y(in\014nite)h (branc)m(h)f(in)h Fv(T)43 b Fx(has)29 b(a)i(name.)40 b(Let)31 b Fv(!)h Fx(b)s(e)e(an)f(in\014nite)h(branc)m(h,)g(so)g Fv(!)3075 3406 y Fu(n)3147 3392 y Fs(2)25 b Fv(T)42 b Fx(for)448 3505 y(all)34 b(su\016cien)m(tly)f(large)g Fv(n)p Fx(.)47 b(F)-8 b(or)33 b(these)g Fv(n)f Fx(let)h Fv(a)2053 3519 y Fu(n)2132 3505 y Fx(denote)g(the)g(lexicographic)h(n)m (um)m(b)s(er)448 3618 y(of)45 b Fv(!)623 3632 y Fu(n)714 3618 y Fx(in)f(the)g(set)h Fv(T)1213 3632 y Fu(n)1305 3618 y Fx(of)f(all)i(strings)e(of)h(length)f Fv(n)g Fx(that)h(are)g(in) f Fv(T)13 b Fx(,)48 b(and)c(let)h Fv(b)3390 3632 y Fu(n)448 3731 y Fx(denote)26 b(the)g(rev)m(erse)g(lexicographic)h(n)m(um)m(b)s (er)e(of)g Fv(!)2212 3745 y Fu(n)2284 3731 y Fx(in)g Fv(T)2438 3745 y Fu(n)2486 3731 y Fx(.)39 b(Let)26 b Fv(a)f Fx(=)g(lim)15 b(sup)f Fv(a)3218 3745 y Fu(n)3291 3731 y Fx(and)448 3844 y Fv(b)26 b Fx(=)f(lim)15 b(sup)f Fv(b)941 3858 y Fu(n)988 3844 y Fx(.)41 b(W)-8 b(e)31 b(claim)h(that)f(for)f(all)h(su\016cien)m(tly)g(large)h Fv(n)p Fx(,)e Fv(h)2755 3859 y Fu(ab)2827 3844 y Fx(\()p Fv(n)p Fx(\))25 b(=)g Fv(!)3130 3858 y Fu(n)3177 3844 y Fx(.)589 3957 y(It)37 b(is)f(easy)h(to)g(see)g(that)g Fv(h)1522 3972 y Fu(ab)1594 3957 y Fx(\()p Fv(i)p Fx(\))g(is)f (de\014ned)f(for)h(all)h Fv(i)p Fx(.)59 b(In)35 b(other)i(w)m(ords,)g (for)f(all)448 4070 y Fv(i)e Fx(there)h(is)e(a)i(pair)e Fv(u;)15 b(v)38 b Fx(satisfying)c(the)g(ab)s(o)m(v)m(e)i(conditions)e (and)f(suc)m(h)h(that)g Fs(j)p Fv(u)p Fs(j)e Fj(>)f Fv(i)p Fx(.)448 4183 y(According)f(to)h(the)e(de\014nition)g(of)h Fv(a)g Fx(and)e Fv(b)i Fx(there)g(are)f(in\014nitely)h(man)m(y)f Fv(n)g Fx(suc)m(h)g(that)448 4296 y Fv(a)496 4310 y Fu(n)584 4296 y Fx(=)40 b Fv(a)f Fx(and)g(in\014nitely)h(man)m(y)f Fv(m)g Fx(suc)m(h)g(that)h Fv(b)2193 4310 y Fu(m)2300 4296 y Fx(=)h Fv(b)p Fx(.)67 b(Cho)s(ose)40 b(a)g(pair)f(of)g(suc)m(h) 448 4408 y Fv(n)e Fx(and)f Fv(m)p Fx(;)41 b(assume)36 b(that)i Fv(i)f Fj(6)f Fv(n)g Fj(6)g Fv(m)p Fx(.)60 b(The)37 b(strings)g Fv(u)f Fx(=)g Fv(!)2678 4422 y Fu(n)2761 4408 y Fx(and)h Fv(v)j Fx(=)c Fv(!)3193 4422 y Fu(m)3296 4408 y Fx(will)448 4521 y(b)s(e)g(disco)m(v)m(ered)i(after)f(all)g (strings)g(of)f(length)h Fv(n)f Fx(and)g Fv(m)g Fx(app)s(ear)g(in)g Fv(T)13 b Fx(.)59 b(These)36 b Fv(u;)15 b(v)448 4634 y Fx(qualify)33 b(all)g(the)f(conditions)h(listed)g(ab)s(o)m(v)m(e.)48 b(\(Note)34 b(that)f(this)f(do)s(es)g(not)g(pro)m(v)m(e)h(that)p 448 4718 1196 4 v 552 4772 a Fk(4)587 4804 y Fp(W)-6 b(e)28 b(sa)n(y)h(that)g Fh(x)f Fp(is)i(less)g(that)f Fh(y)i Fp(w.r.t.)46 b(the)28 b(rev)n(erse)h(lexicographic)i(ordering)f (if)f Fh(y)h(<)c(x)j Fp(w.r.t.)448 4895 y(the)d(normal)g(lexicographic) h(ordering.)1897 5225 y Fx(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 448 573 a Fv(h)500 588 y Fu(ab)573 573 y Fx(\()p Fv(i)p Fx(\))28 b(=)g Fv(!)858 587 y Fu(i)886 573 y Fx(,)k(as)g(w)m(e)h(migh)m(t)f(\014nd)f(a)h(pair)g Fv(u;)15 b(v)35 b Fx(that)e(is)f(di\013eren)m(t)g(from)g(the)g(sp)s (eci\014ed)448 686 y(pair.\))589 799 y(It)g(remains)f(to)g(pro)m(v)m(e) h(that)g(for)f(all)h(su\016cien)m(tly)g(large)g Fv(n)f Fx(w)m(e)g(ha)m(v)m(e)h Fv(h)3012 814 y Fu(ab)3085 799 y Fx(\()p Fv(n)p Fx(\))26 b(=)h Fv(!)3391 813 y Fu(n)448 912 y Fx(pro)m(vided)42 b Fv(h)886 927 y Fu(ab)958 912 y Fx(\()p Fv(n)p Fx(\))g(is)f(de\014ned.)74 b(Fix)42 b Fv(N)52 b Fx(suc)m(h)41 b(that)h Fv(a)2381 926 y Fu(n)2473 912 y Fj(6)h Fv(a)f Fx(and)f Fv(b)2904 926 y Fu(n)2995 912 y Fj(6)j Fv(b)e Fx(for)f(all)448 1024 y Fv(n)48 b Fj(>)f Fv(N)10 b Fx(.)81 b(W)-8 b(e)45 b(will)f(sho)m(w)f(that)i(if)f (b)s(oth)f Fs(j)p Fv(u)p Fs(j)h Fx(and)f Fs(j)p Fv(v)s Fs(j)i Fx(are)f(at)g(least)h Fv(N)54 b Fx(and)43 b Fv(u;)15 b(v)448 1137 y Fx(satisfy)46 b(the)f(conditions)h(listed)g(ab)s(o)m(v)m 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Fx(to)h(b)s(e)f(a)h(pre\014x)f(of)g Fv(v)k Fx(is)448 1702 y(when)d(b)s(oth)f Fv(u)i Fx(and)e Fv(v)1203 1720 y Fy(j)p Fu(u)p Fy(j)1318 1702 y Fx(are)i(pre\014xes)e(of)i Fv(!)s Fx(.)p 3371 1702 4 62 v 3375 1644 55 4 v 3375 1702 V 3429 1702 4 62 v 589 1890 a(W)-8 b(e)34 b(omit)g(the)f(pro)s(of) e(that)j(Lemmas)e(3)h(and)f(4)h(imply)g(Theorem)f(7.)48 b(This)32 b(pro)s(of)448 2002 y(is)44 b(rather)g(straigh)m(tforw)m (ard.)81 b(F)-8 b(or)44 b(the)g(\014rst)f(statemen)m(t)j(of)e(the)f (theorem)h(w)m(e)h(\014x)448 2115 y(Alice's)40 b(strategy)g(in)e(2)1293 2066 y Fl(~)1266 2082 y Fu(M)8 b Fl(\()p Fu(!)r Fl(\)+1)1537 2115 y Fv(;)15 b Fx(2)1622 2082 y Fl(2\()1712 2066 y(~)1684 2082 y Fu(M)9 b Fl(\()p Fu(!)r Fl(\)+1\))1984 2115 y Fx(-game)39 b(\(she)g(includes)f(in)g Fv(T)52 b Fx(all)39 b Fv(x)f Fx(with)448 2228 y Fv(C)7 b Fx(\()p Fv(x)p Fs(j)p Fv(n)p Fx(\))26 b Fj(6)877 2205 y Fx(~)843 2228 y Fv(M)11 b Fx(\()p Fv(!)s Fx(\),)24 b(here)e Fv(n)g Fx(stands)g(for)g(the)g (length)h(of)f Fv(x)p 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y(let)k Fv(c)25 b Fx(=)g(1)p Fv(=p)p Fx(.)589 3618 y(It)34 b(remains)f(to)i(explain)f (ho)m(w)f(to)i(implemen)m(t)f(the)f(random)g(c)m(hoice)j(of)d(a)h (proba-)448 3731 y(bilistic)28 b(mac)m(hine.)40 b(En)m(umerate)27 b(all)h(the)e(probabilistic)i(mac)m(hines)f(in)f(a)h(natural)g(w)m(a)m (y;)448 3844 y(let)32 b Fv(M)668 3858 y Fl(0)707 3844 y Fv(;)15 b(M)835 3858 y Fl(1)875 3844 y Fv(;)g(M)1003 3858 y Fl(2)1043 3844 y Fv(;)g(:)g(:)g(:)47 b