HTTP/1.1 200 OK Date: Sat, 11 Oct 2025 01:30:35 GMT Server: Apache/2.4 Last-Modified: Thu, 28 Oct 2004 12:33:52 GMT ETag: "a9fc2-3e78127a42c00" Accept-Ranges: bytes Content-Length: 696258 Content-Type: application/postscript %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: thesisams.dvi %%Pages: 51 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR8 CMTI8 CMCSC10 CMR7 CMR10 CMTI10 CMMI10 %%+ CMMI7 MSBM7 CMEX10 CMSY7 CMSY10 MSBM10 CMBX7 CMSY5 CMR5 CMMI5 CMTT10 %%+ CMSL10 CMITT10 MSAM10 CMR9 %%EndComments %DVIPSCommandLine: dvips thesisams.dvi -o thesisams.ps %DVIPSParameters: dpi=400 %DVIPSSource: TeX output 1998.10.13:1249 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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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 112 /rightanglenw put dup 113 /rightanglene put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa4f3118e6b59bd7a2c35420557fc465c b2c85252357c57550f33ddf2112e2e8066408349bc421b11418f16b9d52e8dc3 c522008dfdd090915e8cac8e24aee3bfd876093175f04b77c0ce03913680339a d5eb95a9ca66b69f9101bc8a454c64c0d28d082249c3e53409ae286cd509c244 97fec0195ada9bcc2348732c144581ca3786f523ca49f89d0ac19d7587980cb1 d1f8283a1dd55a283547e458e246ff130065d59a2c83af340c39e8c07fd50926 d48dbe070491a603801a9cb666c33571da2f7a877d87fde438a5798e5b8c5939 b99fc45c8835494e9ae0600eb26199834f992b8d6bc8f91590cd3fac9d54c7cd 63b3cee3f30821a0086ab096082e09037379114c7a38e158cfd48e32f5cc6b28 0f56261407f3ec898db9d5610923faa217b7f760a580cf1dcf64ed5083c00620 65781f528aca74a54e5a8689d2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMITT10 %!PS-AdobeFont-1.1: CMITT10 1.0 %%CreationDate: 1991 Aug 18 17:48:50 % 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 (CMITT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch true def end readonly def /FontName /CMITT10 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 65 /A put dup 66 /B put dup 67 /C put dup 75 /K put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 88 /X put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -233 669 696}readonly def /UniqueID 5000779 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % 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 (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 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 101 /e put dup 105 /i put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3036f2ed7bb65d 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4a20da7f373f393defb94e38b3bd6b0d01f5e33b137c7f978502c8b5b9bce9e9 91a8531d23b7cd3fbc125a591f41de35cb5b8a7120a9b77708e95fa483f710b4 f7712f9cab6bc59541bf770dc2cc9b027ce72c180087c448fa60ffa8944833d9 bb9758a510aa91471099ef147bd6f0ea04fa87e5d008506a0c1d42cd824abf53 a329f97f0b2a320fff7fc37a97fc906fa7b6c3611a099f189c7f96bfdfba5d3d 68df55ce93c8db78220405f5dfe889995df3344907f245c858f5146a16df0559 1341ee1e83c6ad61f2c59b1484d6b0af275f0ec24932ffa9c4e23d49587e1987 d3d531e55e1f60d6e25b58d4f158ef0e0328196d9e34ceca3308bd78bd7e36df f93f71c26a1a93600d028ebec3d7318bfc99b6af05146a67b613dbc300cc3124 fa7c372a529b919fb03cdec2768c6e9865c507d68a40356358d695fa3fa8568f a2c8f98ca6d3d7b7f46dacaa05149d6a5db00c8c0729d13d94a684732fecad59 5b13e9929e20872149a5cface5ce22af4d4eb6646ec558fede6b6fdf80f9e0c3 8244b6e7f37adf7ec1f83abbaa7f9b56afe0cd8b75e572580b179c4813695c9a 9bbf3ea5e1a6ecc2c4b52e694947acf2c58b88400faed7592442133efd1188da 354b4e31973b9373e404771bdb563ba9fc0c6af1b80d1ba447d5884dabce0b6d fc96151e74404adc39c863e38ec19f15a0f3e1c79c0b840c5e393ba023129f11 c8ac236cd926784e9713bddcf2d9fdd7c0fea02c87c7b703cfd18242252a0b2a f895d2165b8f248b2c369fd9763a1f42edd835deb3f0bb3d76154c2299c899a2 ab51724d05aeb9167068f1f9296f1905bdfc6239c78f1d45f4ba4a4eda74ea54 e030414ae278913eec5271611798d206d222309dfbef9ab3d848f7fe5093ba55 9d3ee2ff165af6ef72a89933d456744594ec0ee15fea4df156da25bafb7e4d5b 2f1fdb82baf433b4d983e659baef4f8baa7424ac215cf10b086d06abcc991611 911dcd33e863c0d44962669540d7f85afd836b825d909c98f980735fc305018d 0f42bdbe5195f4c97f5f87b9dcdba7f4dca9b018393ca1bbf503868db83aaf86 a0577733dcd7a9da530154749e18b2188f46f08950955595f3df7ffcbfc18644 47627b79b82bafff81cc86050e568f95274245c2c12f0c4e102c6177ca3ea2f9 94c786ca217d40016336cbf1ea89e97f56771cb82c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % 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 (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 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 37 /percent put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 58 /colon put dup 59 /semicolon 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 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 86 /V 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 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 120 /x put dup 121 /y put dup 124 /bar put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueID 5000832 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ecefaacd079732f12c29 315e4b9623a5752c6f1d8145869e120d910b2644887cea7e30b15676a92537c2 9d3aa80dc30082aba94b40990b82fb1a877e805e0c8c48f61e9f2edac05b944e e4d8084ec1d5cc517aaeec5b3ea379dd011eeb454cecab3ad2443c887c582789 72355673e503affe0394fc7db31de364e4f56c24033c7df2265c56445ec63a1d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % 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 (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 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 104 /h put dup 105 /i put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdbae54a44 d411e0ee216ee405bd68ffec694d082ccaf886a16b3edfb240bfb77e86843d8d fb24c80cebb3a5e0c4fb26c974190ae130850a0d4d7c49a0e644dd5f2d03a957 6e064341f89fad712254928be7dc5a38b253b64a3dc6cc3a316ca8c6c7e88b7c 2f734e77acb1a5a7063187fbda19374254c039072cad3926192aef41fe0eb704 5152f118af39c36aacb4496dec42fed0f241ffdc2d571262c0ee2965316eb174 8fdc81502ec12722faf9b323842de7840d3923077be02617dd334b207d2a178e 1f0cf1209c9158c91046421232cb420447afa57d82b0bf5e1f7b531240296e77 e2496571877643b5706bd12d6f85b18a5cc1cf5e86c8d8f6e810e7a71bf6e3e3 594e700bad382703c64bd60088a8951a641a58afe153820f52ddf9a12e6e0589 e8317d27e8ad2fe3ef0e4a82a74da7283354a66348da0e6167236edee4e39e91 3134c0d025276812af0b078338d8b51e397dfec3fab12d39c7b50a272d7ef16c 2e5060220fff9f518e681e4627ad0dbacf248b1f5c35e7139378da0358578965 a06fdeb8323694552713df1ce18c06b1c86a20759431f277289c315ce8e8700c e307ba94fbd62d0a824a6133728402fe38ccad315c0f7dc956a45b60e2ea0e14 5659cb45e3345dad04cc665ebfb3f5aa427019935ee37111ff5befcb7872b01f 7a4cf9f9a3f224a7e6e21337b5a391910242e121667ac34a408b9b3f68fbbd86 1f1d808785c5bb11d985e5f58341b680645d2599b631b4baecae1d76d4c83c66 96aa8fc2c83353141945f9e49f9f27cb4990acb9367ec4305c686338e0470e2c 0546 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b24c5e6c ef3ce751da62120371d4edc733042e2a6661dc6a3111f38f0baec0eb2410e8dc 1605312e5cfd3f3c89cfe4913fe33f71f1546730d31261296f78a1c156b4af7b b27af1a0706f57d0fe0d5c7b18aef611b69c67ea9ad325f159b826404c1e6f08 89105b670bbbef3e3dc50835c71964a2e992354d26571566eb6d7bd7938acb12 486336dcb8ab542223c6811d37c95679b796e22cf608d05b3e0ab856c5fad35d 990d789bef73c3d17dd66b02adca2aae2fef64976a04310219579a91aea4404a e87f66aa06ba008dfbe8c02a0a74e1d0aeb5b3feeb1e5f20d8136c0020a88677 cb72b8e0e7d5c8c2b230baecaea5aecab78e9bd97bb59018a30aaa15cfd0a1f6 fdb877479ac66003d3f3a62f76469169f467078712ae99862e398a6fba3a149f 80aebd858e6482dd5dc8cdae093b74ca98e10683b08b501d21bb5576e6cfad67 a360b4758832c14a056ad4a6528597cd9a09b4048546c50861918d3bca7e47c2 07efce4574d42b7cc18f61b6d57bc513f6a659921daa707c3ae900b85603258b b6ac6343e09f271e5c2194be84d3140266c66c6abd21a461846ce317312274ae 3e6b0e61a432c18dc19a68fb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b6f9385526e98445876548d0f2a09cfea14b84be3721b51afa2185 90c7b9ca643c7f4fe67cd8985f27855c40ae7ea592fadfbba29301071039d18a 8a6880cc0bff22fc8af50fd2802aff5a96381fb4bef240dd26b34a5461188ca0 c85e4b3f803f94dce14e8ddb2284e22abfe148560aff784967ed795332cc0995 80b762f864b2c15bc640a4cd7f6927c7dd95aef777fd226254b8e6e1b8ec6483 2750d9929627c4cecf9f5c8feb57a2c7ee5d1315593f3fa16c7c7292a594ce14 2c396d9b787665569dbd0f710822ae3155a3aa53d6f055cb19e897b235b65bce 4d507a4d41edc6a09dff1f737f2711478e6e998218b7f3e7c1aaf3938f68e9cf 9ea46e4a2904c459281cc156f5ce80d6356e357011cd0bc383318129bc8ce38a 758c05d1205192a457f4b1b31df97baec7611905cac0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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 104 /h put dup 122 /z put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueID 5000765 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b1ee2d9de5f5346241e154574f14a4532b35968596ef7a20 102dad629475cf5d654585d4caac68ba7b2ff21b103a98af3e9ec01fae180804 26b73d62e93f0328ce7775e81898de587dd0fbcfe98934a58afb1aa4dd8c6eec d19373ddd360a5e85cdd4b752b2f2baa13b48a282214a05632d28e915ec90824 21ff8f2166e940c7122c583f235bfd098f8aac0f1dbb9a625b02bd889869664e f1856608899ba509d6eb8ece665afc34a459dd5b54dc7711f6f04a288214efe4 53095e239bf19cd36b7028d46c07a2b8c17d452d8563ea88a3bc5e5ea29a9ea9 a7e38481679f893f9e6a983ccb5512bed4aca5888793526701796b3fcac0fb62 