Fx(b)s(e)30 b(the)g(resulting)h(sequence.)41 b(W)-8 b(e)31 b(toss)g(a)g(coin)g(un)m (til)f(the)h(\014rst)448 3957 y(1)c(app)s(ears.)38 b(Then)26 b(w)m(e)g(sim)m(ulate)i(the)e(mac)m(hine)h Fv(M)2203 3971 y Fu(i)2257 3957 y Fx(where)f Fv(i)h Fx(is)f(the)g(n)m(um)m(b)s (er)f(of)h(zeros)448 4070 y(preceding)31 b(the)f(\014rst)g(1.)p 3371 4070 4 62 v 3375 4012 55 4 v 3375 4070 V 3429 4070 4 62 v 589 4257 a(Fix)39 b(an)m(y)f(maximal)g(lo)m(w)m(er)h (semi-computable)g(semimeasure)f Fv(p)2852 4271 y Fl(0)2891 4257 y Fv(;)15 b(p)2977 4271 y Fl(1)3016 4257 y Fv(;)g(p)3102 4271 y Fl(2)3142 4257 y Fv(;)g(:)g(:)g(:)54 b Fx(on)448 4370 y(natural)33 b(n)m(um)m(b)s(ers.)45 b(W)-8 b(e)34 b(will)e(use)g(the)h(notation)h Fv(m)p Fx(\()p Fv(i)p Fx(\))f(for)f Fv(p)2594 4384 y Fu(i)2654 4370 y Fx(and)g(the)h (notation)g Fv(m)p 448 4454 1196 4 v 552 4508 a Fk(6)587 4540 y Fp(The)c(w)n(ord)g(\\semimeasure")j(ma)n(y)d(lo)r(ok)g(strange,) i(but)e(unfortunately)g(there)f(is)i(no)f(other)g(ap-)448 4631 y(propriate)g(term)f(in)g(the)g(literature.)43 b(Dropping)28 b(semi-computabilit)n(y)h(requiremen)n(t,)f(one)h(can)f(call)448 4723 y(an)n(y)h(function)h Fh(i)f Fe(7!)f Fh(p)1101 4731 y Fi(i)1156 4723 y Fp(with)1335 4667 y Fc(P)1416 4745 y Fi(i)1455 4723 y Fh(p)1494 4731 y Fi(i)1548 4723 y Fb(6)g Fp(1)i(a)g Fg(semime)l(asur)l(e)37 b Fp(on)30 b Fa(N)p Fp(.)46 b(Ev)n(ery)29 b(semimeasure)i(on)f Fa(N)g Fp(de-)448 4814 y(\014nes)e(a)g(probabilit)n(y)g(distribution)g(on)g (the)g(set)g Fa(N)18 b Fe([)h(f?g)28 b Fp(where)h Fe(?)e Fp(is)i(a)f(sp)r(ecial)h(sym)n(b)r(ol)g(meaning)448 4905 y(\\unde\014ned".)34 b(The)26 b(probabilit)n(y)f(of)i(the)e(n)n(um)n(b) r(er)g Fh(i)h Fp(is)g Fh(p)2123 4913 y Fi(i)2174 4905 y Fp(and)g(the)f(probabilit)n(y)h(of)g Fe(?)f Fp(is)h(1)18 b Fe(\000)3231 4849 y Fc(P)3312 4928 y Fi(i)3351 4905 y Fh(p)3390 4913 y Fi(i)3416 4905 y Fp(.)1897 5225 y Fx(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 448 573 a Fx(for)32 b(the)h(semimeasure)f (itself.)47 b(The)31 b(v)-5 b(alue)33 b Fv(m)p Fx(\()p Fv(i)p Fx(\))g(is)f(called)h(the)g Ft(a)h(priori)i(pr)-5 b(ob)g(ability)448 686 y Fx(of)24 b Fv(i)p Fx(.)601 653 y Fl(7)680 686 y Fx(Here)g(is)g(an)g(explanation)h(of)f(this)g(term.)39 b(Assume)23 b(that)i(w)m(e)f(are)h(giv)m(en)g(a)f(device)448 799 y(\(a)37 b(blac)m(k)g(b)s(o)m(x\))f(that)h(after)f(b)s(eing)g (turned)f(on)h(pro)s(duces)e(a)j(natural)f(n)m(um)m(b)s(er.)56 b(F)-8 b(or)448 912 y(eac)m(h)28 b Fv(i)f Fx(w)m(e)g(w)m(an)m(t)g(to)g 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b(e\014x-stable)p Fx(,)32 b(if)f(the)f(follo)m(wing)i(holds)e(for)g (all)h(strings)g Fv(x)p Fx(,)f Fv(y)s Fx(:)676 2959 y Fv(f)10 b Fx(\()p Fv(x)p Fx(\))28 b(is)h(de\014ned)f(and)g Fv(x)h Fx(is)g(a)g(pre\014x)f(of)h Fv(y)89 b Fs(\))d Fv(f)10 b Fx(\()p Fv(y)s Fx(\))29 b(is)g(de\014ned)e(and)i(is)676 3072 y(equal)h(to)i Fv(f)10 b Fx(\()p Fv(x)p Fx(\).)448 3243 y Fm(Theorem)49 b(9.)f Ft(Ther)-5 b(e)45 b(exists)f(an)g(optimal)h (pr)-5 b(e\014x-stable)45 b(de)-5 b(c)g(ompr)g(essor)47 b(\(for)e(the)448 3356 y(family)34 b(of)f(al)5 b(l)33 b(pr)-5 b(e\014x-stable)34 b(de)-5 b(c)g(ompr)g(essors\).)589 3527 y Fx(W)d(e)46 b(omit)e(the)g(pro)s(of)g(of)g(this)g(theorem.)81 b(Let)45 b(us)e(\014x)h(some)g(optimal)h(pre\014x-)448 3640 y(stable)29 b(decompressor)e Fv(D)j Fx(and)d(let)i Fv(K)7 b Fx(\()p Fv(x)p Fx(\))28 b(denote)g Fv(C)2265 3654 y Fu(D)2329 3640 y Fx(\()p Fv(x)p Fx(\).)40 b(W)-8 b(e)29 b(call)g Fv(K)7 b Fx(\()p Fv(x)p Fx(\))28 b(the)g Ft(pr)-5 b(e\014x)448 3753 y(c)g(omplexity)47 b 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Fx(and)h Fv(D)s Fx(\()p Fv(p)p Fx(\))g(is)h(de\014ned,)e(and)h Fv(q)j Fx(is)d(arbitrary)g(string.)3022 766 y Fl(8)3101 799 y Fx(F)-8 b(or)29 b(suc)m(h)448 912 y(inputs)k(w)m(e)h(let)h Fv(D)1080 879 y Fy(0)1103 912 y Fx(\()5 b(^)-50 b Fv(npq)s Fx(\))31 b(=)g Fv(D)s Fx(\()p Fv(p)p Fx(\).)51 b(The)33 b(function)g Fv(D)2348 879 y Fy(0)2405 912 y Fx(is)h(w)m(ell)h (de\014ned:)46 b(if)34 b(a)g(string)448 1024 y(has)i(t)m(w)m(o)h (di\013eren)m(t)f(represen)m(tations)41 b(^)-49 b Fv(n)o(pq)37 b Fx(=)2089 1000 y Fd(b)2076 1024 y Fv(n)2131 998 y Fy(0)2153 1024 y Fv(p)2199 992 y Fy(0)2222 1024 y Fv(q)2266 992 y Fy(0)2325 1024 y Fx(where)e Fv(n)f Fx(=)g Fs(j)p Fv(p)p Fs(j)i Fx(and)f Fv(n)3156 992 y Fy(0)3213 1024 y Fx(=)f Fs(j)p Fv(p)3389 992 y Fy(0)3412 1024 y Fs(j)448 1137 y Fx(then)c Fv(n)25 b Fx(=)g Fv(n)886 1104 y Fy(0)939 1137 y Fx(and)30 b(th)m(us)g Fv(pq)e Fx(=)d Fv(p)1573 1104 y Fy(0)1596 1137 y Fv(q)1640 1104 y Fy(0)1663 1137 y Fx(.)41 b(The)30 b(latter)h(implies)g(that)g Fv(p)25 b Fx(=)g Fv(p)2882 1104 y Fy(0)2905 1137 y Fx(.)589 1250 y(By)39 b(construction)g Fv(D)1347 1217 y Fy(0)1408 1250 y Fx(is)g(a)g(pre\014x-stable)f(function.)64 b(As)39 b(the)f(length)h(of)k(^)-50 b Fv(np)38 b Fx(is)448 1363 y Fs(j)p Fv(p)p Fs(j)21 b Fx(+)f(2)15 b(log)j Fv(n)i Fx(+)g Fv(O)s Fx(\(1\))31 b(w)m(e)g(ha)m(v)m(e)g Fv(C)1641 1378 y Fu(D)1701 1360 y Fq(0)1728 1363 y Fx(\()p Fv(x)p Fx(\))26 b Fj(6)f Fv(C)2037 1377 y Fu(D)2101 1363 y Fx(\()p Fv(x)p Fx(\))20 b(+)g(2)15 b(log)j Fv(C)2592 1377 y Fu(D)2656 1363 y Fx(\()p Fv(x)p Fx(\))j(+)f Fv(O)s Fx(\(1\).)448 1603 y Fm(Pre\014x)36 b(complexit)m(y)g(and)f(a)f(priori)i(probabilit)m (y)448 1775 y Fx(No)m(w)30 b(w)m(e)g(can)f(state)i(the)e(relation)h(b)s (et)m(w)m(een)g(the)f(a)h(priori)f(probabilit)m(y)g(of)h(a)f(string)g Fv(x)448 1888 y Fx(and)h(its)h(complexit)m(y:)448 2076 y Fm(Theorem)k(10.)42 b Fv(K)7 b Fx(\()p Fv(x)p Fx(\))26 b(=)f Fs(\000)15 b Fx(log)i Fv(m)p Fx(\()p Fv(x)p Fx(\))k(+)f Fv(O)s Fx(\(1\))p Ft(.)589 2263 y Fx(W)-8 b(e)44 b(presen)m(t)f(a)g(sk) m(etc)m(h)h(of)f(pro)s(of)f(of)h(this)g(theorem.)78 b(The)42 b(statemen)m(t)j(of)e(the)448 2376 y(theorem)31 b(is)g(a)f(conjunction) h(of)f(t)m(w)m(o)i(inequalities.)771 2580 y Fs(\000)15 b Fx(log)j Fv(m)p Fx(\()p Fv(x)p Fx(\))26 b Fj(6)e Fv(K)7 b Fx(\()p Fv(x)p Fx(\))21 b(+)f Fv(O)s Fx(\(1\))p Fv(;)198 b(K)7 b Fx(\()p Fv(x)p Fx(\))26 b Fj(6)f Fs(\000)15 b Fx(log)i Fv(m)p Fx(\()p Fv(x)p Fx(\))k(+)f Fv(O)s Fx(\(1\))p Fv(:)448 2785 y Fx(The)33 b(\014rst)f(of)h(them)g(follo)m(ws)h (directly)g(from)e(the)h(fact)h(that)g(the)f(function)g(2)3125 2752 y Fy(\000)p Fu(K)5 b Fl(\()p Fu(x)p Fl(\))3376 2785 y Fx(is)448 2897 y(a)29 b(lo)m(w)m(er)h(semi-computable)f(semimeasure,) h(whic)m(h)e(is)g(easy)h(to)g(v)m(erify)-8 b(.)41 b(The)28 b(pro)s(of)g(of)448 3010 y(the)j(con)m(v)m(erse)h(inequalit)m(y)-8 b(,)32 b(uses)e(a)h(certain)g(game.)589 3123 y(The)41 b(semimeasure)g Fv(m)p Fx(\()p Fv(x)p Fx(\))h(is)f(lo)m(w)m(er)h (semi-computable,)j(so)d(w)m(e)f(can)h(generate)448 3236 y(lo)m(w)m(er)35 b(b)s(ounds)c(for)i Fv(m)p Fx(\()p Fv(x)p Fx(\))g(that)h(con)m(v)m(erge)h(to)f Fv(m)p Fx(\()p Fv(x)p Fx(\))f(but)g(no)g(estimates)i(for)e(the)g(ap-)448 3349 y(pro)m(ximation)25 b(error)e(are)g(giv)m(en.)40 b(The)23 b(larger)h Fv(m)p Fx(\()p Fv(x)p Fx(\))g(is,)h(the)e(smaller)h Fv(K)7 b Fx(\()p Fv(x)p Fx(\))24 b(should)e(b)s(e,)448 3462 y(that)29 b(is,)g(the)g(shorter)f(description)g Fv(p)g Fx(w)m(e)h(ha)m(v)m(e)h(to)f(pro)m(vide)f(for)g Fv(x)p Fx(.)40 b(The)28 b(descriptions)448 3575 y(reserv)m(ed)c(for)e (di\013eren)m(t)i(strings)f(m)m(ust)g(b)s(e)f(incompatible.)40 b(\(The)22 b(descriptions)h Fv(p)3228 3589 y Fl(1)3291 3575 y Fx(and)448 3688 y Fv(p)494 3702 y Fl(2)560 3688 y Fx(are)k(incompatible)g(if)g(the)f(in)m(terv)-5 b(als)28 b(\012)1903 3702 y Fu(p)1939 3711 y Fk(1)2003 3688 y Fx(and)e(\012)2242 3702 y Fu(p)2278 3711 y Fk(2)2342 3688 y Fx(are)h(disjoin)m(t.)40 b(Recall)28 b(that)f(the)448 3801 y(in)m(terv)-5 b(al)34 b(\012)845 3815 y Fu(p)917 3801 y Fx(consists)e(of)h(all)g(in\014nite)f(binary)g(sequences)h(b)s (eginning)f(with)g Fv(p)p Fx(.\))47 b(The)448 3914 y(inequalit)m(y)39 b Fs(j)p Fv(p)p Fs(j)d Fj(6)g Fs(\000)15 b Fx(log)1319 3935 y Fl(2)1374 3914 y Fv(m)p Fx(\()p Fv(x)p Fx(\))37 b(means)g(that)h(the)f(measure)f(of)i(the)f(in)m(terv)-5 b(al)38 b(\012)3300 3928 y Fu(p)3376 3914 y Fx(is)448 4027 y(at)e(least)g Fv(m)p Fx(\()p Fv(x)p Fx(\):)50 b(2)1103 3994 y Fy(\000j)p Fu(p)p Fy(j)1270 4027 y Fj(>)33 b Fv(m)p Fx(\()p Fv(x)p Fx(\).)54 b(Th)m(us)34 b(w)m(e)h(ha)m(v)m(e)h(to)g (assign)f(to)h(ev)m(ery)g(string)e Fv(x)h Fx(an)448 4139 y(in)m(terv)-5 b(al)33 b(of)f(measure)g(at)g(least)i Fv(m)p Fx(\()p Fv(x)p Fx(\))e(so)g(that)g(the)g(in)m(terv)-5 b(als)33 b(assigned)f(to)h(di\013eren)m(t)448 4252 y(strings)42 b(do)h(not)f(o)m(v)m(erlap.)78 b(Actually)-8 b(,)47 b(it)c(su\016ces)f (to)h(reserv)m(e)g(an)f(in)m(terv)-5 b(al)43 b(of)g(the)448 4365 y(length)35 b Fv("m)p Fx(\()p Fv(x)p Fx(\))h(rather)e(than)h Fv(m)p Fx(\()p Fv(x)p Fx(\),)h(for)f(some)g(\014xed)f(p)s(ositiv)m(e)i Fv(")p Fx(.)54 b(This)33 b(relaxation)448 4478 y(causes)e(the)g (complexit)m(y)h(increase)f(at)g(most)g(b)m(y)f(a)h(constan)m(t.)p 448 4557 1196 4 v 552 4611 a Fk(8)587 4643 y Fp(Notation)24 b(^)-42 b Fh(n)20 b Fp(refers)h(to)f(the)g(\\self)h(delimiting")g (description)g(of)f(a)g(natural)h(n)n(um)n(b)r(er)d Fh(n)j Fp(in)e(2)14 b(log)h Fh(n)5 b Fp(+)448 4734 y Fh(O)r Fp(\(1\))27 b(bits.)1897 5225 y Fx(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 448 573 a Fm(The)35 b(space)g(allo)s(cation)h (game)448 744 y Fx(So)f(w)m(e)g(arriv)m(e)h(to)f(the)g(follo)m(wing)h (game)g(b)s(et)m(w)m(een)f(t)m(w)m(o)h(pla)m(y)m(ers,)h(called)f(Clien) m(t)g(and)448 857 y(Serv)m(er.)k(Fix)28 b(a)g(p)s(ositiv)m(e)h (rational)f(n)m(um)m(b)s(er)f Fv(")p Fx(.)40 b(Clien)m(t)28 b(de\014nes)f(an)g(in\014nite)h(sequence)448 970 y Fv(a)p Fx(\(1\))p Fv(;)15 b(a)p Fx(\(2\))p Fv(;)g(:)g(:)g(:)58 b Fx(of)38 b(non-negativ)m(e)i(reals.)65 b(A)m(t)39 b(the)g(start)f(of) h(the)f(game)h(all)g Fv(a)p Fx(\()p Fv(i)p Fx(\))h(are)448 1083 y(zero)s(es.)h(In)30 b(her)f(turn)f(Clien)m(t)j(ma)m(y)g(increase) f(an)m(y)g Fv(a)p Fx(\()p Fv(i)p Fx(\))h(b)m(y)f(a)g(rational)h(v)-5 b(alue)30 b(so)g(that)448 1128 y Fd(P)544 1223 y Fu(i)588 1196 y Fv(a)p Fx(\()p Fv(i)p Fx(\))i Fj(6)f Fx(1)j(\(if)h(after)f(some) h(Clien)m(t's)g(mo)m(v)m(e)g(this)f(sum)f(b)s(ecomes)h(greater)i(than)d (1,)448 1309 y(she)d(lo)s(oses)i(immediately\).)589 1422 y(In)37 b(his)g(turn)g(Serv)m(er)g(de\014nes)g(a)g(mapping)g Fv(h)h Fx(:)f Fs(f)p Fx(0)p Fv(;)15 b Fx(1)p Fs(g)2495 1389 y Fy(\003)2574 1422 y Fs(!)37 b(f)p Fx(1)p Fv(;)15 b Fx(2)p Fv(;)g(:)g(:)g(:)j Fs(g)p Fx(.)62 b(A)m(t)39 b(the)448 1535 y(start)25 b(of)g(the)f(game)h Fv(h)g Fx(is)f(unde\014ned)e(on)i(all)i(argumen)m(ts.)39 b(On)23 b(eac)m(h)j(mo)m(v)m(e)f(Serv)m(er)f(ma)m(y)448 1648 y(de\014ne)29 b Fv(h)h Fx(on)f(a)h(new)f(argumen)m(t)h(\(previously)g (de\014ned)e(v)-5 b(alues)30 b(cannot)g(b)s(e)f(c)m(hanged\).)448 1761 y(If)d Fv(h)p Fx(\()p Fv(p)p Fx(\))g(=)f Fv(i)i Fx(then)f(w)m(e)g(sa)m(y)h(that)g(Serv)m(er)f(has)g(allo)s(cated)j (\012)2442 1775 y Fu(p)2507 1761 y Fx(to)e Fv(i)p Fx(th)g(job)e(and)h (the)h(v)-5 b(alue)448 1874 y Fv(a)p Fx(\()p Fv(i)p Fx(\))32 b(will)e(b)s(e)g(called)i(space)f(request)f(for)g Fv(i)p Fx(th)h(job.)589 1986 y(If)23 b(in)m(terv)-5 b(als)24 b(\012)1096 2000 y Fu(p)1159 1986 y Fx(and)e(\012)1394 2000 y Fu(q)1455 1986 y Fx(are)i(allo)s(cated)h(to)f(di\013eren)m(t)g (jobs)e(then)h(\012)2883 2000 y Fu(p)2946 1986 y Fx(and)f(\012)3181 2000 y Fu(q)3242 1986 y Fx(m)m(ust)448 2099 y(b)s(e)30 b(disjoin)m(t)h(\(otherwise)g(Serv)m(er)f(lo)s(oses)h(immediately)h (after)f(that)g(has)f(happ)s(ened\).)589 2212 y(Pla)m(y)m(ers)f(mak)m (e)g(in)f(turn)e(in\014nitely)i(man)m(y)g(mo)m(v)m(es.)41 b(Serv)m(er)28 b(wins)f(the)h(game)g(if)g(\(at)448 2325 y(the)34 b(end)e(of)h(the)h(game\))g(for)f(ev)m(ery)h Fv(i)f Fx(he)g(has)g(allo)s(cated)j(to)d Fv(i)p Fx(th)h(job)e(an)h(in)m (terv)-5 b(al)35 b(\012)3398 2339 y Fu(p)448 2438 y Fx(of)f(measure)f (at)i(least)f Fv("a)p Fx(\()p Fv(i)p Fx(\).)52 b(Let)34 b(us)f(stress)g(that)h(Clien)m(t)h(is)e(in)m(terested)i(not)f(in)f(the) 448 2551 y(total)i(space)f(allo)s(cated)g(to)g(ev)m(ery)g(job,)f(but)g (in)f(the)h(largest)i(in)m(terv)-5 b(al,)35 b(whic)m(h)d(mak)m(es)448 2664 y(\\space)g(managemen)m(t")g(job)e(di\016cult.)589 2777 y(Clearly)-8 b(,)31 b(Clien)m(t)e(wins)f(this)h(\\space)h(allo)s (cation")i(game)d(pro)m(vided)g Fv(")c(>)g Fx(1.)41 b(More-)448 2890 y(o)m(v)m(er,)g(Clien)m(t)e(wins)e(if)g Fv(")h(>)f Fx(1)p Fv(=)p Fx(2.)64 b(Let)38 b(us)f(sho)m(w)g(this,)j(sa)m(y)-8 b(,)41 b(for)c Fv(")h Fx(=)f(9)p Fv(=)p Fx(16.)64 b(Clien)m(t)448 3003 y(requests)26 b(1)p Fv(=)p Fx(9)11 b(+)g Fv(\016)29 b Fx(of)d(space)g(for)f(eac)m(h)i(of)f(the)g(\014rst)e(8)i(jobs,)h (where)e Fv(\016)k Fx(a)d(small)g(n)m(um)m(b)s(er.)448 3116 y(Serv)m(er)35 b(has)f(to)h(allo)s(cate)i(an)d(in)m(terv)-5 b(al)35 b(of)g(measure)f Fv(")p Fx(\(1)p Fv(=)p Fx(9)25 b(+)e Fv(\016)s Fx(\))33 b Fv(>)e Fx(1)p Fv(=)p Fx(16)36 b(to)f(eac)m(h)h(of)448 3228 y(them.)j(So)25 b(eac)m(h)h(of)g(8)f(jobs) g(is)g(allo)s(cated)i(an)e(in)m(terv)-5 b(al)26 b(of)g(measure)f(1/8,)i (whic)m(h)e(means)448 3341 y(the)i(en)m(tire)h(space)g(\012)e(is)h (exhausted.)39 b(If)27 b Fv(\016)j Fx(is)d(small)h(enough,)f(the)g(sum) f(of)h(all)h(requests)448 3454 y(is)h(still)g(less)g(than)f(1)g(and)g (Clien)m(t)h(requests)f(a)h(small)g(amoun)m(t)g(of)f(space)h(for)f(the) h(nin)m(th)448 3567 y(job,)i(whic)m(h)f(Serv)m(er)g(is)g(unable)g(to)h (allo)s(cate.)589 3680 y(In)f(this)g(example)h(Clien)m(t)g(has)f(not)g (used)g(her)f(righ)m(t)i(to)g(increase)g(requests)f(man)m(y)448 3793 y(times.)47 b(Using)32 b(this)g(option)g(Clien)m(t)h(can)f(win)g (the)g(game)h(for)f Fv(")c Fx(=)g(1)p Fv(=)p Fx(2)33 b(and)f(ev)m(en)g(for)448 3906 y(some)f Fv(")26 b(<)f Fx(1)p Fv(=)p Fx(2.)42 b(Ho)m(w)m(ev)m(er)32 b(for)e Fv(")c Fx(=)f(1)p Fv(=)p Fx(4)32 b(the)e(game)h(is)g(w)m(on)f(b)m(y)h (Serv)m(er.)448 4077 y Fm(Lemma)g(6.)40 b Ft(Server)29 b(has)i(a)f(c)-5 b(omputable)32 b(winning)e(str)-5 b(ate)g(gy)31 b(in)f(the)g(describ)-5 b(e)g(d)31 b(game)448 4190 y(for)j Fv(")25 b Fx(=)g(1)p Fv(=)p Fx(4)p Ft(.)448 4362 y(Pr)-5 b(o)g(of.)733 4329 y Fl(9)807 4362 y Fx(The)34 b(main)g(idea)h(is)g(as) f(follo)m(ws:)50 b(Serv)m(er)35 b(tak)m(es)h(in)m(to)f(accoun)m(t)h (only)f(those)448 4474 y(increases)c(of)g(requests)g(when)e Fv(a)p Fx(\()p Fv(i)p Fx(\))i(b)s(ecomes)g(greater)h(than)e(a)h(n)m(um) m(b)s(er)e(of)i(the)g(form)448 4587 y(2)493 4554 y Fy(\000)p Fu(j)585 4587 y Fx(.)40 b(If)29 b(on)g(step)h Fv(t)f Fx(certain)h Fv(a)p Fx(\()p Fv(i)p Fx(\))g(b)s(ecomes)g(greater)g(than) f(2)2524 4554 y Fy(\000)p Fu(j)2645 4587 y Fx(w)m(e)h(allo)s(cate)h(to) f(clien)m(t)448 4700 y Fv(i)37 b Fx(a)g(fresh)f(in)m(terv)-5 b(al)38 b(\(an)f(in)m(terv)-5 b(al)38 b(that)f(is)g(disjoin)m(t)g(with) f(all)i(previously)e(allo)s(cated)448 4813 y(in)m(terv)-5 b(als\))30 b(of)f(measure)g Fv(b)1340 4827 y Fu(t)1395 4813 y Fx(=)c(2)1536 4780 y Fy(\000)p Fu(j)t Fy(\000)p Fl(1)1718 4813 y Fx(.)40 b(\(If)29 b(on)f(step)h Fv(t)g Fx(no)f Fv(a)p Fx(\()p Fv(i)p Fx(\))i(b)s(ecomes)f(greater)h(than)p 448 4891 1196 4 v 552 4945 a Fk(9)587 4976 y Fp(The)25 b(pro)r(of)i(is)f(similar)i(to)e(the)f(pro)r(of)i(of)f(the)f(so)i (called)f(Kraft{Chaitin)h(lemma,)g(see)f([4)q(].)1897 5225 y Fx(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 448 573 a Fx(a)31 b(n)m(um)m(b)s(er)d(of)i(the) g(form)f(2)1372 540 y Fy(\000)p Fu(j)1464 573 y Fx(,)h(then)g(w)m(e)g (let)h Fv(b)2029 587 y Fu(t)2084 573 y Fx(=)25 b(0)30 b(and)f(do)h(not)g(allo)s(cate)i(an)m(y)e(space)448 686 y(on)h(that)g(step.