f69c66d0688eede0fefd590fbce954b2cdd2abc9017390077a2cf6f6fa7d0db2 dcb4b0963e3dfbf02be1dca9cad104f3d29495d19337fbffdd745e1729e7c0a6 41fdd3bf42c30d715670b3ade44f481d795071f1b24f4f03dff59d91bf9d678b 4560da27ad826c9fc8f8cd9a333248cb2007b4ac0a9e2d5c044774124f1ee23c 0e3e005a4a16e1d0cba15797fc42fa75cb45b24fcf8a8cf41de8e06388b5f995 04454646c3246074f495564cfc8ad75a949cc477ecf03f0511b90b6a7456b730 cf09773b5544f6ec17a9a179714e47e2eb1142d8005bd228affaa375eed7fb1b 075b3023be6386f3805aee05b47c7f370e8ad410110429d2ee8801d0d2da9e68 a453a323c147716ca588fbf563175e5f47d1086a21b04102cba338a3580225ce 00dfb1784fbcad0fcd11434afcc2ec13337ee24775c2952be02e20572aaab17f d21287775e4a4d9fbf877a487b252b5c880463541489f5578c7472d39fc27320 2e4ec86f9935ab2ded14ab9ea36770d79a1a72161102412bc57c587426a39c77 b24a0746f71f62d109484272b90d2588232925f239183d68842015db2cae2c38 cbf862632683fb120f22802344e8345bb7dfd941f7c1f2b4768de7110dca8071 9ea2b23f0bd4ea358f6ae7bd3077c58fcf81b3d1c1412ac4dc0440eea46f6591 19d0b5a60d0c2a737b0de5e3674e5302833e7b710c2c7bc9cc774e2fa2b84502 7cd437efdc99f957d7cb47c421db259ac8abb35233a68428cc21cc5747bb1774 30ad370a598a0d05ad7d898c4ff76beb167683c9956870f55da2f3e1933baf81 be5afe73b41ca451ea1621562285d5256bcbee9ca084791e5709841f59b14e4d 038eeb7d23ca66f492f3e04d69c753aa0efbeba2a0dbaa5e2c9abb8008e01d72 6edee271cb9ab9fc67eb982938c2b8eac4fd2f7d94a09b247a6e916ab2675818 5c092e1acb9011398db560d045b56e22e25b0ecede8aebc789711823c53c0920 5ae44ca8d1dd7c0f9a940d5f8a55cc4831cf84f6f547d7b7f1ae482677281fee d4bbd80171cfb3a1df4a56c96748f0cffa9ce0c8403ee4e42e2d527ae9fbbcdd be47402022c1a759b53030e5cabff24b7eeb9b3e6eac 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c45f1fd9d21e6db10d6ceaf7928ae8b00f2dded80901954281 b65ab541041dfa20060b17b3605e518db67cffc097304ff8c72cfe9384604791 84ca6277a7cbad21da97cd16159f554d84359224a09f5cb6887f9222cefa2a8d 9a9fe908199abe4e89680314002cc52583f20ef690437e5261ddefee7fb13574 05d0a308bff33aa4667d85a98e7522b851ddab7c86db0940801f0ea717de2cbd fe7cd8111194d27fee3f86791b23ed5152711af9d51514d88f63857bbdb18974 7bcc25078299cd15cfc6e3d8a9dc61105c162570007a49e5a84795838351299c 1888f70c110e7bfce223a0e6fdc2498bb1357376eb5f5e5eb2da3e2f9a8ade3a ce391bf14312338274a305069f33249211e229710e7d15a722c45b151525683e 1d15ee1b3a73db987b0a91b90db404c42f48c234c4f6646cbe63fd04138842bf 1f16c4cdeca6942c374d13024d1e122d199eb93254f47186b2915388e5bd4a02 eef57ebde639f2cb2c4df5fa9a85033240297b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 2 /multiply put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 26 /propersubset put dup 27 /propersuperset put dup 30 /precedes put dup 32 /arrowleft put dup 33 /arrowright put dup 50 /element put dup 52 /triangle put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 58 /logicalnot put dup 59 /emptyset put dup 62 /latticetop put dup 80 /P put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 96 /turnstileleft 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 /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 18 /reflexsubset put dup 21 /greaterequal put dup 48 /prime put dup 50 /element put dup 59 /emptyset put dup 91 /union put dup 106 /bar put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b87124d146d8836482f73a24389afb8a74364f22077cc3fd6015cd4469 c34470c86925433a77fb3c838a1a7158139e96980de23ff461382df32c6cb4c8 59d292d187a7c919a1c3e51647cecd04d7e091b936aaf37b758d1de4f6846958 e6703fd60aeca4e5c38b34574b55b0e2671154366e5554dc12a98bd9581eec29 a866523ed5b825c2ede44f7369230a595721fa307787764add7beda87195e1ac bd494f6474d023bc194cded7c514920260f6af49909f64b316c6e7d0c9924fa4 50f4025d038ee4402adedb44aaab577adc7137de131167d35a1fe4c6dd18c577 9cdede365f8b102f3882f92b2ac3479fd2811bdec9d36e8d02e68781dfa05e4b 059cb3d8a2e1609ae17a1f4268ca66623fc5ef9edb6a4a643f6015a2b6afb34c 8d085138fe34fbe8632874127af55a64188dd05d589d27c9a62c2d4c0687232b 1c8d1651c779edd5935640ab0901bb0480b1195ef69e59d89a81d19dce10db37 64b73d166f7634e93c64f4e803ee3d7a1424454115a7963499330938ce937c2d 53f33445183eeb0697f47c5791649e55681bc166995506f9b0ae185969058a1b 42ff78f72b2eaba14b5e395c07f6c2afa04e5738b90d1f3484707f7540b70a09 b1e2f7dd510a69b13e2590371d0a40ccfbb2e32458f32dfeb4eacce05fb02388 dcee559bd778377e30e70003069755dc6e99cf589b391dfd4fbe8be4059f62e7 d856066220ec3c5a215d7ae1f5a362a89f6c80d35477c289232a154467352358 ae0f5c13606e12c3ddd2a27761217501733d524221856f5d425003ad424406c9 7d1dd1e46d0c2e906b6564b6759233a0ae72d3b32388912b192eb6317365019b da05c3416962ed20d60b89af9a77fe44d31acb26d776ec5654c4ad869421b171 4b5305d4e97bb32844a72e872b50b0960f60f4e0cf35ac99468030437468975d 13c6cb07222d225b8c16a5ac64f77ba435deff5735ef2cd048bb8f7170818937 9ffa56dc7db4c2bf6e139085e5b93f57325462f54dc04bad4fe950c28c22171f 1c9c3c6bca60b6d428b7dc23a3e404514b3b58d110b88c10dd0784ecd7e93678 c789418ad9b171663de6c6ecdc774dc37b2c67fd236819cd6d21d17b813156d6 523c4f3d01a3d4a282bfd65fd35ba7e3b3ef9854611a63c0a93c8cd41c7015ff 48dbecaf698b2d8ba7e323c63e40f0c0e9386153f0735769aca8831ad14d382f 5822af3443b928f9ced92b991e6b5378b274cab153ae4aa344622af4b239abe0 5591ea08322412833ca32375d2ad5d234acbd5b889d0a2ef9cbe67ecbc34363d e9194e6270ee55d9dce35e8f28c072721b2a42c8a017ff198b43665e372ecbd4 1bdfe9720a3b8305337f737e6399b9aaa6271aa0bec6731f23308549f48d2f45 f58baa59bd4c0be88de06584099be0a40b7cabbdc5ec3ee571d6bb9286ad7628 6e258720ef917a6468ec33e33fd272923df4485e5d06803e5feb1e721a14c42e 7dfc667f0f67306f0f6dee0d0dca9055aae291431b5b7dc41668ec96b697d21d ef0acc316200c98a56b1d75da42e72fb463f012ca7178486152f23a5145956f3 10b1a6f3c73ebe2379ed56db692c11e7c996f86bdc9166461d11d65e8659a38b a0d3054547e55592c9cb275ec0da1bdd841ff435f2133e1c3b61f90c2b78d0e5 d214879a35b13686c3ce0dbe584d318031a4f7d63eb4f07b6cedec4e8daa6b34 8fe7029bb806673c898c29c30884c36bab0144e974f5714f6a4e2047719ebdd1 a6db1e3ac39d80ca97fa0258d88c7259568aad6a20b77ea315aee0a2d91ddcf2 1a38a88efe32a2a3f5e1e1f01f04f9f6d6144979cbd7bdd825e41a7603e682fe f11daec6b3fdb4fd4d3b66fdf36c9a550f549a924ba6200792db3697247bbe85 0ebe2a48e904c7e6305f6f7dec4f076ba22684cfc2bb2bfeb94bb08af0973318 1b03a86a13884670894013c003019e7961ebef55b5a8cfcde6de 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 83 /uniontext put dup 88 /summationdisplay put dup 91 /uniondisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 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4427a22cdc505f7b727a5f5369c5aba5f6fdee46cc5d7b319659b98ef65094a1 2f615398207424fd5eb11d61d192639cc84bb80a85f9df366f8789f798642604 e9b91ceb53ae910a8e7070c21cf1e9d18eaffb97eb6b8cd86fe331d99029097a ded39454e5310a00083eea4470e70c12a27e58ff7e20dfd4f82a0fec1a83aa32 b4474aec11370fe07c007e08ca573d2140eb7c2eed6c9b24946ca2641fac4772 a110b362bb91e63e1a21e37b8990bd78815d18d22e591170e97256524aaf3d2e c668697291e94fc6b8e84014423a67dba4153fb493856208ecf3816a86d175ac 64984383816e3bdbbbdf1ee2c323d1133da63c90bf2a18c28f44a5c8753594d9 822f22197e363f15ea457882db3a05a7e4f696db28bb27b588ebb205caf7cdd0 bfa1fb6bd55cd8728d72e56a8965fa6b1db563c9a7206543a3614c90a8a2c962 3fa703e281d3162cd577155c24f9b505bb759e65f4283fef25ff8367e727ee89 c731d74d99e0126319baf0566d351e090e771a714e70ec64f899f31e492d2f87 9f0a5b63b290a6be0994d3ef426fb9689b0965d87f15247a20d18d79ff728e54 6c0191b095dbf3b1f82c2e1e388679aadae910e0f582e37267e515bc96473f3c 8baae12c075486f891aefa28be3c4173705963439e3c3bd8e7c298dc89c80035 477dff8942eea7714d20f6e7c2c76dd27b78046b59d1d1c902696b22dcfbd257 8917deca9d9b9764fe6e47e96cb1907fc275b2fcb990105c6e2f800f897e3d36 9d55faf14cc9a1ad87e2aaefc47b547a3065f1e28bb70fb5f620336e261e1b10 ce6d52b08031a768c4b243045eaf3a9c8647f87ebcc86fcc861a4b3b506f966d 691634e1ef2da8b0028ab30724d223db4ddaa2c6e7361f4bd41236feaa29ab11 2a5a2231481dd3b864e231 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 /UniqueID 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcf3fe48b07398eec882c325567e28b9eb4d699111343b1910da9117d 74d6e9de75247ad908f8ce2cfc49e0faa22dbf4d1abe2c035b8f8eee2d1967e0 7cb9b5d8de5e0b1c3a86c7a08a580e51be75a8afdaba2cda08f87d352652a3c7 9bda6bcd32fd59e8079fee3fe7fa161ff626461e85ff8b9b60e2e6bc72884c6f dbaa5e8ad5912206cdd26b204cb34a5403c4fe9cc6d0a2e6805a98624c56baf0 3e673abaa926e941739ba23126dd78ca82499f376d15bfc7cc70f1586f5f10a1 fe99eed1a1f44ace1e3392335eea8cd2e1d09075415ff9382dd812456ca3c6dd bc8c180322518bc8baff72f1367ee7e30c44048dd9c7a82f57d35575e569a561 29f7d2f3da7149f2147e1565f14304aa020b506478e4758d10bbf9950f87a5d1 af646d8454ce4223a8706c6009a08f2da8dc401107b8d0273be0133e23743ca3 2b5c4ee63b3f41b58321e8556ee5aa6ae5dde7cdab9749185da031a2cbbb14b1 a3b22b8f7a3a785ecb4b62b6a50424c4b4022893660fa66ae0a28f0383369a02 5c0771325ab47c96188a9a52d51b3627735f5e7ef25513a708625d4d38f8bf69 bce988960621a1047be63b708c7c1571e27a527e2a804bfd061024ce9bbfd937 e969f28e1326c816ca7b86e1d17e220dc6044c94f5c59a3460da217683ef3cb6 276a560b97b063d6a12f0c00aeeeb6a3d4e10d3c282a2f1011722c8d730e8ece 357bdb82b74f27887f35a3b3bf75e30b8f31bfe85072abd4c0d3af100cf2fdef ba07d0fec1202d5d4fbb947bc8999338 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /gamma put dup 59 /comma put dup 68 /D put dup 70 /F put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 99 /c 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 