\))589 799 y(Before)37 b(to)f(describ)s(e)f(the)h (allo)s(cation)h(strategy)g(note)f(that)h(the)e(total)j(space)e(w)m(e) 448 912 y(will)31 b(allo)s(cate)i(to)e(job)f Fv(i)h Fx(do)s(es)f(not)g (exceed)i(the)e(sum)1105 1101 y(2)1150 1064 y Fy(\000)p Fu(j)t Fy(\000)p Fl(1)1352 1101 y Fx(+)20 b(2)1488 1064 y Fy(\000)p Fu(j)t Fy(\000)p Fl(2)1691 1101 y Fx(+)f(2)1826 1064 y Fy(\000)p Fu(j)t Fy(\000)p Fl(3)2029 1101 y Fx(+)h Fs(\001)15 b(\001)g(\001)26 b Fx(=)f(2)2392 1064 y Fy(\000)p Fu(j)2509 1101 y Fv(<)g(a)p Fx(\()p Fv(i)p Fx(\))p Fv(;)448 1291 y Fx(where)34 b Fv(j)40 b Fx(is)34 b(the)h(in)m(teger)g(n)m(um)m (b)s(ers)e(with)h(2)1979 1258 y Fy(\000)p Fu(j)2103 1291 y Fv(<)d(a)p Fx(\()p Fv(i)p Fx(\))i Fj(6)f Fx(2)2535 1258 y Fy(\000)p Fu(j)t Fl(+1)2717 1291 y Fx(.)52 b(This)33 b(sho)m(ws)h(that)448 1336 y Fd(P)559 1404 y Fv(b)598 1418 y Fu(t)678 1404 y Fj(6)50 b Fx(1)45 b(pro)m(vided)1278 1336 y Fd(P)1374 1431 y Fu(i)1417 1404 y Fv(a)p Fx(\()p Fv(i)p Fx(\))51 b Fj(6)f Fx(1)45 b(\(Serv)m(er)h(will)f(not)g(run)f (out)i(of)f(space\).)86 b(In)448 1517 y(particular,)31 b(at)g(an)m(y)g(momen)m(t)g(of)f(the)h(game,)g(the)g(total)h(measure)e Fv(b)2816 1531 y Fu(t)p Fl(+1)2956 1517 y Fx(+)19 b Fv(b)3085 1531 y Fu(t)p 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b(of)g(free)g(in)m (terv)-5 b(als)35 b(is)f(at)h(least)g Fv(w)i Fx(pro)m(vided)2364 2352 y Fd(P)2460 2447 y Fu(i)2503 2420 y Fv(a)p Fx(\()p Fv(i)p Fx(\))c Fj(6)e Fx(1.)51 b(If)34 b(all)h(the)f(free)448 2533 y(in)m(terv)-5 b(als)40 b(had)e(measures)g(smaller)i(than)e Fv(w)r Fx(,)k(their)c(total)j(measure)d(w)m(ould)h(b)s(e)f(less)448 2646 y(than)c Fv(w)i Fx(since)e(they)g(ha)m(v)m(e)g(di\013eren)m(t)h (measures)e(and)g(the)h(sum)f(of)g(p)s(o)m(w)m(ers)h(of)g(2)g(less)448 2759 y(than)c Fv(w)j Fx(is)e(less)f(than)g Fv(w)r Fx(.)589 2872 y(If)41 b(there)h(is)g(a)g(free)f(in)m(terv)-5 b(al)43 b(in)e(the)h(list)g(that)g(has)f(measure)g(exactly)j Fv(w)r Fx(,)h(our)448 2985 y(task)32 b(is)f(simple.)42 b(W)-8 b(e)32 b(just)e(allo)s(cate)j(this)e(in)m(terv)-5 b(al)32 b(and)e(delete)j(it)e(from)f(the)h(free)g(list)448 3097 y(\(main)m(taining)h(the)f(in)m(v)-5 b(arian)m(t)31 b(relation\).)589 3210 y(Assume)37 b(that)g(this)f(is)h(not)g(the)g (case.)60 b(Then)36 b(w)m(e)h(ha)m(v)m(e)h(some)f(in)m(terv)-5 b(als)38 b(in)e(the)448 3323 y(list)c(that)g(are)f(bigger)h(than)f (requested.)42 b(Using)32 b(the)f(b)s(est)g(\014t)f(strategy)-8 b(,)34 b(w)m(e)d(tak)m(e)i(the)448 3436 y(smallest)28 b(among)f(these)g(in)m(terv)-5 b(als.)41 b(Let)27 b Fv(w)1937 3403 y Fy(0)1986 3436 y Fv(>)e(w)k Fx(b)s(e)d(its)h(length.)40 b(Then)25 b(w)m(e)i(split)g(free)448 3549 y(in)m(terv)-5 b(al)39 b(of)e(measure)g Fv(w)1322 3516 y Fy(0)1383 3549 y Fx(in)m(to)i(in)m(terv)-5 b(als)38 b(of)g(measure)f Fv(w)r(;)15 b(w)r(;)g Fx(2)p Fv(w)r(;)g Fx(4)p Fv(w)r(;)g Fx(8)p Fv(w)r(;)g(:)g(:)h(:)21 b(;)15 b(w)3322 3516 y Fy(0)3346 3549 y Fv(=)p Fx(2)448 3662 y(\(note)34 b(that)e Fv(w)24 b Fx(+)e Fv(w)i Fx(+)d(2)p Fv(w)j Fx(+)d(4)p Fv(w)k Fx(+)c(8)p Fv(w)j Fx(+)e Fv(:)15 b(:)g(:)22 b Fx(+)f Fv(w)2215 3629 y Fy(0)2239 3662 y Fv(=)p Fx(2)30 b(=)e Fv(w)2525 3629 y Fy(0)2548 3662 y Fx(.)47 b(The)32 b(\014rst)f(in)m(terv)-5 b(al)34 b(\(of)448 3775 y(measure)26 b Fv(w)r Fx(\))g(is)g(allo)s(cated,)j(all)d(the)g(other)g(in)m(terv)-5 b(als)27 b(are)f(added)f(to)h(the)g(free)g(list.)40 b(W)-8 b(e)448 3888 y(ha)m(v)m(e)38 b(to)f(c)m(hec)m(k)h(out)e(the)h(in)m(v)-5 b(arian)m(t)37 b(relation:)54 b(all)37 b(new)f(in)m(terv)-5 b(als)37 b(in)f(the)h(list)g(ha)m(v)m(e)448 4001 y(di\013eren)m(t)30 b(measures)e(starting)i(with)e Fv(w)j Fx(and)d(up)g(to)i Fv(w)2315 3968 y Fy(0)2338 4001 y Fv(=)p Fx(2;)h(old)e(free)g(in)m (terv)-5 b(als)29 b(cannot)448 4114 y(ha)m(v)m(e)j(this)e(measure)g (since)h Fv(w)1479 4081 y Fy(0)1533 4114 y Fx(w)m(as)g(the)g(b)s(est)f (\014t)g(in)g(the)g(list.)589 4227 y(Let)h(us)e(pro)m(v)m(e)i(that)g (Serv)m(er)f(wins.)40 b(Let)30 b Fv(a)p Fx(\()p Fv(i)p Fx(\))h(denote)g(the)f(space)h(request)f(for)g Fv(i)p Fx(th)448 4339 y(job)35 b(at)h(the)g(end)f(of)g(the)h(game.)56 b(Assume)35 b(that)h Fv(a)p Fx(\()p Fv(i)p Fx(\))g(is)g(p)s(ositiv)m(e) g(and)f(let)h Fv(j)41 b Fx(b)s(e)35 b(the)448 4452 y(in)m(teger)h(n)m (um)m(b)s(ers)d(with)h(2)1380 4419 y Fy(\000)p Fu(j)1504 4452 y Fv(<)e(a)p Fx(\()p Fv(i)p Fx(\))h Fj(6)f Fx(2)1937 4419 y Fy(\000)p Fu(j)t Fl(+1)2119 4452 y Fx(.)53 b(A)m(t)35 b(some)g(time)g(in)f(the)h(game)g Fv(a)p Fx(\()p Fv(i)p Fx(\))448 4565 y(b)s(ecomes)j(larger)f(than)g(2)1345 4532 y Fy(\000)p Fu(j)1474 4565 y Fx(and)g(Serv)m(er)g(then)f(allo)s (cates)k(an)d(in)m(terv)-5 b(al)38 b(of)f(measure)448 4678 y(2)493 4645 y Fy(\000)p Fu(j)t Fy(\000)p Fl(1)701 4678 y Fj(>)25 b Fv(a)p Fx(\()p Fv(i)p Fx(\))p Fv(=)p Fx(4)32 b(to)f Fv(i)p Fx(th)f(job.)p 3371 4678 4 62 v 3375 4620 55 4 v 3375 4678 V 3429 4678 4 62 v 589 4863 a(Let)37 b(us)e(\014nish)f(the)i(pro)s(of)f(of)h(Theorem)g(10.)57 b(Let)37 b(Clien)m(t)f(lo)m(w)m(er)h(semi-compute)448 4976 y(the)42 b(a)f(priori)g(probabilit)m(y)h Fv(m)p Fx(\()p Fv(i)p Fx(\))g(so)f(that)h(after)g(an)m(y)g(her)e(mo)m(v)m(e)j Fv(a)p Fx(\()p Fv(i)p Fx(\))f(is)g(equal)f(to)1897 5225 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 448 573 a Fx(the)39 b(b)s(est)e(lo)m(w)m(er)j (b)s(ound)c(of)i Fv(m)p Fx(\()p Fv(i)p Fx(\))h(disco)m(v)m(ered)g(up)e (to)i(that)g(time.)65 b(Let)38 b(Serv)m(er)g(use)448 686 y(the)29 b(computable)f(strategy)i(of)e(Lemma.)40 b(Then)27 b(he)h(de\014nes)f(a)i(computable)f(function)448 799 y Fv(h)p Fx(\()p Fv(p)p Fx(\))36 b(suc)m(h)e(that)h Fv(C)1127 814 y Fu(h)1171 799 y Fx(\()p Fv(i)p Fx(\))f Fj(6)d Fs(\000)15 b Fx(log)j Fv(m)p Fx(\()p Fv(i)p Fx(\))23 b(+)g Fv(O)s Fx(\(1\).)54 b(That)34 b(function)g(can)h(b)s(e)f (extended)448 912 y(to)h(a)e(computable)i(pre\014x)d(stable)i (decompressor)g Fv(D)i Fx(b)m(y)d(letting)i Fv(D)s Fx(\()p Fv(pq)s Fx(\))c(=)f Fv(h)p Fx(\()p Fv(p)p Fx(\))k(for)448 1024 y(all)e Fv(p;)15 b(q)s Fx(.)40 b(Th)m(us)30 b(w)m(e)g(ha)m(v)m(e)i