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 readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star 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 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 86 /V put dup 88 /X put dup 89 /Y put dup 90 /Z 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 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{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a 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e862e90d79b07763dc20f41eb394c4ed352f5343f5ab9755b32cd6c9aa2e706d 5b1252547e6883789021fb5358325918c6792459cf384bb5cf9bb653d341401c a872c7fac51028d6595993a3d139748cc13d9629483073f49bc4ed2400bb71e3 d15b006fddf3ba0f3a115f9a53a0e847bf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /ff put dup 12 /fi put dup 13 /fl put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z 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 readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 5 /Pi put dup 6 /Sigma put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 34 /quotedblright put dup 37 /percent 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 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 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 96 /quoteleft 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 readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 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af0140731625e8163e10f1fb7d1d16b514c541bd5c57eac0b7a2eb0bb4f2ac3c e83a4944abc804ad1e3ae49f7f40f8e9e00d91d455421cfcf805d520e4e6fb26 f754d4c2030f664085332b06e22146ba5290487c3a54af6f4fcc0704239fa7e4 a097494ae59dd95df833d24938ba2840bad0bbc19021a5b8d573b9a96fd960ad 7e8f116f94d9ec1ddb4a4818414b93fc4237d3766065064f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 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 56 /eight 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 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W 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 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{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 6 /Sigma put dup 35 /numbersign put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus 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 61 /equal put dup 65 /A 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 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 89 /Y put dup 103 /g put dup 108 /l put dup 111 /o put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 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 50 /two put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 58 /colon 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 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 86 /V put dup 89 /Y 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{-35 -250 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36f6372968dbd0df8505aef5b22e57f2b9834fef2c2798f9d715aa323 3e09b183f8d92797bbd3a990980982018887612cdc5507a47581f07a1cf99a5c 587021d6b67b0524e780517e7a088a6388097ced01e5a59aa8779c6ad82af704 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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 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute 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 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 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 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright 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 readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 39 /quoteright put dup 40 /parenleft put dup 41 /parenright 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 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 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 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z 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 /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 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y(ev)n(ery)21 b(free)e(pattern)g(v)m (ariable)h Fy(y)i Fs(o)r(ccurring)c(in)k Fq(p)f Fs(has)f(b)r(een)f (replaced)g(with)i Fq(s)p Fs(\()p Fy(y)q Fs(\))f(is)h(called)304 2406 y(the)c(result)g(of)h Fr(applying)25 b Fq(s)18 b Fs(to)g Fq(p)p Fs(.)25 b(W)-5 b(e)18 b(also)g(denote)f(this)h(pattern)d (b)n(y)k([)p Fq(s)p Fs(])p Fq(p)p Fs(.)370 2472 y(A)f(substitution)e Fq(s)j Fs(is)f Fr(idemp)m(otent)27 b Fs(if)19 b([)p Fq(s)p Fs(])p Fq(p)d Fs(=)f([)p Fq(s)p Fs(][)p Fq(s)p Fs(])p Fq(p)20 b Fs(for)e(ev)n(ery)g Fq(p)d Fl(2)g Fq(P)8 b(at)1913 2452 y Fp(Y)1913 2487 y Fu(\006)p Fp(X)1986 2472 y Fs(.)370 2539 y(A)18 b(substitution)e(is)j Fr(minimal)25 b Fs(if)19 b(\()p Fy(y)q Fq(;)9 b Fy(y)q Fs(\))20 b Fq(=)-34 b Fl(2)15 b Fq(s)j Fs(for)g(ev)n(ery)g Fy(y)e Fl(2)f Fq(Y)e Fs(.)304 2638 y Fr(R)m(emark)28 b Fs(1.4.2)p Fr(.)f Fs(Note)17 b(that)g Fq(s)h Fs(is)h(idemp)r(oten)n(t)d(only)j(if)1038 2737 y Fq(Z)d Fl(\\)1153 2684 y Fn([)1141 2803 y Fj(z)p Fm(2)p Fp(Z)1235 2737 y Fq(F)8 b(r)r Fs(\()p Fq(s)p 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y(renaming)c(to)g(a)i(term)e(is)i(a)f Fr(r)m(enaming)25 b Fs(of)18 b(the)g(term.)304 3411 y Fr(R)m(emark)28 b Fs(1.4.4)p Fr(.)f Fs(It)22 b(is)h(customary)d(to)i(assume)f(that)g (appropriate)e(renamings)i(can)h(alw)n(a)n(ys)304 3477 y(b)r(e)17 b(c)n(hosen)g(to)h(a)n(v)n(oid)h(unin)n(ten)n(tional)e (con\015icts)g(b)r(et)n(w)n(een)f(v)m(ariable)i(names)g(in)g(terms.)p eop %%Page: 5 7 5 6 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)569 b(5)304 283 y Fs(1.5.)32 b Fy(P)n(attern)21 b(matc)n(hing)h(and)f (generalization.)304 385 y(De\014nition)g(1.5.1.)26 b Fs(A)20 b(substitution)d Fq(s)j Fr(matches)26 b Fs(pattern)18 b Fq(p)i Fs(to)f(pattern)e Fq(q)22 b Fs(if)e Fq(q)f Fs(=)f([)p Fq(s)p Fs(])p Fq(p)p Fs(.)30 b(If)304 451 y(there)16 b(is)j(suc)n(h)f(a)g(substitution,)e(w)n(e)j(sa)n(y)g Fq(p)f Fs(matc)n(hes)f Fq(q)j Fs(under)d Fq(s)p Fs(.)370 518 y(The)i(w)n(eak)h(pattern)d(ordering)h Fl(\026)i Fs(is)g(de\014ned)e(so)i(that)e Fq(p)g Fl(\026)f Fq(q)22 b Fs(means)d(that)f Fq(p)i 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Fs(1.5.2)p Fr(.)f Fs(Matc)n(hing)21 b(is)h(a)g(sp)r(ecial)f(case)f(of)i Fr(uni\014c)m(ation)p Fs(,)j(the)c(problem)g(of)g(\014nding)h(a)304 1285 y(substitution)f Fq(s)j Fs(for)f(t)n(w)n(o)h(patterns)d Fq(p)i Fs(and)g Fq(q)j Fs(suc)n(h)d(that)f([)p Fq(s)p Fs(])p Fq(p)j Fs(=)f([)p Fq(s)p Fs(])p Fq(q)r Fs(.)41 b(There)23 b(are)f(kno)n(wn)304 1352 y(e\016cien)n(t)15 b(algorithms)h(for)g(\014nding)f(most)h (general)f(uni\014ers[2)o(,)i(20)o(].)25 b(Matc)n(hing)15 b(can)h(b)r(e)g(imple-)304 1418 y(men)n(ted)h(in)i Fq(O)r Fs(\()p Fq(n)p Fs(\))e(time,)h(where)g Fq(n)g Fs(is)h(the)e(size)h(of)g (the)f(input)h(patterns)e(or)i(terms.)304 1520 y Fr(R)m(emark)28 b Fs(1.5.3)p Fr(.)f Fs(Observ)n(e)21 b(that)e(a)i(most)g(general)e (matc)n(hing)h(substitution)f(nev)n(er)i(con)n(tains)304 1586 y Fy(y)30 b Fl(7!)f Fy(y)f Fs(for)e(an)n(y)h Fy(y)j Fl(2)f Fq(Y)13 b Fs(,)29 b(since)d(if)h(there)e(w)n(ere)i(suc)n(h)f(a)g (substitution)f Fq(s)p Fs(,)k(there)c(w)n(ould)304 1652 y(b)r(e)18 b(another)e(matc)n(hing)i(substitution)e Fq(s)p Fl(nf)p Fs(\()p Fy(y)q Fq(;)9 b Fy(y)q Fs(\))p Fl(g)p Fs(,)17 b(and)i(this)f(substitution)f(w)n(ould)h(b)r(e)g(more)304 1719 y(general)f(than)g Fq(s)p Fs(.)25 b(Therefore,)16 b(all)j(most)e(general)g(matc)n(hing)h(substitutions)d(are)j(minimal.) 304 1820 y Fy(De\014nition)j(1.5.4.)26 b Fs(A)15 b(pattern)e Fq(q)k Fs(is)f(a)f Fr(gener)m(alization)23 b Fs(of)15 b(patterns)e Fq(p)1823 1828 y Fu(1)1863 1820 y Fs(and)i Fq(p)1996 1828 y Fu(2)2036 1820 y Fs(if)h(there)d(are)304 1887 y(substitutions)j Fq(s)660 1895 y Fu(1)703 1887 y Fs(and)i Fq(s)837 1895 y Fu(2)880 1887 y Fs(suc)n(h)g(that)f Fq(p)1153 1895 y Fu(1)1193 1887 y Fs(=)e([)p Fq(s)1292 1895 y Fu(1)1317 1887 y Fs(])p Fq(q)21 b Fs(and)c Fq(p)1513 1895 y Fu(2)1553 1887 y Fs(=)f([)p Fq(s)1653 1895 y Fu(2)1678 1887 y Fs(])p Fq(q)r Fs(.)370 1953 y(P)n(atterns)i(ma)n(y)j(also)f(ha)n (v)n(e)g(more)g(than)f(one)g(generalization.)28 b(Supp)r(ose)18 b(b)r(oth)g Fq(q)2092 1961 y Fu(1)2137 1953 y Fs(and)i Fq(q)2272 1961 y Fu(2)304 2019 y Fs(generalize)15 b(patterns)g Fq(p)804 2027 y Fu(1)846 2019 y Fs(and)h Fq(p)980 2027 y Fu(2)1005 2019 y Fs(.)25 b(W)-5 b(e)17 b(sa)n(y)g(that)f Fq(q)1377 2027 y Fu(1)1419 2019 y Fs(is)i Fr(mor)m(e)i(sp)m(e)m (ci\014c)i Fs(than)16 b Fq(q)1955 2027 y Fu(2)1997 2019 y Fs(if)i Fq(q)2072 2027 y Fu(2)2112 2019 y Fl(\036)d Fq(q)2195 2027 y Fu(1)2220 2019 y Fs(.)24 b(If)304 2086 y(for)17 b(ev)n(ery)i(other)d(generalization)g Fq(q)r Fs(,)j Fq(p)c Fl(\026)g Fq(q)r Fs(,)k(then)e Fq(p)h Fs(is)h Fr(most)h(sp)m(e)m(ci\014c)p Fs(.)304 2187 y Fr(R)m(emark)28 b Fs(1.5.5)p Fr(.)f Fs(Generalization)21 b(is)h(also)g(sometimes)e (called)i Fr(anti-uni\014c)m(ation)p Fs(,)k(since)21 b(it)h(is)304 2254 y(the)c(dual)g(op)r(eration)e(to)i(uni\014cation.)25 b(There)18 b(are)g(kno)n(wn)g(e\016cien)n(t)g(algorithms)g(for)g (\014nding)304 2320 y(most)23 b(sp)r(eci\014c)e(an)n(ti-uni\014ers)i ([2,)h(17)o(].)41 b(Generalization)22 b(is)i Fq(O)r Fs(\()p Fq(n)1697 2300 y Fu(2)1721 2320 y Fs(\))g(if)g Fq(n)g Fs(is)f(the)g(size)g(of)g(the)304 2386 y(input)17 b(patterns.)304 2488 y(1.6.)32 b Fy(Substitution)21 b(matc)n(hing)h(and)g (generalization.)304 2589 y(De\014nition)f(1.6.1.)26 b Fs(The)21 b Fr(c)m(omp)m(osition)29 b Fs(of)22 b(substitution)d Fq(s)j Fl(2)f Fq(S)s(ub)1782 2569 y Fp(Z)1782 2605 y Fu(\006)p Fp(X)1877 2589 y Fs(with)h(substitution)304 2662 y Fq(t)15 b Fl(2)g Fq(S)s(ub)484 2642 y Fp(Z)517 2625 y Fi(0)484 2677 y Fu(\006)p Fp(X)558 2662 y Fs(,)j(is)h(the)e (substitution)g Fq(s)12 b Fl(\016)g Fq(t)j Fl(2)g Fq(S)s(ub)1309 2642 y Fp(Z)s Fm([)p Fp(Z)1405 2625 y Fi(0)1309 2677 y Fu(\006)p Fp(X)1441 2662 y Fs(de\014ned)h(so)i(that)854 2806 y(\()p Fq(s)12 b Fl(\016)g Fq(t)p Fs(\)\()p Fy(y)q Fs(\))h(=)1147 2712 y Fn(\()1191 2768 y Fs([)p Fq(s)p Fs(]\()p Fq(t)p Fs(\()p Fy(y)q Fs(\)\))54 b(if)19 b Fy(y)d Fl(2)f Fq(Z)1639 2748 y Fm(0)1191 2848 y Fq(s)p Fs(\()p Fy(y)q Fs(\))148 b(if)19 b Fy(y)d Fl(2)f Fq(Z)t Fl(n)p Fq(Z)1709 2828 y Fm(0)1724 2848 y Fq(:)370 2946 y Fs(A)h(substitution)e Fq(t)i Fr(matches)23 b Fq(s)1011 2954 y Fu(1)1052 2946 y Fs(to)15 b Fq(s)1143 2954 y Fu(2)1185 2946 y Fs(if)h Fq(t)8 b Fl(\016)g Fq(s)1323 2954 y Fu(1)1363 2946 y Fs(=)15 b Fq(s)1447 2954 y Fu(2)1472 2946 y Fs(.)25 b(If)16 b Fq(s)1591 2954 y Fu(1)1632 2946 y Fs(and)f Fq(s)1763 2954 y Fu(2)1804 2946 y Fs(matc)n(h)h(then)f(w)n(e)h(write)304 3012 y Fq(s)330 3020 y Fu(1)370 3012 y Fl(\026)f Fq(s)454 3020 y Fu(2)479 3012 y Fs(.)370 3079 y(If)20 b Fq(s)e Fl(\026)f Fq(t)j Fs(and)f Fq(t)f Fl(\026)g Fq(s)i Fs(then)e(w)n(e)i (write)f Fq(s)f Fl(\021)f Fq(t)p Fs(,)k(or)e Fq(s)h Fs(is)g(equiv)m (alen)n(t)f(to)g Fq(t)p Fs(.)30 b(Again)18 b(note)h(that)304 3145 y(then)e(the)g(matc)n(hing)h(substitutions)e(are)i(in)n(v)n (ertible)g(and)g(in)n(v)n(erses)h(of)f(eac)n(h)f(other.)24 b(Observ)n(e)304 3211 y(that)14 b Fl(\001)i Fs(is)g(an)g(in)n(v)n (ertible)g(substitution)d(whic)n(h)j(is)g(its)g(o)n(wn)f(in)n(v)n(erse) h(and)f(that)f(ev)n(ery)i(in)n(v)n(ertible)304 3278 y(substitution)g Fq(s)j Fs(satis\014es)e Fq(s)e Fl(\021)h(\001)p Fs(,)j(since)f Fl(\001)e(\026)g Fq(s)i Fs(\()p Fq(s)12 b Fl(\016)g(\001)k Fs(=)g Fq(s)p Fs(\))i(and)g Fq(s)d Fl(\026)h(\001)j Fs(\()p Fq(s)1826 3258 y Fm(\000)p Fu(1)1897 3278 y Fl(\016)12 b Fq(s)k Fs(=)f Fl(\001)p Fs(\).)26 b(W)-5 b(e)18 b(will)304 3344 y(therefore)d(write)j Fq(s)d Fl(\021)h(\001)i Fs(for)g(\\)p Fq(s)g Fs(is)h(in)n(v)n(ertible".)370 3411 y(W)-5 b(e)19 b(also)f(de\014ne)f(the)h(concepts)e(of)j Fr(mor)m(e)i(gener)m(al)27 b Fs(and)18 b Fr(most)j(gener)m(al)27 b Fs(matc)n(hing)18 b(substi-)304 3477 y(tutions)f(analogously)g(to)g(the)h(de\014nitions)f (for)g(pattern)f(matc)n(hing.)p eop %%Page: 6 8 6 7 bop 304 151 a Fu(6)252 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)370 283 y Fs(A)e(substitution)e Fq(t)j Fs(is)g(a)f Fr(gener)m(alization)26 b Fs(of)18 b(t)n(w)n(o)g(substitutions)e Fq(s)1758 291 y Fu(1)1802 283 y Fs(and)h Fq(s)1935 291 y Fu(2)1979 283 y Fs(under)g(substi-)304 350 y(tutions)g Fq(u)524 358 y Fu(1)567 350 y Fs(and)g Fq(u)706 358 y Fu(2)749 350 y Fs(if)i Fq(s)826 358 y Fu(1)866 350 y Fs(=)c Fq(u)956 358 y Fu(1)993 350 y Fl(\016)d Fq(t)19 b Fs(and)e Fq(s)1205 358 y Fu(2)1245 350 y Fs(=)f Fq(u)1336 358 y Fu(2)1373 350 y Fl(\016)c Fq(t)p Fs(.)370 416 y Fr(Mor)m(e)20 b(sp)m(e)m(ci\014c)h Fs(and)c Fr(most)i(sp)m(e)m (ci\014c)j Fs(generalizations)14 b(of)j(substitutions)d(are)i (de\014ned)f(anal-)304 483 y(ogously)i(to)h(the)f(concepts)f(for)i (pattern)e(generalization.)370 549 y(W)-5 b(e)13 b(sa)n(y)h(that)e Fq(s)h Fs(is)h(more)e(general)g(\(or)g(more)h(sp)r(eci\014c\))e(than)h Fq(t)h Fs(if)h Fq(s)h Fl(\036)h Fq(t)d Fs(\(or)g Fq(t)i Fl(\036)g Fq(s)p Fs(\).)23 b(Observ)n(e)304 616 y(that)18 b(the)h(concept)f(\\more)g(general")g(de\014ned)h(here)f(is)i(the)f (same)g(as)h(that)e(of)i(Section)e(1.5)i(if)304 682 y(restricted)15 b(to)j(matc)n(hing)f(substitutions.)964 795 y(2.)31 b Ft(Additional)20 b(Concepts)370 895 y Fs(Here)c(w)n(e)i(in)n(tro)r (duce)d(some)i(less)g(familiar)g(ideas)g(that)f(extend)g(the)g(basic)h (ideas)g(discussed)304 961 y(ab)r(o)n(v)n(e)h(in)i(w)n(a)n(ys)g(that)e (will)i(pro)n(v)n(e)f(helpful)g(to)g(our)f(discussion.)27 b(Sp)r(eci\014cally)-5 b(,)19 b(w)n(e)g(will)h(in)n(tro-)304 1028 y(duce)i(analogoues)f(to)h(patterns)f(and)h(substitutions)f(using) i(term-p)r(osition)e(notation,)i(and)304 1094 y(sho)n(w)i(ho)n(w)g(to)f (use)g(these)g(concepts)e(to)i(de\014ne)g(sp)r(eci\014c)f(rewritings)h (of)h(terms.)43 b(Some)25 b(of)304 1160 y(these)16 b(ideas)i(are)g(dra) n(wn)f(from)h([4,)h(3].)304 1260 y(2.1.)32 b Fy(T)-5 b(erm)21 b(p)r(ositions)g(and)h(replacemen)n(ts.)304 1360 y(De\014nition)f(2.1.1.)26 b Fs(W)-5 b(e)26 b(de\014ne)e(a)h(set)g Fq(P)8 b(os)24 b Fs(of)i Fr(term)i(p)m(ositions)p Fs(.)47 b(The)25 b(set)g(of)g(p)r(ositions)304 1426 y(presen)n(t)18 b(in)i(a)g(term)e Fq(t)i Fs(is)g(denoted)e Fq(P)8 b(os)p Fs(\()p Fq(t)p Fs(\).)28 b(W)-5 b(e)19 b(also)h(de\014ne)e(functions)g Fq(oper)h Fs(:)f Fq(T)8 b(er)r(m)2167 1434 y Fu(\006)p Fp(X)2253 1426 y Fl(\002)304 1493 y Fq(P)g(os)14 b Fl(!)i Fs(\006)8 b Fl([)g Fq(X)21 b Fs(and)15 b Fq(subter)r(m)f Fs(:)h Fq(T)8 b(er)r(m)1141 1501 y Fu(\006)p Fp(X)1222 1493 y Fl(\002)g Fq(P)g(os)15 b Fl(!)g Fq(T)8 b(er)r(m)1597 1501 y Fu(\006)p Fp(X)1686 1493 y Fs(suc)n(h)16 b(that)f Fq(oper)r Fs(\()p Fq(t;)9 b(p)p Fs(\))15 b(is)h(the)304 1559 y(op)r(erand)g(or)j(v)m(ariable)g(that)e(o)r(ccupies)g(p)r (osition)h Fq(p)h Fs(in)h Fq(t)f Fs(and)f Fq(subter)r(m)p Fs(\()p Fq(t;)9 b(p)p Fs(\))17 b(is)j(the)e(subterm)304 1625 y(starting)e(at)i(p)r(osition)f Fq(p)h Fs(in)h Fq(t)p Fs(.)370 1692 y(P)n(ositions)14 b(ma)n(y)h(b)r(e)f(represen)n(ted)d(as) j(sequences)f(of)h(nonzero)e(in)n(tegers)i(\(in)g(Dew)n(ey)h(decimal) 304 1758 y(notation\),)20 b(with)h(the)f(empt)n(y)h(sequence)e (denoting)g(the)h(ro)r(ot)g(p)r(osition,)h(and)f(the)g(sequence)304 1825 y(\()p Fq(i)345 1833 y Fu(1)369 1825 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b(i)520 1833 y Fp(n)551 1825 y Fs(\))21 b(denoting)e(the)i(p)r (osition)f(found)g(b)n(y)i(follo)n(wing)g(the)e Fq(i)1729 1833 y Fp(j)1753 1825 y Fs(th)h(subp)r(osition)e(of)i(eac)n(h)304 1891 y(p)r(osition)c(starting)f(at)i(the)f(ro)r(ot.)370 1958 y(The)h(relation)g Fl(\024)731 1966 y Fp(P)768 1958 y Fs(=)e Fl(f)p Fs(\()p Fq(l)q(;)9 b(m)p Fs(\))15 b Fl(j)h Fq(l)k Fs(is)e(a)g(pre\014x)g(of)g Fq(m)o Fl(g)h Fs(is)g(an)g(ordering) e(on)h Fq(P)8 b(os)p Fs(.)25 b(The)19 b(strict)304 2024 y(relation)e Fq(<)550 2032 y Fp(P)605 2024 y Fs(is)i(de\014ned)e(so)g (that)g(if)i Fq(l)d(<)1166 2032 y Fp(P)1218 2024 y Fq(m)i Fs(then)g Fq(l)e Fl(6)p Fs(=)f Fq(m)j Fs(and)g Fq(l)e Fl(\024)1754 2032 y Fp(P)1806 2024 y Fq(m)p Fs(.)370 2090 y(F)-5 b(or)18 b(con)n(v)n(enience,)g(w)n(e)g(will)i(refer)d(to)g Fq(oper)r Fs(\()p Fq(t;)9 b Fs(\(\)\))16 b(as)i(the)g(ro)r(ot)e(op)r (erator)f(of)j Fq(t)p Fs(.)304 2190 y Fy(Example)j(2.1.2.)26 b Fs(Consider)18 b(the)h(term)g Fq(t)f Fs(=)g Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(a;)9 b(b)p Fs(\))p Fq(;)g(c)p 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y(replacing)17 b(the)g(subterm)g(at) g Fq(p)i Fs(in)f Fq(t)p Fs(.)304 2723 y Fy(Example)j(2.1.4.)26 b Fs(Consider)21 b(again)i Fq(t)g Fs(=)g Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(a;)9 b(b)p Fs(\))p Fq(;)g(c)p Fs(\),)23 b(and)f(also)h Fq(u)g Fs(=)g Fq(\015)s Fs(\()p Fq(x;)9 b(y)r Fs(\))22 b(and)g Fq(v)j Fs(=)304 2790 y Fq(\033)r Fs(\()p Fq(d;)9 b(c)p Fs(\).)22 b(W)-5 b(e)13 b(then)f(ha)n(v)n(e)i Fl(h)p Fq(u)p Fl(i)905 2806 y Fu(\(1)p Fp(;)p Fu(2\))999 2790 y Fq(t)i Fs(=)f Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(a;)9 b(\015)s Fs(\()p Fq(x;)g(y)r Fs(\)\))p Fq(;)g(c)p Fs(\))i(and)i Fl(h)p Fq(u)p Fl(i)1698 2806 y Fu(\(1\))1757 2790 y Fq(t)i Fs(=)h Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(x;)9 b(y)r Fs(\))p Fq(;)g(c)p Fs(\).)21 b(Also,)304 2863 y Fq(v)f Fs(is)f(a)f(con)n(text)f(for)g(the)h(latter)e(replacemen) n(t)h(at)g(\(1\))g(in)i Fq(t)p Fs(.)25 b(Note)17 b(that)g Fl(h)p Fq(u)p Fl(i)1899 2880 y Fu(\(1\))1959 2863 y Fq(v)g Fs(=)e Fl(h)p Fq(u)p Fl(i)2136 2880 y Fu(\(1\))2196 2863 y Fq(t)p Fs(.)304 2966 y Fr(R)m(emark)28 b Fs(2.1.5)p Fr(.)f Fs(If)22 b Fq(u)g Fs(is)g(a)g(con)n(text)e(for)h(a)h(replacemen) n(t)e(at)h Fq(p)h Fs(in)g Fq(t)p Fs(,)h(then)e(for)g(an)n(y)h(term)f Fq(s)p Fs(,)304 3033 y Fl(h)p Fq(s)p Fl(i)373 3049 y Fp(p)398 3033 y Fq(u)15 b Fs(=)h Fl(h)p Fq(s)p Fl(i)573 3049 y Fp(p)598 3033 y Fq(t)p Fs(.)304 3134 y(2.2.)32 b Fy(T)-5 b(erm)21 b(Rewriting)g(Systems.)304 3234 y(De\014nition)g (2.2.1.)26 b Fs(A)18 b Fr(r)m(ewrite)k(r)m(elation)j Fs(is)18 b(a)g(relation)g Fl(!)g Fs(o)n(v)n(er)h Fq(P)8 b(at)1820 3214 y Fp(Y)1820 3249 y Fu(\006)p Fp(X)1911 3234 y Fs(whic)n(h)19 b(is:)370 3319 y(1.)28 b(closed)17 b(under)g(substitution:)23 b(if)18 b Fq(q)g Fl(!)d Fq(r)20 b Fs(then)d([)p Fq(s)p Fs(])p Fq(q)h Fl(!)e Fs([)p Fq(s)p Fs(])p Fq(r)r Fs(,)j(for)f(an)n(y)g Fq(s)d Fl(2)h Fq(S)s(ub)2088 3293 y Fp(Z)s Fm(\022)p Fp(Y)2088 3335 y Fu(\006)p Fp(X)2193 3319 y Fs(.)370 3386 y(2.)28 