Fv(K)7 b Fx(\()p Fv(i)p Fx(\))26 b Fj(6)f Fv(C)1716 1038 y Fu(D)1780 1024 y Fx(\()p Fv(i)p Fx(\))c(+)f Fv(O)s Fx(\(1\))26 b Fj(6)f Fs(\000)15 b Fx(log)j Fv(m)p Fx(\()p Fv(i)p Fx(\))j(+)f Fv(O)s Fx(\(1\).)448 1311 y Fw(7)135 b(Solo)l(v)-7 b(a)l(y's)46 b(theorem)g(and)f(\\cats)h(vs.)60 b(an)l(ts")46 b(game)448 1514 y Fm(Pre\014x)36 b(complexit)m(y)g(and)f (a)f(priori)i(probabilit)m(y)f(of)g(en)m(umerable)g(sets)448 1685 y Fx(Kolmogoro)m(v)d(complexit)m(y)g(\(as)f(w)m(ell)g(as)f (pre\014x)f(complexit)m(y)j(and)d(a)i(priori)e(probabil-)448 1798 y(it)m(y\))j(can)g(b)s(e)e(de\014ned)f(for)i(algorithm)h(problems) e(of)h(a)g(v)m(ery)g(general)h(t)m(yp)s(e.)42 b(Roughly)448 1911 y(sp)s(eaking,)29 b(Kolmogoro)m(v)h(complexit)m(y)h(of)d(an)g (algorithmic)i(problem)e(is)g(the)h(minimal)448 2024 y(length)37 b(of)g(a)g(program)g(that)g(solv)m(es)h(that)f(problem,)h (with)e(resp)s(ect)h(to)h(an)e(optimal)448 2137 y(programming)f (language.)55 b(F)-8 b(rom)36 b(this)f(viewp)s(oin)m(t)g(the)g(plain)f (complexit)m(y)j Fv(C)7 b Fx(\()p Fv(x)p Fx(\))35 b(is)448 2250 y(the)c(complexit)m(y)h(of)f(the)f(algorithmic)i(problem)e(\\prin) m(t)g Fv(x)p Fx(".)589 2363 y(In)h(this)h(section,)h(w)m(e)f(will)g(fo) s(cus)f(on)g(algorithmic)i(problems)e(of)h(the)g(t)m(yp)s(e)f(\\en)m (u-)448 2476 y(merate)45 b(the)g(set)f Fv(A)p Fx(")h(\(where)f Fv(A)h Fx(is)f(a)g(computably)h(en)m(umerable)f(set)h(of)f(natural)448 2589 y(n)m(um)m(b)s(ers\).)f(The)31 b(follo)m(wing)i(de\014nitions)e (are)h(attributed)g(b)m(y)f(Solo)m(v)-5 b(a)m(y)33 b(in)f([12)q(])f(to) i(G.)448 2702 y(Chaitin)e(\(w)m(ee)g(k)m(eep)g(Solo)m(v)-5 b(a)m(y's)33 b(notations\).)589 2815 y(First)k(w)m(e)f(de\014ne)f (pre\014x)g(complexit)m(y)j Fv(I)7 b Fx(\()p Fv(A)p Fx(\))37 b(of)f(the)g(problem)f(\\en)m(umerate)i Fv(A)p Fx(".)448 2927 y(Let)30 b Fv(M)39 b Fx(b)s(e)29 b(an)g(algorithm)h(with)f(one)h (in\014nite)f(input)f(tap)s(e)h(and)g(one)g(in\014nite)g(output)448 3040 y(tap)s(e.)41 b(A)m(t)30 b(the)g(start)g(the)f(input)g(tap)s(e)g (con)m(tains)i(an)e(in\014nite)g(binary)f(string)i Fv(!)i Fx(called)448 3153 y(the)40 b(input)f(to)i Fv(M)10 b Fx(.)69 b(The)39 b(output)h(tap)s(e)g(is)f(empt)m(y)i(at)f(the)g (start.)70 b(W)-8 b(e)41 b(sa)m(y)f(that)h(a)448 3266 y(program)27 b Fv(p)f Ft(enumer)-5 b(ates)35 b Fx(a)28 b(set)f Fv(A)e Fs(\032)g Ff(N)g Fx(=)g Fs(f)p Fx(1)p Fv(;)15 b Fx(2)p Fv(;)g(:)g(:)g(:)j Fs(g)28 b Fx(with)e(resp)s(ect)h (to)g Fv(M)10 b Fx(,)28 b(if)e(in)h(the)448 3379 y(run)e(on)h(ev)m(ery) h(input)f Fv(!)j Fx(extending)d Fv(p)g Fx(the)g(algorithm)i Fv(M)36 b Fx(prin)m(ts)26 b(all)h(the)f(elemen)m(ts)i(of)448 3492 y Fv(A)g Fx(in)g(some)g(order)g(and)f(no)h(other)g(elemen)m(ts.)41 b(W)-8 b(e)29 b(do)f(not)g(require)f Fv(M)38 b Fx(to)29 b(halt)f(in)g(the)448 3605 y(case)i(when)e Fv(A)h Fx(is)g(\014nite.) 1289 3572 y Fl(10)1404 3605 y Fx(Let)g Fv(I)1605 3619 y Fu(M)1684 3605 y Fx(\()p Fv(A)p Fx(\))h(denote)g(the)f(minimal)g (length)g(of)g(a)g(program)448 3718 y(en)m(umerating)i Fv(A)p Fx(.)589 3831 y(There)37 b(is)f(an)h(algorithm)h Fv(M)1598 3845 y Fl(0)1674 3831 y Fx(\(called)g(a)f Ft(universal)47 b Fx(algorithm\))38 b(suc)m(h)e(that)h(for)448 3944 y(ev)m(ery)31 b(other)g(algorithm)g Fv(M)41 b 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b(is)f(a)h(constan)m(t)h Fv(c)e Fx(suc)m(h)h(that)1552 2357 y Fv(c)21 b Fs(\001)f Fv(p)1703 2371 y Fu(M)1771 2380 y Fk(1)1809 2357 y Fx(\()p Fv(A)p Fx(\))27 b Fj(>)e Fv(p)2116 2371 y Fu(M)2194 2357 y Fx(\()p Fv(A)p Fx(\))448 2538 y(for)33 b(all)h Fv(A)c Fs(\032)f Ff(N)p Fx(.)49 b(This)32 b(is)h(pro)m(v)m(ed)g(just)g(as)g(the)g (theorem)h(on)e(existence)j(of)e(maximal)448 2651 y(lo)m(w)m(er)f (semi-computable)f(semimeasure)g(on)f(natural)h(n)m(um)m(b)s(ers.)589 2788 y(Fix)k(an)m(y)e(optimal)i Fv(M)1355 2802 y Fl(1)1428 2788 y Fx(and)e(call)i Fv(m)p Fx(\()p Fv(A)p Fx(\))2027 2732 y Fu(def)2047 2788 y Fx(=)50 b Fv(p)2214 2802 y Fu(M)2282 2811 y Fk(1)2320 2788 y Fx(\()p Fv(A)p Fx(\))35 b(the)e Ft(a)j(priori)45 b Fx(probabilit)m(y)448 2901 y(of)34 b(en)m(umerating)g Fv(A)p Fx(.)51 b(The)33 b(a)g(priori)h (distribution)e(th)m(us)i(dep)s(ends)d(on)j(the)g(c)m(hoice)h(of)448 3014 y(the)i(optimal)g(algorithm)g(and)f(is)g(de\014ned)g(up)f(to)i(a)g (m)m(ultiplicativ)m(e)i(constan)m(t.)60 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Fx(3)p Fs(g)31 b Fx(and)448 4317 y(Bob)g(wins.)589 4430 y(On)21 b(the)h(other)g(hand,)h(it)f(turns)e (out)i(that)h(Alice)g(can)f(win)f(with)g Fv(O)s Fx(\(1)p Fv(=")2987 4397 y Fl(2)3028 4430 y Fx(\))h(cats,)j(and)448 4543 y(her)f(winning)f(strategy)i(is)f(computable.)39 b(Theorem)24 b(11)h(is)f(basically)h(a)f(reform)m(ulation)448 4656 y(of)31 b(the)f(existence)i(of)f(suc)m(h)f(strategy)-8 b(.)448 4841 y Fm(Lemma)50 b(7.)f Ft(A)n(lic)-5 b(e)43 b(has)j(a)f(c)-5 b(omputable)46 b(winning)f(str)-5 b(ate)g(gy)46 b(in)e Fv(";)15 b(O)s Fx(\(1)p Fv(=")3095 4808 y Fl(2)3137 4841 y Fx(\))p Ft(-game.)448 4954 y(\(The)33 b(winning)g(algorithm)i(r) -5 b(e)g(c)g(eives)33 b Fv(")g Ft(as)g(an)g(input.