b(closed)e(under)f(con)n(text)h (replacemen)n(t:)40 b(if)27 b Fq(q)32 b Fl(!)e Fq(r)e Fs(then)e Fl(h)p Fq(u)p Fl(i)1796 3402 y Fp(p)1821 3386 y Fq(q)32 b Fl(!)e(h)p Fq(u)p Fl(i)2038 3402 y Fp(p)2063 3386 y Fq(r)f Fs(for)d(an)n(y)441 3462 y Fq(u)15 b Fl(2)g Fq(P)8 b(at)633 3442 y Fp(Y)633 3477 y Fu(\006)p Fp(X)725 3462 y Fs(and)17 b Fq(p)f Fl(2)f Fq(P)8 b(os)p Fs(\()p Fq(u)p Fs(\).)p eop %%Page: 7 9 7 8 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)569 b(7)304 283 y Fy(De\014nition)21 b(2.2.2.)26 b Fs(A)17 b Fr(r)m(ewrite)j(system)i Fs(is)17 b(a)g(relation)f Fq(R)h Fs(o)n(v)n(er)g Fq(P)8 b(at)1774 263 y Fp(Y)1774 299 y Fu(\006)p Fp(X)1865 283 y Fs(suc)n(h)16 b(that)g(for)g(eac)n(h) 304 350 y(\()p Fq(p;)9 b(q)r Fs(\))26 b Fl(2)g Fq(R)p Fs(,)i Fq(F)8 b(r)r Fs(\()p Fq(q)r Fs(\))24 b Fl(\022)i Fq(F)8 b(r)r Fs(\()p Fq(p)p Fs(\).)43 b(The)25 b(rewrite)e(relation)h Fl(!)1617 358 y Fp(R)1679 350 y Fs(is)h(the)f Fl(\022)p Fs(-minimal)j(rewrite)304 416 y(relation)17 b(con)n(taining)g Fq(R)p Fs(.)25 b(T)-5 b(erm)19 b Fq(s)f Fs(is)h(said)f(to)f Fr(r)m(ewrite)25 b Fs(to)17 b Fq(t)i Fs(if)g Fq(s)c Fl(!)1729 424 y Fp(R)1781 416 y Fq(t)p Fs(.)304 526 y Fy(De\014nition)21 b(2.2.3.)26 b Fs(A)13 b(rewrite)f(system)g Fq(R)i Fs(is)g Fr(terminating)20 b Fs(if)14 b(there)e(is)h(no)g(in\014nite)f(sequence) 304 592 y(of)18 b(terms)f Fq(t)541 600 y Fp(i)578 592 y Fs(\()p Fq(i)e Fl(2)g Fk(N)p Fs(\))j(with)g Fq(t)912 600 y Fu(0)952 592 y Fl(!)1007 600 y Fp(R)1059 592 y Fq(t)1079 600 y Fu(1)1120 592 y Fl(!)1175 600 y Fp(R)1227 592 y Fq(t)1247 600 y Fu(2)1281 592 y Fl(\001)9 b(\001)g(\001)h Fs(.)370 659 y(A)17 b(rewrite)e(system)i Fq(R)g Fs(is)h Fr(c)m(on\015uent)23 b Fs(if)18 b(whenev)n(er)d Fq(t)h Fl(!)1535 667 y Fp(R)1587 659 y Fq(u)h Fs(and)f Fq(t)f Fl(!)1832 667 y Fp(R)1884 659 y Fq(v)r Fs(,)j(there)d(exists)i(an)304 725 y Fq(x)h Fs(suc)n(h)g(that)f Fq(u)e Fl(!)701 733 y Fp(R)753 725 y Fq(x)j Fs(and)f Fq(v)h Fl(!)1010 733 y Fp(R)1062 725 y Fq(x)p Fs(.)370 792 y(A)g(rewrite)f(system)h(is)g Fr(c)m(omplete)26 b Fs(if)18 b(it)h(is)f(terminating)f(and)h (con\015uen)n(t.)370 858 y(W)-5 b(e)20 b(sa)n(y)f(that)f(a)i(rewrite)e (system)g(is)i Fr(de)m(cr)m(e)m(asing)28 b Fs(with)19 b(resp)r(ect)d(to)j Fq(f)k Fs(:)18 b Fq(T)8 b(er)r(m)2042 866 y Fu(\006)p Fp(X)2132 858 y Fl(!)17 b Fk(N)j Fs(if)304 925 y(for)d(ev)n(ery)i Fq(s;)9 b(t)19 b Fs(suc)n(h)e(that)g Fq(s)f Fl(!)965 933 y Fp(R)1017 925 y Fq(t)p Fs(,)j Fq(f)6 b Fs(\()p Fq(t)p Fs(\))14 b Fq(<)h(f)6 b Fs(\()p Fq(s)p Fs(\).)370 991 y(A)16 b(rewrite)e(system)h(is)h Fr(left-line)m(ar)24 b Fs(if)17 b(for)e(eac)n(h)g(\()p Fq(p;)9 b(q)r Fs(\))14 b Fl(2)i Fq(R)p Fs(,)h Fq(p)f Fs(is)g(linear.)23 b(A)16 b(rewrite)e(system)304 1057 y(is)k Fr(left-atomic)24 b Fs(if)19 b(for)f(eac)n(h)f(\()p Fq(p;)9 b(q)r Fs(\))15 b Fl(2)g Fq(R)p Fs(,)k Fq(p)g Fs(is)f(atomic.)370 1124 y(A)g(left-linear)g(rewrite)g(system)g Fq(R)h Fs(is)g Fr(ortho)m(gonal)27 b Fs(if)19 b(no)f(left-hand)g(side)g(of)h(a)f(rule) g(uni\014es)304 1190 y(with)c(a)g(renamed)e(non-v)m(ariable)h(subterm)g (\(other)f(than)h(itself)t(\))h(of)g(an)n(y)g(rule's)g(left-hand)f (side.)304 1257 y(Tw)n(o)21 b(patterns)e Fq(p)681 1265 y Fu(1)728 1257 y Fs(and)i Fq(p)867 1265 y Fu(2)913 1257 y Fr(unify)26 b Fs(if)c(there)e(is)h(a)h(substitution)d Fq(s)j Fs(suc)n(h)f(that)f([)p Fq(s)p Fs(])p Fq(p)2062 1265 y Fu(1)2108 1257 y Fs(=)h([)p Fq(s)p Fs(])p Fq(p)2256 1265 y Fu(2)2281 1257 y Fs(.)304 1323 y(F)-5 b(ormally)g(,)23 b Fq(R)e Fs(is)g(orthogonal)e(if)i(for)f(ev)n(ery)h(rule)f Fq(r)h Fs(=)f(\()p Fq(p;)9 b(q)r Fs(\),)21 b(for)f(ev)n(ery)h (left-hand)f(size)g(of)h(a)304 1389 y(rule)g Fq(s)h Fs(=)g(\()p Fq(p)581 1369 y Fm(0)596 1389 y Fq(;)9 b(q)647 1369 y Fm(0)663 1389 y Fs(\),)23 b(for)f(ev)n(ery)g(p)r(osition)f Fq(l)h Fl(2)g Fq(P)8 b(os)p Fs(\()p Fq(p)1422 1369 y Fm(0)1436 1389 y Fs(\))22 b(with)g Fq(oper)r Fs(\()p Fq(p)1768 1369 y Fm(0)1781 1389 y Fq(;)9 b(l)q Fs(\))28 b Fq(=)-34 b Fl(2)22 b Fq(X)t Fs(,)i(if)e Fq(r)i Fl(6)p Fs(=)e Fq(r)2210 1369 y Fm(0)2247 1389 y Fs(or)304 1456 y Fq(l)16 b Fl(6)p Fs(=)f(\(\))j(then)f(for)g(ev)n(ery)i(renaming)e Fq(t)h Fs(and)g(substitution)e Fq(s)p Fs(,)j([)p Fq(s)p Fs(])p Fq(p)d Fl(6)p Fs(=)f([)p Fq(s)p Fs(]\([)p Fq(t)p Fs(])p Fq(p)1885 1436 y Fm(0)1901 1456 y Fs(\).)370 1522 y(The)20 b(relation)g Fq(R)734 1502 y Fm(\000)p Fu(1)813 1522 y Fs(=)g Fl(f)p Fs(\()p Fq(q)r(;)9 b(p)p Fs(\))18 b Fl(j)i Fs(\()p Fq(p;)9 b(q)r Fs(\))19 b Fl(2)g Fq(R)p Fl(g)j Fs(is)f(the)f Fr(r)m(everse)27 b Fs(of)20 b Fq(R)p Fs(.)33 b(W)-5 b(e)21 b(denote)e Fl(!)2208 1533 y Fp(R)2241 1522 y Fi(\000)p Fh(1)304 1589 y Fs(b)n(y)g Fl( )436 1597 y Fp(R)472 1589 y Fs(.)25 b(A)18 b(rewrite)f(system)h Fq(R)h Fs(is)f Fr(r)m(eversible)26 b Fs(if)18 b Fq(R)1400 1569 y Fm(\000)p Fu(1)1479 1589 y Fs(is)g(a)h(rewrite)d(system.)370 1655 y(W)-5 b(e)20 b(sa)n(y)g(that)f(a)h(rewrite)e(system)h Fq(R)i Fs(is)f Fr(or)m(der)m(able)28 b Fs(if)20 b(there)e(is)j(an)e (ordering)f(of)i(the)f(rules)304 1722 y(of)g Fq(R)i Fs(suc)n(h)e(that)f (for)h(eac)n(h)g Fq(i)p Fs(,)i(ev)n(ery)f(op)r(erator)c(o)r(ccurring)i (in)i(the)e Fq(i)p Fs(th)i(rule's)f(left-hand)g(side)304 1788 y(o)r(ccurs)d(only)i(in)h(that)e(pattern)f(and)h(in)i(the)e(righ)n (t-hand)h(sides)f(of)h(later)g(rules.)370 1854 y(If)f Fq(t)e Fl(!)514 1862 y Fp(R)566 1854 y Fq(u)i Fs(implies)h Fq(t)d Fs(=)g Fq(u)i Fs(then)f Fq(t)h Fs(is)g(in)g Fq(R)p Fr(-normal)j(form)p Fs(.)25 b(If)17 b Fq(s)f Fl(!)1778 1862 y Fp(R)1830 1854 y Fq(t)h Fs(and)f Fq(t)h Fs(is)g(in)g(normal)304 1921 y(form)h(then)f Fq(t)h Fs(is)h(a)f Fq(R)p Fs(-normal)g(form)g (\(or)g(normalization\))e(of)i Fq(s)p Fs(.)304 2030 y Fy(Lemma)j(2.2.4.)27 b Fr(A)21 b(r)m(ewrite)i(system)e Fq(R)g Fr(is)g(r)m(eversible)j(if)d(and)h(only)f(if)g Fq(F)8 b(r)r Fs(\()p Fq(p)p Fs(\))15 b(=)j Fq(F)8 b(r)r Fs(\()p Fq(q)r Fs(\))19 b Fr(for)304 2106 y(e)m(ach)i Fs(\()p Fq(p;)9 b(q)r Fs(\))15 b Fl(2)g Fq(R)p Fr(,)20 b(and)h(if)f Fq(R)g Fr(is)f(r)m(eversible)k(then)e Fq(R)1390 2086 y Fm(\000)p Fu(1)1470 2106 y Fr(is)e(r)m(eversible)k(and)d Fq(R)1923 2086 y Fm(\000)p Fu(1)1983 2075 y Fm(\000)p Fu(1)2058 2106 y Fs(=)15 b Fq(R)p Fr(.)304 2216 y(Pr)m(o)m(of.)28 b Fs(T)-5 b(rivial.)p 2256 2216 3 38 v 2259 2181 34 3 v 2259 2216 V 2291 2216 3 38 v 304 2341 a Fy(Lemma)21 b(2.2.5.)27 b Fr(De)m(cr)m(e)m(asing)22 b(systems)e(ar)m(e)h (terminating.)304 2450 y(Pr)m(o)m(of.)28 b Fs(If)19 b(there)d(w)n(ere)i (a)g(system)g(decreasing)e(with)i(resp)r(ect)d(to)j(some)g Fq(f)24 b Fs(con)n(taining)17 b(a)i(non-)304 2517 y(terminating)14 b(deriv)m(ation)g Fq(t)880 2525 y Fu(0)921 2517 y Fl(!)976 2525 y Fp(R)1028 2517 y Fq(t)1048 2525 y Fp(i)1081 2517 y Fl(!)1136 2525 y Fp(R)1188 2517 y Fl(\001)9 b(\001)g(\001)i Fs(,)17 b(then)e Fq(f)6 b Fs(\()p Fq(t)1493 2525 y Fu(0)1517 2517 y Fs(\))15 b Fl(\025)h Fq(f)6 b Fs(\()p Fq(t)1688 2525 y Fu(1)1712 2517 y Fs(\))15 b Fl(\025)g(\001)9 b(\001)g(\001)27 b Fs(w)n(ould)16 b(constitute)304 2583 y(an)i(in\014nite)f(decreasing)f (sequence)h(of)h Fk(N)p Fs(.)p 2256 2583 V 2259 2548 34 3 v 2259 2583 V 2291 2583 3 38 v 304 2708 a Fy(Lemma)j(2.2.6.)27 b Fr(If)c(a)h(r)m(ewrite)i(system)d Fq(R)h Fr(is)f(left-atomic)j(and)f (left-line)m(ar,)h(and)e(its)g(rules)304 2775 y(have)d(distinct)g (left-hand)h(r)m(o)m(ot)f(op)m(er)m(ators)i(\(that)d(is,)g(no)g(two)g (left-hand)i(sides)942 2882 y Fq(\033)r Fs(\()p Fy(y)1032 2890 y Fu(1)1056 2882 y Fq(;)9 b(:)g(:)g(:)20 b(;)9 b Fy(y)1221 2890 y Fp(n)1252 2882 y Fs(\))p Fr(,)19 b Fq(\033)1344 2859 y Fm(0)1359 2882 y Fs(\()p Fy(z)1409 2890 y Fu(1)1434 2882 y Fq(;)9 b(:)g(:)g(:)20 b(;)9 b Fy(z)1593 2890 y Fp(m)1636 2882 y Fs(\))304 2989 y Fr(ar)m(e)21 b(such)f(that)h Fq(\033)c Fs(=)e Fq(\033)778 2969 y Fm(0)793 2989 y Fr(\),)20 b(then)h Fq(R)f Fr(is)g(ortho)m(gonal.)304 3099 y(Pr)m(o)m(of.)28 b Fs(The)21 b(only)h(non)n(trivial)g(subterm)e(of)h(an)g(atomic)g(term) f(is)i(the)f(term)g(itself.)34 b(No)21 b(t)n(w)n(o)304 3165 y(terms)15 b(with)h(di\013eren)n(t)g(ro)r(ot)e(op)r(erators)f(can) j(b)r(e)f(uni\014ed.)23 b(Since)16 b Fq(R)h Fs(is)f(left-linear,)h(it)f (m)n(ust)g(b)r(e)304 3232 y(orthogonal.)p 2256 3232 V 2259 3197 34 3 v 2259 3232 V 2291 3232 3 38 v 304 3357 a Fy(Theorem)22 b(2.2.7)e Fs(\(Rosen)d([23)o(]\))p Fy(.)28 b Fr(Ortho)m(gonal)23 b(systems)c(ar)m(e)i(c)m(on\015uent.)304 3466 y Fy(Lemma)g(2.2.8.)27 b Fr(Finite)20 b(or)m(der)m(e)m(d)j (systems)d(ar)m(e)h(terminating.)p eop %%Page: 8 10 8 9 bop 304 151 a Fu(8)252 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)304 283 y Fr(Pr)m(o)m(of.)28 b Fs(W)-5 b(e)17 b(pro)r(ceed)c(b)n(y)k(induction)e(on)h(the)f(n)n(um)n (b)r(er)h(of)g(rules.)24 b(Supp)r(ose)14 b(that)h(there)f(are)i(no)304 350 y(rules.)24 b(The)17 b(empt)n(y)g(system)g(is)h(ordered)d(and)i(it) h(terminates.)k(Supp)r(ose)16 b(that)g(ev)n(ery)i(ordered)304 416 y(system)k(with)g(few)n(er)g(than)f Fq(n)i Fs(rules)f(terminates.) 35 b(Let)22 b Fq(R)h Fs(b)r(e)e(an)i(ordered)d(rewrite)h(system)304 483 y(with)e Fq(n)g Fs(rules)g(\()p Fq(r)664 491 y Fu(1)688 483 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b(r)845 491 y Fp(n)875 483 y Fs(\))19 b(in)g(order.)26 b(Put)19 b(\()p Fq(p;)9 b(q)r Fs(\))15 b(=)i Fq(r)1483 491 y Fp(n)1513 483 y Fs(.)28 b(Assume)18 b(that)g Fq(t)1903 491 y Fu(0)1944 483 y Fl(!)1999 491 y Fp(R)2052 483 y Fq(t)2072 491 y Fu(1)2114 483 y Fl(!)2169 491 y Fp(R)2222 483 y Fl(\001)9 b(\001)g(\001)304 549 y Fs(is)26 b(an)g(in\014nite)f(deriv)m(ation)g (of)h(a)g(pattern)e Fq(t)p Fs(.)49 b(If)26 b(there)e(w)n(ere)i (\014nitely)g(man)n(y)h(uses)e(of)h Fq(r)2251 557 y Fp(n)2281 549 y Fs(,)304 616 y(then)19 b(for)g(some)h Fq(m)p Fs(,)h Fq(t)764 624 y Fp(m)824 616 y Fl(!)879 624 y Fp(R)935 616 y Fq(t)955 624 y Fp(m)p Fu(+1)1071 616 y Fl(!)1126 624 y Fp(R)1182 616 y Fl(\001)9 b(\001)g(\001)31 b Fs(w)n(ould)20 b(b)r(e)f(an)h(in\014nite)g(deriv)m(ation)f(using)g(only)304 682 y(the)e(\014rst)h Fq(n)13 b Fl(\000)f Fs(1)19 b(rules,)f(con)n (tradicting)e(the)i(induction)f(h)n(yp)r(othesis.)24 b(Therefore)17 b(there)g(m)n(ust)304 748 y(b)r(e)24 b(in\014nitely)i (man)n(y)g(uses)f(of)g Fq(r)1004 756 y Fp(n)1060 748 y Fs(in)h(the)f(in\014nite)g(deriv)m(ation.)46 b(Ho)n(w)n(ev)n(er,)27 b(there)e(can)f(b)r(e)304 815 y(at)f(most)h(\014nitely)g(man)n(y)h(o)r (ccurrences)20 b(of)k Fq(p)h Fs(in)f Fq(t)p Fs(,)j(and)c(no)h(more)g (can)f(b)r(e)h(in)n(tro)r(duced)e(b)n(y)304 881 y(the)i(earlier)g (rules)g(b)r(ecause)e(they)i(cannot)f(con)n(tain)h(an)n(y)h(of)g(the)f (op)r(erators)d(in)k Fq(p)g Fs(in)g(their)304 948 y(righ)n(t-hand)d (sides.)40 b(Consequen)n(tly)-5 b(,)24 b(there)e(cannot)g(b)r(e)g (in\014nitely)i(man)n(y)g(uses)e(of)h(the)g(last)304 1014 y(rule)d(in)h(an)n(y)g(deriv)m(ation.)30 b(Therefore)18 b(it)j(is)g(imp)r(ossible)f(for)g(an)n(y)g(term)g(to)g(ha)n(v)n(e)h(an) f(in\014nite)304 1080 y(deriv)m(ation,)i(so)f(the)g Fq(R)i Fs(terminates.)34 b(Consequen)n(tly)21 b(b)n(y)h(induction,)g(ev)n(ery) g(\014nite)g(ordered)304 1147 y(system)17 b(terminates.)p 2256 1147 3 38 v 2259 1112 34 3 v 2259 1147 V 2291 1147 3 38 v 304 1251 a Fy(Lemma)k(2.2.9.)27 b Fr(If)17 b Fq(R)g Fr(is)g(c)m(omplete)i(then)f(every)h(term)f Fq(s)f Fr(has)g(exactly)h (one)g Fq(R)p Fr(-normal)g(form.)304 1353 y(Pr)m(o)m(of.)28 b Fs(Since)20 b Fq(R)i Fs(is)g(complete,)e(it)h(terminates)e(and)i(is)g (con\015uen)n(t.)31 b(Clearly)21 b Fq(s)g Fs(has)g(at)f(least)304 1419 y(one)c(normal)h(form)g(b)r(ecause)e Fq(R)j Fs(terminates.)k(Supp) r(ose)15 b Fq(s)g Fl(!)1592 1427 y Fp(R)1644 1419 y Fq(t)j Fs(and)e Fq(s)g Fl(!)1885 1427 y Fp(R)1937 1419 y Fq(u)h Fs(and)g(b)r(oth)e(are)304 1486 y(in)h(normal)g(form.)24 b(By)17 b(con\015uence,)d(there)g(exists)i Fq(x)g Fs(suc)n(h)g(that)f Fq(t)g Fl(!)1739 1494 y Fp(R)1791 1486 y Fq(x)i Fs(and)e Fq(u)g Fl(!)2047 1494 y Fp(R)2099 1486 y Fq(x)p Fs(.)24 b(Then)304 1552 y Fq(t)15 b Fs(=)h Fq(x)f Fs(=)g Fq(u)p Fs(.)p 2256 1552 V 2259 1517 34 3 v 2259 1552 V 2291 1552 3 38 v 304 1657 a Fy(Theorem)22 b(2.2.10.)92 b Fs(1.)28 b Fr(If)20 b Fq(R)g Fr(c)m(onver)m(ges)j(then)e(every)g(term)h Fq(s)d Fr(and)i Fq(t)f Fr(with)g Fq(t)15 b Fl( )2091 1665 y Fp(R)2143 1657 y Fq(s)20 b Fr(have)441 1723 y(the)h(same)f Fq(R)p Fr(-normal)i(form.)370 1789 y Fs(2.)28 b Fr(If)18 b Fq(s)g Fr(is)g(in)g Fq(R)p Fr(-normal)i(form)e(and)h Fq(t)c Fl( )1243 1797 y Fp(R)1295 1789 y Fq(s)j Fr(and)h Fq(R)g Fr(c)m(onver)m(ges)i(then)e Fq(s)f Fr(is)f(the)j Fq(R)p Fr(-normal)441 1856 y(form)h(of)f Fq(t)p Fr(.)304 1957 y(Pr)m(o)m(of.)95 b Fs(1.)27 b(Supp)r(ose)20 b Fq(x)i Fs(is)g(an)g Fq(R)p Fs(-normal)g(form)f(of)h Fq(t)p Fs(.)36 b(Clearly)22 b Fq(s)f Fl(!)1851 1965 y Fp(R)1909 1957 y Fq(t)h Fl(!)2006 1965 y Fp(R)2064 1957 y Fq(x)p Fs(,)h(so)e Fq(x)h Fs(is)441 2024 y(also)e(an)f Fq(R)p Fs(-normal)i(form)f(for)f Fq(s)p Fs(.)31 b(Because)18 b Fq(R)j Fs(con)n(v)n(erges,)e(b)n(y)i (2.2.9)f Fq(x)g Fs(is)h(the)e(unique)441 2090 y(normal)f(form)f(for)h (b)r(oth)f Fq(s)h Fs(and)g Fq(t)p Fs(.)370 2157 y(2.)28 b(A)18 b(corollary)f(of)h(1.)p 2256 2223 V 2259 2188 34 3 v 2259 2223 V 2291 2223 3 38 v 304 2327 a Fr(R)m(emark)28 b Fs(2.2.11)p Fr(.)f Fs(There)13 b(is,)i(of)f(course,)g(m)n(uc)n(h)g (more)g(to)f(rewrite)g(systems)g(than)g(this,)i(but)e(w)n(e)304 2394 y(will)19 b(only)g(need)e(to)h(study)f(simple)i(rewriting)e (systems)g(for)h(whic)n(h)g(no)g(adv)m(anced)e(discussion)304 2460 y(is)h(required.)23 b(The)17 b(ab)r(o)n(v)n(e)g(lemmas)g(su\016ce) f(to)g(sho)n(w)h(that)f(the)g(rewrite)g(systems)g(w)n(e)i(will)g(use) 304 2527 y(to)i(enco)r(de)f(terms)g(terminate,)h(and)g(that)g(those)f (w)n(e)i(use)g(to)f(deco)r(de)f(terms)g(con)n(v)n(erge,)i(and)304 2593 y(moreo)n(v)n(er)c(that)g(deco)r(ding)f(an)n(y)j(enco)r(ding)d(of) i(a)g(term)g(restores)e(the)h(term.)304 2695 y(2.3.)32 b Fy(P)n(ositions)20 b(and)i(substitutions.)304 2796 y(De\014nition)f(2.3.1.)26 b Fs(Giv)n(en)c(a)f(term)f(or)h(pattern)d Fq(t)p Fs(,)23 b(a)d Fr(\(p)m(ositional\))k(substitution)i Fs(with)21 b(re-)304 2863 y(sp)r(ect)16 b(to)i Fq(t)g Fs(o)n(v)n(er)h(v)m(ariable)e(set)h Fq(Z)k Fs(is)d(a)f(relation)f Fq(s)e Fl(\022)h Fq(Z)g Fl(\002)c Fq(P)c(os)p Fs(\()p Fq(t)p Fs(\))16 b(suc)n(h)i(that)370 2942 y(1.)28 b Fq(subter)r(m)p Fs(\()p Fq(t;)9 b(l)q Fs(\))k(=)i Fq(subter)r(m)p Fs(\()p Fq(t;)9 b(l)1110 2922 y Fm(0)1124 2942 y Fs(\))15 b(for)h(ev)n(ery)g Fq(l)q(;)9 b(l)1448 2922 y Fm(0)1479 2942 y Fl(2)15 b Fq(P)8 b(os)p Fs(\()p Fq(t)p Fs(\))14 b(and)h Fy(z)h Fl(2)f Fq(Z)20 b Fs(with)c(\()p Fy(z)p Fq(;)9 b(l)q Fs(\))15 b Fl(2)g Fq(s)441 3009 y Fs(and)i(\()p Fy(z)p Fq(;)9 b(l)640 2989 y Fm(0)656 3009 y Fs(\))15 b Fl(2)g Fq(s)370 3075 y Fs(2.)28 b(for)18 b(ev)n(ery)g Fy(z)d Fl(2)h Fq(Z)t Fs(,)i(there)f(exists)h(a)g Fq(l)e Fl(2)f Fq(P)8 b(os)p Fs(\()p Fq(t)p Fs(\))17 b(suc)n(h)g(that)g(\()p Fy(z)p Fq(;)9 b(l)q Fs(\))15 b Fl(2)h Fq(s)p Fs(.)304 3155 y(The)i(set)g(of)g (p)r(ositional)f(substitutions)g(with)h(resp)r(ect)e(to)i Fq(t)h Fs(o)n(v)n(er)g Fq(Z)j Fs(is)d(denoted)d Fq(P)8 b(S)s(ub)2157 3135 y Fp(Z)2157 3169 y(t)2192 3155 y Fs(.)26 b(As)304 3221 y(for)17 b(v)m(ariable)h(substitutions,)e(w)n(e)j(ma)n(y) g(use)f(the)f(notation)998 3314 y Fq(s)e Fs(=)h Fy(z)1126 3322 y Fu(1)1166 3314 y Fl(7!)g Fq(l)1254 3322 y Fu(1)1278 3314 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b Fy(z)1438 3322 y Fp(n)1484 3314 y Fl(7!)16 b Fq(l)1572 3322 y Fp(n)304 3407 y Fs(for)981 3477 y Fq(s)g Fs(=)f Fl(f)p Fs(\()p Fy(z)1159 3485 y Fu(1)1183 3477 y Fq(;)9 b(l)1224 3485 y Fu(1)1249 3477 y Fs(\))p Fq(;)g(:)g(:)g(:)20 b(;)9 b Fs(\()p Fy(z)1452 3485 y Fp(n)1483 3477 y Fq(;)g(l)1524 3485 y Fp(n)1554 3477 y Fs(\))p Fl(g)p Fq(:)p eop %%Page: 9 11 9 10 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)569 b(9)370 283 y Fs(Giv)n(en)22 b(a)f(p)r(ositional)f(substitution)e Fq(s)i Fl(2)g Fq(P)8 b(S)s(ub)1389 263 y Fp(Z)1389 297 y(t)1445 283 y Fs(and)20 b Fy(z)g Fl(2)g Fq(Z)t Fs(,)i(the)e(set)g Fl(f)p Fq(subter)r(m)p Fs(\()p Fq(t;)9 b(l)q Fs(\))17 b Fl(j)304 350 y Fs(\()p Fy(z)p Fq(;)9 b(l)q Fs(\))15 b Fl(2)g Fq(s)p Fl(g)j Fs(has)g(only)g(one)g(mem)n(b)r(er,)f(whic)n(h)i (w)n(e)f(will)i(denote)c(b)n(y)j Fq(s)1732 358 y Fp(t)1751 350 y Fs(\()p Fy(z)p Fs(\).)370 416 y(The)j(result)g(of)g(applying)h(a) f(p)r(ositional)g(substitution)e(to)i(a)h(pattern)d Fq(p)j Fs(is)g(a)g(pattern)d Fq(p)2281 396 y Fm(0)304 483 y Fs(iden)n(tical)13 b(to)g Fq(p)g Fs(except)f(that)g(ev)n(ery)h(o)r (ccurrence)d(of)j(a)g(v)m(ariable)g Fy(z)j Fl(2)f Fq(Z)i Fs(in)d Fq(p)g Fs(has)e(b)r(een)g(replaced)304 549 y(b)n(y)19 b Fq(s)407 557 y Fp(t)426 549 y Fs(\()p Fy(z)p Fs(\).)24 b(W)-5 b(e)19 b(also)f(denote)e(application)h(b)n(y)i([)p Fq(s)p Fs(])1341 557 y Fp(t)1361 549 y Fq(p)p Fs(.)304 649 y Fy(Example)i(2.3.2.)26 b Fs(Consider)17 b Fq(t)f Fs(=)g Fq(\033)r Fs(\()p Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(x;)9 b(a)p Fs(\))p Fq(;)g(y)r Fs(\))p Fq(;)g(\015)s Fs(\()p Fq(x;)g(a)p Fs(\)\).)23 b(Supp)r(ose)16 b Fq(s)g Fs(=)g Fy(y)g Fl(7!)g Fs(\(1\))p Fq(;)9 b Fy(z)16 b Fl(7!)304 716 y Fs(\(2\).)27 b(Then)19 b Fq(s)590 724 y Fp(t)609 716 y Fs(\()p Fy(y)q Fs(\))d(=)i Fq(\033)r Fs(\()p Fq(\015)s Fs(\()p Fq(x;)9 b(a)p Fs(\))p Fq(;)g(y)r Fs(\))18 b(and)h 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Fs(\()p Fy(z)p Fs(\))h(for)f(eac)n(h)h Fy(z)f Fl(2)f Fq(Z)t Fs(.)24 b(In)18 b(other)e(w)n(ords,)304 1807 y(if)i (substitution)f Fq(s)h Fs(matc)n(hes)f(p)r(ositional)g(substitution)f Fq(u)i Fs(under)f(p)r(ositional)g(substitution)f Fq(v)r Fs(,)304 1873 y(then)h(for)h(eac)n(h)f Fy(z)i Fs(in)f(the)g(domain)f (of)i Fq(v)r Fs(,)f Fq(u)g Fs(matc)n(hes)f Fq(s)p Fs(\()p Fy(z)p Fs(\))h(and)g Fq(v)1683 1881 y Fp(t)1702 1873 y Fs(\()p Fy(z)p Fs(\).)304 1973 y Fy(Example)j(2.3.4.)26 b Fs(Consider)e(again)h Fq(t)h Fs(as)g(ab)r(o)n(v)n(e.)47 b(The)25 b(pattern)e Fq(p)28 b Fs(=)g Fq(\033)r Fs(\()p Fy(y)q Fq(;)9 b Fy(z)p Fs(\))25 b(matc)n(hes)304 2040 y Fq(t)g Fs(at)g(\(\))f(under)g Fq(s)h Fs(as)g(ab)r(o)n(v)n(e,)i(and)e (at)f(\(1\))g(under)g Fl(f)p Fs(\()p Fy(y)q Fq(;)9 b Fs(\(1)p Fq(;)g Fs(1\)\))p Fq(;)g Fs(\()p Fy(z)p Fq(;)g Fs(\(1)p Fq(;)g Fs(2\)\))p Fl(g)p Fs(.)43 b(The)25 b(pattern)304 2106 y Fq(p)332 2086 y Fm(0)362 2106 y Fs(=)16 b Fq(\033)r Fs(\()p Fq(\033)r Fs(\()p Fy(y)q Fq(;)9 b(y)r 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Fq(\033)r Fs(\()p Fy(y)q Fq(;)9 b Fy(z)p Fs(\))24 b(in)i Fq(t)p Fs(,)h(namely)f Fq(o)1355 3419 y Fu(1)1406 3411 y Fs(=)h(\(\(\))p Fq(;)9 b Fy(y)27 b Fl(7!)g Fs(\(1\))p Fq(;)9 b Fy(z)27 b Fl(7!)g Fs(\(2\)\))d(and)h Fq(o)2202 3419 y Fu(2)2253 3411 y Fs(=)304 3477 y(\(\(1\))p Fq(;)9 b Fy(y)19 b Fl(7!)i Fs(\(1)p Fq(;)9 b Fs(1\))p Fq(;)g Fy(z)20 b Fl(7!)g Fs(\(1)p Fq(;)9 b Fs(2\)\).)32 b(These)20 b(o)r(ccurrences)e(co)n(v)n(er)j Fl(f)p Fs(\(\))p Fl(g)e Fs(and)i Fl(f)p Fs(\(1\))p Fl(g)e Fs(resp)r(ectiv)n(ely)-5 b(,)p eop %%Page: 10 12 10 11 bop 304 151 a Fu(10)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)304 283 y Fs(so)27 b(they)f(are)g(indep)r(enden)n(t.)49 b(There)26 b(is)h(also)g(an)g(o)r (ccurrence)c(of)k Fq(p)1800 263 y Fm(0)1846 283 y Fs(=)j Fq(\033)r Fs(\()p Fq(\033)r Fs(\()p Fy(y)q Fq(;)9 b(y)r Fs(\))p Fq(;)g Fy(y)q Fs(\))25 b(at)304 350 y Fq(o)331 358 y Fu(3)371 350 y Fs(=)15 b(\(\(1\))p Fq(;)9 b Fy(y)15 b Fl(7!)h Fs(\(1)p Fq(;)9 b Fs(1\))p Fq(;)g Fy(y)15 b 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Fs(whic)n(h)i(coun)n(ts)f(the)g(n)n(um)n(b)r(er)h(of)g(sym)n(b)r(ols) g(o)r(ccurring)304 3411 y(in)22 b(a)g(term)g(is)h(a)f(term)f(cost)g (metric.)36 b(It)22 b(is)h(also)f(based)f(on)h(the)f(op)r(erator)e (cost)j(function)e Fq(c)304 3477 y Fs(whic)n(h)e(assigns)g(a)g(cost)f (of)h(1)g(to)g(eac)n(h)f(op)r(erator,)f(constan)n(t,)g(and)i(v)m (ariable.)p eop %%Page: 12 14 12 13 bop 304 151 a Fu(12)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)370 283 y Fs(Similarly)-5 b(,)21 b(the)d(dictionary)f(cost)h(metric)g Fq(N)1308 291 y Fu(#)1366 283 y Fs(sums)h(the)e(n)n(um)n(b)r(er)h(of)h(sym)n(b)r (ols)g(of)f(all)h(the)304 350 y(en)n(tries)e(in)i(the)e(dictionary)-5 b(.)304 449 y Fy(De\014nition)21 b(1.1.6.)26 b Fs(Giv)n(en)c(a)f(term)f (cost)g(metric)g Fq(M)6 b Fs(,)22 b(dictionary)e(cost)f(metric)i Fq(N)6 b Fs(,)22 b(and)f(a)304 516 y(m)n(ultiset)15 b(of)h(\006)p Fq(X)t Fs(-terms)g Fl(f)p Fq(t)885 524 y Fu(1)909 516 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b(t)1061 524 y Fp(n)1092 516 y Fl(g)p Fs(,)16 b(the)f Fr(term)k(c)m(ompr)m(ession)h(pr)m(oblem)j Fs(is)17 b(the)e(problem)g(of)304 582 y(\014nding)k(a)h(dictionary)e Fq(D)k Fs(o)n(v)n(er)e(\006)1030 562 y Fm(0)1066 582 y Fs(and)f(a)h(m)n(ultiset)g(of)f(\006)14 b Fl([)f Fs(\006)1647 562 y Fm(0)1663 582 y Fq(X)t Fs(-terms)19 b Fl(f)p Fq(t)1935 562 y Fm(0)1935 596 y Fu(1)1960 582 y Fq(;)9 b(:)g(:)g(:)20 b(;)9 b(t)2111 562 y Fm(0)2111 596 y Fp(n)2142 582 y Fl(g)20 b Fs(suc)n(h)304 649 y(that)d Fq(t)444 628 y Fm(0)444 663 y Fp(i)477 649 y Fl( )532 657 y Fp(D)588 649 y Fq(t)608 657 y Fp(i)645 649 y Fs(for)h(eac)n(h)f Fq(i)p Fs(,)i(\(1)c Fl(\024)g Fq(i)h Fl(\024)f Fq(n)p Fs(\),)k(and)921 783 y Fq(N)6 b Fs(\()p Fq(D)r Fs(\))11 b(+)1156 714 y Fp(n)1129 731 y Fn(X)1133 848 y Fp(i)p Fu(=1)1219 783 y Fq(M)6 b Fs(\()p Fq(t)1321 760 y Fm(0)1321 797 y Fp(i)1338 783 y Fs(\))15 b Fq(<)1460 714 y Fp(n)1433 731 y Fn(X)1438 848 y Fp(i)p Fu(=1)1523 783 y Fq(M)6 b Fs(\()p Fq(t)1625 791 y Fp(i)1642 783 y Fs(\))p Fq(:)304 921 y Fs(An)19 b Fr(optimal)27 b Fs(solution)18 b(to)h(the)g(term)f (compression)g(problem)g(is)i(a)g(dictionary)e(and)h(a)g(set)g(of)304 988 y(enco)r(dings)d(suc)n(h)i(that)802 946 y Fn(P)861 960 y Fp(n)861 1004 y(i)p Fu(=1)945 988 y Fq(M)6 b Fs(\()p Fq(T)1067 967 y Fm(0)1059 1002 y Fp(i)1081 988 y Fs(\))18 b(is)h(the)e(smallest)h(among)f(all)i(solutions.)304 1087 y Fr(R)m(emark)28 b Fs(1.1.7)p Fr(.)f Fs(There)22 b(are)g(three)g(parts)f(of)i(the)f(compression)f(system:)34 b(dictionary)22 b(con-)304 1153 y(struction,)16 b(enco)r(ding,)g(and)h (deco)r(ding.)22 b(Dictionary)c(construction)c(consists)i(of)i(a)g (searc)n(h)e(for)304 1220 y(appropriate)g(patterns)h(whic)n(h)j(are)e (rep)r(eated)f(in)j(the)e(input.)28 b(These)18 b(patterns)f(b)r(ecome)g (the)304 1286 y(righ)n(t)24 b(sides)g(of)g(rules)g(in)h(a)f(dictionary) g Fq(D)r Fs(.)43 b(Enco)r(ding)22 b(corresp)r(onds)f(to)j Fl( )1980 1294 y Fp(D)2021 1286 y Fs(-deriv)m(ation,)304 1353 y(with)f(the)f(goal)h(of)g(\014nding)g(deriv)m(ations)e(whic)n(h)j (minimize)g(the)e Fq(M)6 b Fs(-v)m(alues)23 b(of)g(the)f(results.)304 1419 y(Deco)r(ding)16 b(corresp)r(onds)f(to)j Fl(!)974 1427 y Fp(D)1014 1419 y Fs(-normalization.)370 1485 y(Dictionary)13 b(construction)d(is)j(complex)g(and)f(will)j(b)r(e)d(studied)g(more)g (closely)h(in)h(the)e(section)304 1552 y(on)24 b(complexit)n(y)-5 b(,)26 b(along)e(with)g(enco)r(ding.)40 b(Enco)r(ding)22 b(and)i(deco)r(ding)e(ha)n(v)n(e)j(the)e(follo)n(wing)304 1618 y(prop)r(erties:)304 1718 y Fy(Theorem)f(1.1.8.)k Fr(Enc)m(o)m(ding)c(a)e Fs(\006)p Fq(X)t Fr(-term)h(with)f(r)m(esp)m(e) m(ct)i(to)f(dictionary)h Fq(D)f Fr(terminates.)304 1817 y(Pr)m(o)m(of.)28 b Fs(The)13 b(relation)e Fl( )838 1825 y Fp(D)892 1817 y Fs(built)i(from)f(the)g(rev)n(erse)g(of)g(pattern)f (dictionary)h Fq(D)i Fs(is)f(decreasing,)304 1884 y(so)18 b(is)g(guaran)n(teed)e(to)i(terminate.)p 2256 1884 3 38 v 2259 1848 34 3 v 2259 1884 V 2291 1884 3 38 v 304 1983 a Fy(Theorem)k(1.1.9.)k Fr(De)m(c)m(o)m(ding)20 b(any)d(enc)m(o)m(ding)j(of)d(a)h Fs(\006)p Fq(X)t Fr(-term)g Fq(t)f Fr(with)h(r)m(esp)m(e)m(ct)h(to)f(dictionary)304 2050 y Fq(D)j Fr(terminates)h(and)f(pr)m(o)m(duc)m(es)g Fq(t)p Fr(.)304 2149 y(Pr)m(o)m(of.)28 b Fs(According)21 b(to)i(Lemma)f(2.2.6)h(of)g(P)n(art)f(1,)j(pattern)c(dictionaries)g (are)h(orthogonal,)304 2215 y(and)g(therefore)d(con\015uen)n(t)i(b)n(y) i(Theorem)e(2.2.7)i(of)f(P)n(art)g(1.)37 b(Since)22 b(dictionaries)f (are)g(\014nite)304 2282 y(and)j(orderable,)h(they)g(terminate.)43 b(Therefore)22 b Fq(D)27 b Fs(is)f(complete.)43 b(As)25 b(a)g(result,)h(the)e Fl(!)2256 2290 y Fp(D)304 2348 y Fs(relation)19 b(is)h(guaran)n(teed)d(to)i(terminate)f(and)i(pro)r (duce)d(a)j(unique)f(normal)h(form.)29 b(Since)19 b(the)304 2415 y(term)j Fq(t)i Fs(cannot)e(con)n(tain)h(an)n(y)h(sym)n(b)r(ols)f (of)h(\006)1308 2395 y Fm(0)1323 2415 y Fs(,)i(it)d(is)h(in)g Fq(D)r Fs(-normal)f(form,)i(so)f(if)g Fq(t)2117 2395 y Fm(0)2156 2415 y Fl(!)2211 2423 y Fp(D)2276 2415 y Fq(t)304 2481 y Fs(then)15 b(the)h Fq(D)r Fs(-normal)f(form)h(of)h Fq(t)984 2461 y Fm(0)1016 2481 y Fs(is)g Fq(t)p Fs(.)24 b(This)17 b(pro)n(v)n(es)f(that)f(deco)r(ding)f(an)i(enco)r(ding)f(of)h (a)g(term)304 2547 y(pro)r(duces)f(the)j(original)g(term.)p 2256 2547 V 2259 2512 34 3 v 2259 2547 V 2291 2547 3 38 v 304 2647 a Fy(Example)j(1.1.10.)26 b Fs(If)17 b(the)g(pattern)e Fq(p)g Fs(=)h Fq(\033)r Fs(\()p Fq(\033)r Fs(\()p Fq(x;)9 b Fy(y)1424 2655 y Fu(1)1447 2647 y Fs(\))p Fq(;)g Fy(y)1527 2655 y Fu(2)1551 2647 y Fs(\))17 b(o)r(ccurs)f(man)n(y)i(times)f(in)h (a)f(term,)304 2713 y(w)n(e)f(can)e(use)h(a)h(new)f(binary)g(op)r (erator)e Fq(\033)1155 2693 y Fm(0)1191 2713 y Fq(=)-34 b Fl(2)15 b Fs(\006.)24 b(An)15 b(appropriate)e(dictionary)i(w)n(ould)g (con)n(tain)304 2780 y(a)i(rewrite)g(rule)g Fq(\033)681 2760 y Fm(0)696 2780 y Fs(\()p Fy(y)752 2788 y Fu(1)776 2780 y Fq(;)9 b Fy(y)834 2788 y Fu(2)859 2780 y Fs(\))15 b Fl(!)h Fq(\033)r Fs(\()p Fq(\033)r 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y(problem)j(in)n(to)h(t)n(w)n(o)g(phases:)23 b(searc)n(hing)15 b(for)i(patterns,)e(and)h(enco)r(ding)f(the)h(term)g (with)h(these)304 3211 y(patterns.)k(P)n(attern)13 b(searc)n(h)h(tak)n (es)h(a)f(set)h(of)g(terms)f(as)g(input)h(and)f(yields)i(a)f(pattern)d (m)n(ultiset,)304 3278 y(with)22 b(m)n(ultiplicities)i(indicating)d (the)h(n)n(um)n(b)r(er)g(of)h(o)r(ccurrences)c(seen)i(in)i(the)f(term.) 38 b(T)-5 b(erm)304 3344 y(enco)r(ding)15 b(algorithms)h(are)h(giv)n (en)h(a)f(set)g(of)g(terms)f(and)h(a)g(pattern)f(m)n(ultiset)h(and)g (pro)r(duce)e(a)304 3411 y(pattern)f(dictionary)h(and)h(a)g(set)f(of)h (enco)r(ded)e(terms.)23 b(W)-5 b(e)17 b(will)g(presen)n(t)e(algorithms) g(for)h(b)r(oth)304 3477 y(these)g(phases)h(in)i(the)e(the)h(last)f(t)n (w)n(o)i(sections)e(of)h(P)n(art)f(2.)p eop %%Page: 13 15 13 14 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(13)370 283 y Fs(The)22 b(algorithm)e(for)i(deco)r(ding)d(an)j(enco)r (ded)d(term)i(using)g(a)h(dictionary)f(is)h(straigh)n(tfor-)304 350 y(w)n(ard,)c(so)g(w)n(e)h(will)g(not)e(dev)n(ote)h(an)n(y)g (discussion)g(to)f(it.)304 443 y(1.2.)32 b Fy(Sequen)n(tial)e (Compression.)e Fs(In)d(standard)f(sequen)n(tial)h(compression)f (algorithms)304 510 y(data)15 b(is)i(represen)n(ted)d(as)i(a)g (sequence)f Fq(s)h Fs(of)h(sym)n(b)r(ols)f(in)h(some)f(alphab)r(et)e (\000.)24 b(A)17 b(sequence)d(can)304 576 y(b)r(e)19 b(represen)n(ted)e(as)j(a)g(term,)g(with)g(unary)f(op)r(erators)e Fq(\033)1512 584 y Fp(\015)1561 576 y Fs(for)i(eac)n(h)h Fq(\015)h Fl(2)d Fs(\000)i(and)f(a)h(constan)n(t)304 643 y Fq(\017)27 b(=)-34 b Fl(2)21 b Fs(\000.)34 b(The)21 b(constan)n(t)e Fq(\017)j Fs(represen)n(ts)d(the)i(empt)n(y)g(string,)h (while)g(if)f Fq(s)1815 623 y Fm(0)1853 643 y Fs(represen)n(ts)e Fq(s)h Fl(2)h Fs(\000)2256 623 y Fm(\003)2281 643 y Fs(,)304 709 y(then)11 b Fq(\033)456 717 y Fp(\015)484 709 y Fs(\()p Fq(s)532 689 y Fm(0)547 709 y Fs(\))h(represen)n(ts)f(the)g(string)h Fq(\015)s(s)p Fs(.)23 b(This)12 b(transformation)e(can)i(also)g(b)r(e)g (easily)g(rev)n(ersed.)304 775 y(The)18 b(follo)n(wing)g(algorithm)f (formalizes)h(this)g(pro)r(cedure.)304 869 y Fy(Algorithm)i(1.2.1)h Fs(\()p Fy(trans)p Fs(\))p Fy(.)304 935 y Fs(Giv)n(en)e(alphab)r(et)d (\000)i(of)h(sym)n(b)r(ols,)f(de\014ne)f(op)r(erator)f(domain)i(\(\006) p Fq(;)9 b(\036)p Fs(\))18 b(so)g(that)f(\006)e(=)h Fl(f)p Fq(\033)2138 943 y Fp(\015)2181 935 y Fl(j)g Fq(\015)i Fl(2)304 1002 y Fs(\000)p Fl(g)11 b([)h(f)p Fq(\017)p Fl(g)19 b Fs(where)e Fq(\017)22 b(=)-34 b Fl(2)15 b Fs(\000,)j(and)g Fq(\036)p Fs(\()p Fq(\033)1037 1010 y Fp(\015)1064 1002 y Fs(\))d(=)h(1)i(for)g Fq(\015)g Fl(2)d Fs(\000,)j(and)g Fq(\036)p Fs(\()p Fq(\017)p Fs(\))d(=)g(0.)370 1071 y(De\014ne)j Fy(trans)684 1050 y Fu(\006)684 1085 y(\000)737 1071 y Fs(as)g(follo)n(ws:)370 1150 y(1.)28 b Fy(trans)583 1129 y Fu(\006)583 1164 y(\000)618 1150 y Fs(\()p Fq(e)p Fs(\))14 b(=)h Fq(\017)k Fs(if)g Fq(e)f Fs(is)h(the)e(empt)n(y)h 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y(that)h(tree,)i(there)e(can)h(b)r(e)g(at)g (most)g(2)1125 1757 y Fp(n)1177 1777 y Fs(di\013eren)n(t)g(patterns)e (matc)n(hing)i(an)n(y)h(term)f(b)r(ecause)304 1844 y(there)14 b(can)h(b)r(e)g(at)g(most)g(2)841 1824 y Fp(n)887 1844 y Fs(di\013eren)n(t)g(subsets)f(\(connected)e(or)k(otherwise\))d(of)j (v)n(ertices)f(of)g(the)304 1910 y(tree.)32 b(So)20 b(at)h(w)n(orst)f (the)g(n)n(um)n(b)r(er)h(of)g(patterns)d(can)i(b)r(e)h Fq(O)r Fs(\(2)1614 1890 y Fp(n)1643 1910 y Fs(\),)h(and)e(no)h(smaller) g(b)r(ound)e(is)304 1977 y(p)r(ossible)e(b)r(ecause)f(in)i(the)g(case)f (of)h(complete)f(binary)h(terms)f(the)g(m)n(um)n(b)r(er)h(of)g (patterns)e(can)304 2043 y(b)r(e)h(prop)r(ortional)f(to)h(2)793 2023 y Fp(n)823 2043 y Fs(.)370 2109 y(By)k(an)g(argumen)n(t)e(similar) i(to)f(that)f(for)h(sequen)n(tial)g(enco)r(dings)f(w)n(e)i(can)f(sho)n (w)g(that)f(the)304 2176 y(n)n(um)n(b)r(er)f(of)g(p)r(ossible)g(enco)r (dings)e(is)j Fq(O)r Fs(\(\()p Fq(T)1208 2184 y Fu(log)7 b Fp(n)1302 2176 y Fs(\))1324 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b(,)20 b(b)r(ecause)d(a)i(non-ro)r(ot)f(v)n (ertex)h(with)g(only)g(one)g(c)n(hild)h(can)f(alw)n(a)n(ys)h(b)r(e)e (optimized)304 3344 y(out)23 b(b)n(y)h(concatenating)c(the)j(c)n (hild-edge)h(and)f(paren)n(t-edge's)e(substitutions.)40 b(T)-5 b(erms)23 b(can)304 3411 y(b)r(e)17 b(inserted)f(in)n(to,)i(lo)r (ok)n(ed)f(up,)h(and)f(remo)n(v)n(ed)h(from)f(substitution)f(trees)g (e\016cien)n(tly)-5 b(,)19 b(and)e(so)304 3477 y(they)g(are)h(used)f (for)h(term)f(indexing)h(in)h(automated)c(theorem)i(pro)n(v)n(ers.)p eop %%Page: 15 17 15 16 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(15)1111 811 y @beginspecial 43 @llx 661 @lly 114 @urx 765 @ury 710 @rwi @setspecial %%BeginDocument: figures/pst1.eps %! %%BoundingBox: 43 661 114 765 %%Title: pst1 %%CreationDate: Mon Aug 17 11:47:01 1998 %%Creator: Tgif-3.0-p17 by William Chia-Wei Cheng (william@cs.UCLA.edu) /tgifdict 4 dict def tgifdict begin /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { 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y(c)n(ho)r(ose)16 b Fq(w)23 b(=)-34 b Fl(2)15 b Fq(V)e Fs(\()p Fq(P)8 b Fs(\))16 b(and)i(return)690 2393 y Fq(P)734 2371 y Fm(0)764 2393 y Fs(=)d(\()p Fq(r)n(;)9 b(u;)g(V)26 b Fl([)12 b(f)p Fq(w)q Fl(g)p Fq(;)d(E)16 b Fl([)c(f)p Fs(\()p Fq(v)r(;)d(w)q Fs(\))p Fl(g)p Fq(;)g(\030)15 b Fl([)d(f)p Fs(\(\()p Fq(v)r(;)d(w)q Fs(\))p Fq(;)g(s)p Fs(\))p Fl(g)p Fs(\))p Fq(:)304 2496 y Fs(Complexit)n(y:)22 b(O\(1\))12 b(time.)23 b(This)12 b(is)h(simply)h(insertion)d(in)n(to)i (the)f(tree)f(and)h(lab)r(eling)g(structure.)304 2603 y Fy(Algorithm)20 b(2.2.2)h Fs(\()p Fy(get)p 876 2603 20 3 v 22 w(sub)p Fs(\))p Fy(.)304 2670 y Fs(Input:)i(Tw)n(o)c(v)n (ertices)e Fq(v)k Fs(and)c Fq(w)j Fs(of)e Fq(P)25 b Fs(suc)n(h)18 b(that)f Fq(v)k Fs(is)d(an)g(ancestor)e(of)i Fq(w)q Fs(.)304 2736 y(Output:)26 b(A)19 b(substitution)e Fq(s)j Fs(that)e(is)i(the)f (concatenation)d(of)j(all)h(the)f(substitutions)e(on)i(the)304 2802 y(path)e(from)h Fq(v)i Fs(to)e Fq(w)q Fs(.)304 2869 y(Pro)r(cedure)d Fy(get)p 658 2869 V 22 w(sub)p Fs(\()p Fq(P)r(;)9 b(v)r(;)g(w)q Fs(\):)370 2953 y(1.)28 b(If)18 b Fq(v)f Fs(=)f Fq(w)j Fs(then)e(return)g Fl(\001)370 3020 y Fs(2.)28 b(Otherwise,)17 b(return)g([)p Fq(\030)s Fs(\()p Fq(p;)9 b(w)q Fs(\)])p Fy(get)p 1171 3020 17 3 v 19 w(sub)q Fs(\()p Fq(P)r(;)g(v)r(;)g(p)p Fs(\),)18 b(where)f Fq(p)h Fs(is)h(the)e(paren)n(t)g(of)h Fq(w)q Fs(.)304 3104 y(Complexit)n(y:)34 b Fq(O)r Fs(\()p Fq(h)p Fs(\()p Fq(P)8 b Fs(\))p Fq(sub)p 902 3104 V 17 w(max)p Fs(\()p Fq(P)g Fs(\)\))20 b(time.)38 b(A)n(t)22 b(most)g Fq(h)p Fs(\()p Fq(P)8 b Fs(\))21 b(comp)r(ositions)g(are)h(needed,)304 3171 y(where)17 b Fq(h)p Fs(\()p Fq(P)8 b Fs(\))16 b(is)j(the)e(heigh)n (t)h(of)g Fq(P)8 b Fs(.)304 3278 y Fy(Algorithm)20 b(2.2.3)h Fs(\()p Fy(get)p 876 3278 20 3 v 22 w(pat)p Fs(\))p Fy(.)304 3344 y Fs(Input:)i(A)18 b(v)n(ertex)g Fq(v)i Fs(of)e Fq(P)8 b Fs(.)304 3411 y(Output:)23 b(The)18 b(pattern)e Fq(pat)910 3419 y Fp(P)947 3411 y Fs(\()p Fq(v)r Fs(\))h(asso)r(ciated) f(with)i Fq(v)i Fs(in)f Fq(P)8 b Fs(.)304 3477 y(Pro)r(cedure)15 b Fy(get)p 658 3477 V 22 w(pat)p Fs(\()p Fq(P)r(;)9 b(v)r Fs(\):)p eop %%Page: 17 19 17 18 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(17)304 283 y Fs(Return)16 b([)p Fy(get)p 596 283 17 3 v 20 w(sub)q Fs(\()p Fq(P)r(;)9 b(r)r Fs(\()p Fq(P)f Fs(\))p Fq(;)h(v)r Fs(\)])p Fy(u)p Fs(\()p Fq(P)f Fs(\).)304 350 y(Complexit)n(y:)25 b Fq(O)r Fs(\()p Fl(j)p Fq(P)8 b Fl(j)p Fq(sub)p 847 350 V 19 w(max)p Fs(\()p Fq(P)g Fs(\)\))15 b(time.)304 456 y Fy(Algorithm)20 b(2.2.4)h Fs(\()p Fy(\014nd)p 895 456 20 3 v 23 w(matc)n(hing)p 1173 456 V 23 w(sub)p Fs(\))p Fy(.)304 523 y Fs(Input:)i(A)18 b(v)n(ertex)g Fq(v)i Fs(of)e Fq(P)26 b Fs(and)18 b(a)g(substitution)e Fq(s)p Fs(.)304 589 y(Output:)35 b(A)24 b(v)n(ertex)g Fq(v)803 569 y Fm(0)843 589 y Fs(of)g Fq(P)32 b Fs(and)23 b(a)h(substitution)f Fq(s)1486 569 y Fm(0)1526 589 y Fs(suc)n(h)h(that)f Fq(v)j Fs(corresp)r(onds)21 b(to)i(the)304 655 y(most)e(sp)r(eci\014c)f(pattern)f(in)j(the)f(tree)f(that)g(matc)n (hes)h([)p Fq(s)p Fs(])p Fq(pat)1594 663 y Fp(P)1631 655 y Fs(\()p Fq(v)r Fs(\),)h(and)f Fq(s)1878 635 y Fm(0)1916 655 y Fs(is)h(the)e(matc)n(hing)304 722 y(substitution.)304 788 y(Pro)r(cedure)15 b Fy(\014nd)p 677 788 V 24 w(matc)n(hing)p 956 788 V 23 w(sub)p Fs(\()p Fq(P)r(;)9 b(v)r(;)g(s)p Fs(\):)370 872 y(1.)28 b(If)15 b(there)e(is)i(a)f(c)n(hild)h Fq(w)h Fs(of)e Fq(v)j Fs(suc)n(h)d(that)f Fq(\030)s Fs(\()p Fq(v)r(;)c(w)q Fs(\))15 b Fl(\026)h Fq(s)e Fs(under)g(substitution)e Fq(t)j Fs(then)e(return)441 939 y Fy(\014nd)p 549 939 17 3 v 21 w(matc)n(hing)p 826 939 V 21 w(sub)q Fs(\()p Fq(P)r(;)c(w)q(;)g(t)p Fs(\).)370 1005 y(2.)28 b(Otherwise)17 b(return)f(\()p Fq(v)r(;)9 b(s)p Fs(\).)304 1089 y(Complexit)n(y:)25 b Fq(O)r Fs(\()p Fl(j)p Fq(P)8 b Fl(j)h Fs(max)o(\()p Fl(j)p Fq(s)p Fl(j)p Fq(;)g(sub)p 1060 1089 V 21 w(max)p Fs(\()p Fq(P)f Fs(\)\)\))15 b(time.)304 1195 y Fy(Algorithm)20 b(2.2.5)h Fs(\()p Fy(lo)r(okup)p Fs(\))p Fy(.)304 1261 y Fs(Input:)i(A)18 b(pattern)e Fq(p)j Fs(and)e(substitution)f(tree)h Fq(P)8 b Fs(.)304 1328 y(Output:)22 b(A)16 b(v)n(ertex)g Fq(v)774 1308 y Fm(0)806 1328 y Fs(and)f(substitution)f Fq(s)1243 1308 y Fm(0)1275 1328 y Fs(suc)n(h)i(that)f Fq(v)1545 1308 y Fm(0)1577 1328 y Fs(corresp)r(onds)e(to)i(a)i(most)e (sp)r(eci\014c)304 1394 y(matc)n(h)i(in)i Fq(P)26 b Fs(of)18 b Fq(p)g Fs(and)g Fq(s)841 1374 y Fm(0)874 1394 y Fs(is)h(the)e(matc)n (hing)h(substitution.)304 1461 y(Pro)r(cedure)d Fy(lo)r(okup)p Fs(\()p Fq(P)r(;)9 b(p)p Fs(\):)304 1527 y(Return)16 b Fy(\014nd)p 600 1527 V 21 w(matc)n(hing)p 877 1527 V 22 w(sub)p Fs(\()p Fq(P)r(;)9 b(r)r Fs(\()p Fq(P)f Fs(\))p Fq(;)h Fy(u)p Fs(\()p Fq(P)f Fs(\))k Fl(7!)k Fq(p)p Fs(\).)304 1594 y(Complexit)n(y:)25 b Fq(O)r Fs(\()p Fl(j)p Fq(P)8 b Fl(j)h Fs(max)o(\()p Fq(sub)p 980 1594 V 19 w(max)p Fs(\()p Fq(P)f Fs(\))p Fq(;)h Fl(j)p Fq(p)p Fl(j)p Fs(\)\))17 b(time.)304 1700 y Fy(Algorithm)j(2.2.6)h Fs(\()p Fy(split)p Fs(\))p Fy(.)304 1766 y Fs(Input:)i(A)18 b(v)n(ertex)g Fq(v)i Fs(of)e Fq(P)26 b Fs(and)18 b(a)g(substitution)e Fq(s)p Fs(.)304 1833 y(Output:)22 b(A)15 b(v)n(ertex)g Fq(w)i Fs(and)e(pattern)e(substitution)g(tree)h Fq(P)1556 1813 y Fm(0)1587 1833 y Fs(iden)n(tical)h(to)g Fq(P)23 b Fs(except)14 b(p)r(ossibly)304 1899 y(mo)r(di\014ed)j(so)h(that)f([)p Fq(s)p Fs(])p Fq(pat)851 1907 y Fp(P)888 1899 y Fs(\()p Fq(v)r Fs(\))e(=)g Fq(pat)1111 1907 y Fp(P)1148 1899 y Fs(\()p Fq(w)q Fs(\))f Fl(2)i Fq(P)8 b Fs(.)304 1965 y(Pro)r(cedure)15 b Fy(split)p Fs(\()p Fq(P)r(;)9 b(v)r(;)g(s)p Fs(\):)370 2049 y(1.)28 b(If)20 b(there)f(is)h(a)g(c)n(hild)h Fq(w)g Fs(of)f Fq(v)i Fs(suc)n(h)e(that)f Fq(t)f Fs(=)h Fq(\030)s Fs(\()p Fq(v)r(;)9 b(w)q Fs(\))20 b(and)f Fq(s)h Fs(ha)n(v)n(e)h(a)f(non)n(trivial)g(most)441 2116 y(sp)r(eci\014c)14 b(generalization)g Fq(u)i Fs(and)g(that)f Fq(u)1287 2124 y Fu(1)1319 2116 y Fl(\016)8 b Fq(u)15 b Fs(=)h Fq(t)g Fs(and)g Fq(u)1635 2124 y Fu(2)1667 2116 y Fl(\016)8 b Fq(u)15 b Fs(=)h Fq(s)p Fs(.)24 b(Then)15 b(c)n(ho)r(ose)g(suc)n(h) 441 2182 y(a)j Fq(w)i Fs(and)446 2249 y(\(a\))27 b(if)19 b Fq(