\))1897 5225 y Fx(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 448 573 a Ft(Pr)-5 b(o)g(of.)733 540 y Fl(12)837 573 y Fx(Let)29 b Fv(K)j Fx(=)25 b(1)p Fv(=")p Fx(.)41 b(Alice)30 b(will)f(ha)m(v)m(e)h Fv(K)7 b Fx(\()p Fv(K)24 b Fx(+)17 b(1\))p Fv(=)p Fx(2)30 b(cats.)42 b(After)29 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b(sets)h(of)g(an)m(ts)g(attended) g(b)m(y)g(cats)g(of)g(the)g(same)g(rank)f(are)h(pair-wise)g(dis-)676 1763 y(join)m(t.)40 b(\(As)26 b(there)h(are)g Fv(K)33 b Fx(cats)27 b(of)g(rank)e Fv(K)7 b Fx(,)27 b(this)g(item)g(implies)f (that)h(ev)m(ery)g(an)m(t)676 1876 y(is)j(attended)h(b)m(y)f(a)h(cat)g (of)g(rank)f Fv(K)7 b Fx(.\))559 2064 y(3.)47 b(All)35 b(an)m(ts)h(attended)g(b)m(y)f(a)h(cat)g(are)g(at)g(its)f(reac)m(h)h (\(the)g(cat)h(can)e(mo)m(v)m(e)i(to)f(the)676 2177 y(no)s(de)29 b(where)h(that)h(an)m(t)g(sits\).)559 2364 y(4.)47 b(There)28 b(is)i(a)g(cat)g(in)f(ev)m(ery)h(no)s(de)f Fv(A)h Fx(where)e(the)i (amoun)m(t)g(of)f(an)m(ts)h(is)g(at)g(least)g Fv(")p Fx(.)589 2577 y(A)m(t)24 b(the)e(start)h(all)g Fv(K)7 b Fx(\()p Fv(K)k Fx(+)t(1\))p Fv(=)p Fx(2)24 b(cats)f(are)g(in)f(the)g (no)s(de)g Fs(;)h Fx(and)e(ranks)h(and)g(attended)448 2690 y(an)m(ts)31 b(are)g(assigned)g(so)f(that)h(items)g(1)g(and)f(2)g (b)s(e)g(true.)41 b(Items)30 b(3)h(and)f(4)g(follo)m(w.)589 2803 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Fv(m)p Fx(\()p Fv(A)p Fx(\))i(on)e(all)448 2452 y(p)s(ossible)g(random)f(inputs)g(for) h Fv(t)e Fx(=)g(1)p Fv(;)15 b Fx(2)p Fv(;)g(:)g(:)g(:)45 b Fx(steps.)39 b(Ev)m(ery)27 b(in\014nite)g(sequence)g Fv(!)i Fs(2)24 b Fx(\012)448 2565 y(is)37 b(considered)f(b)m(y)h(Bob)g (as)f(an)h(in\014nitesimal)g(an)m(t.)60 b(An)m(ts)36 b(mo)m(v)m(e)i(so)f(that)g(after)g Fv(t)p Fx(th)448 2678 y(Bob's)c(mo)m(v)m(e)h(an)e(an)m(t)h Fv(!)i Fx(sits)d(in)g(a)h(no)s(de) e Fv(A)i Fx(i\013)f(the)g(univ)m(ersal)h(mac)m(hine)g(on)f(input)f Fv(!)448 2790 y Fx(in)f Fv(t)h Fx(steps)f(has)g(prin)m(ted)g(all)h (elemen)m(ts)h(of)e Fv(A)h Fx(and)f(no)g(other)g(elemen)m(ts.)589 2903 y(More)35 b(sp)s(eci\014cally)-8 b(,)35 b(let)f Fv(t)f Fx(b)s(e)g(a)h(natural)f(n)m(um)m(b)s(er)f(and)h Fv(p)g Fx(a)h(string)f(of)h(length)f(at)448 3016 y(least)27 b Fv(t)p Fx(.)39 b(Let)26 b Fv(A)979 2983 y Fu(t)1009 3016 y Fx(\()p Fv(p)p Fx(\))g(denote)g(the)g(output)f(of)g(the)h (algorithm)h(on)e(inputs)g(with)g(pre\014x)f Fv(p)448 3129 y 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Fx(+)f(log)d Fv(j)26 b Fx(+)20 b Fv(O)s Fx(\(1\))26 b Fj(6)f Fx(2)15 b(log)k Fv(i)h Fx(+)g Fv(\013i)h Fx(+)f Fv(O)s Fx(\(1\))448 4626 y(and)30 b(b)m(y)g(univ)m(ersalit)m(y) 1043 4810 y Fv(I)7 b Fx(\()p Fv(C)1190 4824 y Fu(j)1227 4810 y Fx(\))26 b Fj(6)f Fv(I)1424 4825 y Fu(M)1499 4806 y Fq(0)1525 4810 y Fx(\()p Fv(C)1625 4824 y Fu(j)1662 4810 y Fx(\))20 b(+)g Fv(O)s Fx(\(1\))27 b Fj(6)e Fv(\013i)c Fx(+)f(2)15 b(log)j Fv(i)i Fx(+)g Fv(O)s Fx(\(1\))p 448 4891 1196 4 v 522 4945 a Fk(13)587 4976 y Fp(The)25 b(winning)i (algorithm)g(receiv)n(es)f Fh(")f Fp(as)h(an)g(input.)1897 5225 y Fx(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 448 573 a Fx(for)30 b(all)i Fv(i)p Fx(.)589 686 y(Assume)37 b(no)m(w)f(that)i Fv(m)p Fx(\()p Fv(A)p Fx(\))e Fj(>)g Fx(2)1738 653 y Fy(\000)p Fu(i)p Fl(+1)1911 686 y Fx(.)60 b(Starting)37 b(from)f(some)i(Bob's)f(mo)m(v)m (e)h(the)448 799 y(fraction)27 b(of)f(an)m(ts)h(in)e(the)i(no)s(de)e Fv(A)h Fx(will)g(b)s(e)g(at)h(least)g(2)2280 766 y Fy(\000)p Fu(i)2389 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Fx(steps)448 2433 y(all)k(elemen)m (ts)g(from)f(a)g(set)g Fv(A)h Fx(and)e(no)h(other)g(elemen)m(ts,)i(she) d(mo)m(v)m(es)j(\\cat)f Fv(p)p Fx(")f(to)h(the)448 2546 y(no)s(de)c Fv(A)p Fx(.)41 b(This)29 b(strategy)j(guaran)m(tees)g(the)f (follo)m(wing)g(prop)s(ert)m(y:)676 2734 y(If)k Fv(A)i Fx(is)f(a)h(\014nite)f(set)h(suc)m(h)f(that)h Fv(I)7 b Fx(\()p Fv(A)p Fx(\))36 b Fv(<)f(\013i)25 b Fs(\000)f Fv(c)36 b Fx(then)g(starting)h(from)676 2847 y(some)30 b(time)h(there)g(is)f(a)h(cat)h(in)e(the)g(no)s(de)g Fv(A)p Fx(.)589 3034 y(Apply)j(Bob's)g(winning)f(strategy)j(against)f (this)f(Alice's)h(strategy)-8 b(.)51 b(W)-8 b(e)34 b(can)f(as-)448 3147 y(sume)28 b(that)h(Bob's)g(strategy)h(is)e(\\lazy".)42 b(That)28 b(is,)h(if)g(in)f(Bob's)h(turn)e(there)i(is)f(a)h(no)s(de)448 3260 y(where)34 b(the)g(amoun)m(t)h(of)f(an)m(ts)h(is)f(at)h(least)g(2) 1987 3227 y Fy(\000)p Fu(i)2105 3260 y Fx(but)e(no)h(cat)h(sits)f(in)g (that)h(no)s(de)e(then)448 3373 y(the)28 b(strategy)h(just)e(passes)g (\(do)s(es)h(nothing\).)40 b(Call)28 b Fv(t)f Ft(go)-5 b(o)g(d)39 b Fx(if)27 b(this)h(happ)s(ens)d(after)j Fv(t)p Fx(th)448 3486 y(Alice's)k(mo)m(v)m(e.)589 3599 y(W)-8 b(e)32 b(are)f(giv)m(en)g(that)g(Bob)g(wins)e(the)i(game.)42 b(This)29 b(means)h(that)h(there)g(is)f Fv(T)43 b Fx(suc)m(h)448 3712 y(that)i(all)g Fv(t)j(>)f(T)57 b Fx(are)44 b(go)s(o)s(d.)82 b(On)43 b(all)i(mo)m(v)m(es)g Fv(t)j(>)f(T)57 b Fx(an)m(ts)44 b(do)g(not)h(mo)m(v)m(e.)83 b(Let)448 3825 y Fv(A)516 3839 y Fl(1)556 3825 y Fv(;)15 b(:)g(:)g(:)i(;)e(A)826 3839 y Fu(N)932 3825 y Fx(b)s(e)39 b(all)g(no)s(des)f(where)h(the)g (amoun)m(t)g(of)h(an)m(ts)f(on)g(mo)m(v)m(es)h Fv(t)f(>)h(T)51 b Fx(is)39 b(at)448 3938 y(least)27 b(2)701 3905 y Fy(\000)p Fu(i)785 3938 y Fx(.)39 b(F)-8 b(or)26 b(ev)m(ery)g Fv(t)g(>)f(T)38 b Fx(at)26 b(least)h(one)f(of)g(the)f(no)s(des)g Fv(A)2533 3952 y Fl(1)2573 3938 y Fv(;)15 b(:)g(:)g(:)h(;)f(A)2842 3952 y Fu(N)2936 3938 y Fx(has)25 b(no)h(cat)g(in)448 4051 y(time)k Fv(t)p Fx(.)41 b(Therefore,)29 b(there)h(is)f(a)h(no)s (de)f Fv(A)1873 4065 y Fu(j)1939 4051 y Fx(that)h(has)f(no)h(cat)g (in\014nitely)g(man)m(y)f(times,)448 4164 y(whic)m(h)h(implies)h Fv(I)7 b Fx(\()p Fv(A)1167 4178 y Fu(j)1204 4164 y Fx(\))26 b Fj(>)f Fv(\013i)c Fs(\000)f Fv(c)p Fx(.)p 448 4252 1196 4 v 522 4305 a Fk(14)587 4337 y Fp(Indeed,)h(let)g Fh(\014)26 b Fp(denote)21 b(the)f(amoun)n(t)h(of)h(an)n(ts)f(that)g(ha) n(v)n(e)g(visited)g(the)g(no)r(de)g Fh(A)f Fp(during)h(the)g(en)n(tire) 448 4428 y(game.)43 b(There)29 b(is)g Fh(t)f Fp(suc)n(h)g(that)g(the)g (amoun)n(t)g(of)h(an)n(ts)f(that)g(ha)n(v)n(e)f(visited)i Fh(A)f Fp(in)g(\014rst)g Fh(t)g Fp(mo)n(v)n(es)g(is)h(at)448 4520 y(least)f Fh(\014)22 b Fe(\000)17 b Fh(")p Fp(.)38 b(On)26 b(steps)h Fh(t)17 b Fp(+)h(1)p Fh(;)13 b(t)18 b Fp(+)f(2)p Fh(;)d(:)f(:)g(:)40 b Fp(at)27 b(most)g Fh(")f Fp(of)i(an)n(ts)e(in)h(total)g(can)g(arriv)n(e)g(in)g Fh(A)p Fp(,)g(hence)f(on)448 4611 y(ev)n(ery)f(step)h Fh(t)842 4579 y Fq(0)885 4611 y Fb(>)21 b Fh(t)26 b Fp(the)f(amoun)n(t) g(of)i(an)n(ts)e(in)h Fh(A)f Fp(is)h(at)g(least)h Fh(m)p Fp(\()p Fh(A)p Fp(\))16 b Fe(\000)h Fh(")k Fb(>)g Fp(2)2662 4579 y Fq(\000)p Fi(i)p Fk(+1)2831 4611 y Fe(\000)16 b Fh(")21 b Fp(=)g Fh(")p Fp(.)522 4671 y Fk(15)587 4702 y Fp(Indeed,)g(assume)h(that)f(Bob)g(has)h(a)f(computable)h(winning)f (strategy)h(in)f Fh(";)13 b(\016)s(=")2879 4671 y Fi(\013)2923 4702 y Fp(-game.)34 b(Let)21 b Fh(i)g Fp(b)r(e)448 4794 y(the)26 b(in)n(teger)g(suc)n(h)f(that)g(2)1212 4762 y Fq(\000)p Fi(i)1308 4794 y Fb(6)c Fh(")g(<)g Fp(2)1565 4762 y Fq(\000)p Fi(i)p Fk(+1)1717 4794 y Fp(.)35 b(Then)25 b(the)g(same)i(strategy)f(wins)g(2)2808 4762 y Fq(\000)p Fi(i)2883 4794 y Fh(;)13 b(\016)s Fp(2)2992 4762 y Fi(\013)p Fk(\()p Fi(i)p Fq(\000)p Fk(1\))3185 4794 y Fp(-game.)1897 5225 y Fx(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 589 573 a Fx(As)42 b(mo)m(v)m(emen)m(t)h(of)e (an)m(ts)h(is)f(computable)h(\(giv)m(en)g Fv(i)p Fx(\),)j(for)c(ev)m (ery)h Fv(i)g Fx(there)f(is)g(an)448 686 y(algorithm)32 b Fv(M)951 700 y Fu(i)1010 686 y Fx(suc)m(h)f(that)g Fv(m)1493 700 y Fu(M)1561 710 y Fi(i)1591 686 y Fx(\()p Fv(A)p Fx(\))h(coincides)g(with)e(the)h(limit)h(of)f(fraction)g(of)g (an)m(ts)448 799 y(in)f(no)s(de)g Fv(A)h Fx(for)f(ev)m(ery)h(\014nite)f (set)h Fv(A)p Fx(.)589 912 y(The)37 b(algorithm)h Fv(M)1292 926 y Fu(i)1358 912 y Fx(dep)s(ends)d(on)i Fv(i)p Fx(.)62 b(Ho)m(w)m(ev)m(er,)41 b(all)d Fv(M)2588 926 y Fu(i)2654 912 y Fx(can)f(b)s(e)g(merged)g(in)m(to)448 1024 y(one)31 b(algorithm)g Fv(M)41 b Fx(suc)m(h)30 b(that)1492 1271 y Fv(m)1572 1285 y Fu(M)1650 1271 y Fx(\()p Fv(A)p Fx(\))d(=)1910 1185 y Fd(X)1964 1381 y Fu(i)2067 1210 y Fv(m)2147 1224 y Fu(M)2215 1234 y Fi(i)2245 1210 y Fx(\()p Fv(A)p Fx(\))p 2067 1251 317 4 v 2081 1334 a Fv(i)p Fx(\()p Fv(i)21 b Fx(+)f(1\))448 1568 y(\(c)m(ho)s(ose)37 b Fv(i)e Fx(with)g (probabilit)m(y)1614 1532 y Fl(1)p 1535 1547 194 4 v 1535 1600 a Fu(i)p Fl(\()p Fu(i)p Fl(+1\))1773 1568 y 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y(22,)i(1975,)g(pp.)e(329-340.)494 4831 y([5])46 b(Kolmogoro)m(v)36 b(A.)e(N.,)i(Three)d(approac)m(hes)i (to)f(the)h(quan)m(titativ)m(e)h(de\014nition)e(of)635 4943 y(information.)d Ft(Pr)-5 b(oblems)34 b(Inform.)f(T)-7 b(r)i(ansmission)p Fx(,)34 b(1\(1\):1{7,)f(1965.\))1897 5225 y(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 494 573 a Fx([6])46 b(K.)22 b(de)h(Leeu)m(w,)h (E.)e(F.)h(Mo)s(ore,)i(C.)d(E.)h(Shannon,)f(and)g(N.)h(Shapiro,)g (Computabil-)635 686 y(it)m(y)42 b(b)m(y)e(probabilistic)i(mac)m (hines,)i(In:)60 b(C.)41 b(E.)g(Shannon)e(and)h(J.)g(McCarthey)635 799 y(\(Eds.\),)30 b(Automata)h(Studies,)e(Princeton)g(Univ)m(ersit)m (y)i(Press,)e(Princeton,)h(New)635 912 y(Jersey)-8 b(,)31 b(1956,)i(183{212.)494 1099 y([7])46 b(M.)32 b(Li,)f(P)-8 b(.)31 b(Vitan)m(yi.)i(An)d(In)m(tro)s(duction)h(to)h(Kolmogoro)m(v)g (Complexit)m(y)g(and)f(its)635 1212 y(Applications.)h(Second)e (edition.)h(Springer)e(V)-8 b(erlag,)33 b(1997.)494 1400 y([8])46 b(D.W.)g(Lo)m(v)m(eland.)g(\\A)g(v)-5 b(arian)m(t)45 b(of)g(Kolmogoro)m(v)i(concept)e(of)g(Complexit)m(y",)635 1513 y Ft(Information)35 b(and)f(Contr)-5 b(ol,)32 b Fx(15:510{526,)j(1969.)494 1700 y([9])46 b(D.A.)32 b(Martin,)f(Borel)g (indeterminacy)-8 b(,)31 b(Ann.)f(Math.)h(102)h(\(1978\))h(363{371.)448 1888 y([10])47 b(An.A.)34 b(Muc)m(hnik,)g(On)f(basic)h(structures)e(of) i(the)g(descriptiv)m(e)g(theory)g(of)f(algo-)635 2001 y(rithms.)d Ft(Soviet)j(Math.)g(Dokl.)p Fx(,)e Fm(32)p Fx(,)g(671{674)i(\(1985\))448 2188 y([11])47 b(R.J.)38 b(Solomono\013.)h(\\A)f(formal)g(theory)g(of)g(inductiv)m(e)h (inference,)h(part)d(1)i(and)635 2301 y(part)30 b(2,")i Ft(Information)j(and)e(Contr)-5 b(ol,)33 b Fx(7:1{22,)g(224-254,)g (1964.)448 2489 y([12])47 b(R.M.)d(Solo)m(v)-5 b(a)m(y)d(,)49 b(In:)66 b(A.I.)44 b(Arruda,)i(N.C.A.)e(da)f(Costa,)48 b(R.)43 b(Chaqui)g(\(Eds.\))635 2602 y(On)29 b(Random)h(R.E.)g(Sets,)g (Non-Classical)i(Logics,)g(Mo)s(del)e(Theory)g(and)f(Com-)635 2715 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b(Zv)m(onkin,)h(L.A.)e(Levin.)g(\\The)g(complexit)m(y)i(of)e(\014nite)h (ob)5 b(jects)38 b(and)g(the)635 4218 y(dev)m(elopmen)m(t)32 b(of)f(the)g(concepts)g(of)g(information)f(and)g(randomness)g(b)m(y)g (means)635 4330 y(of)h(theory)f(of)h(algorithms.")h Ft(R)n(ussian)h (Math.)g(Surveys)p Fx(,)e(25\(6\):83{124,)k(1970.)1897 5225 y(35)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF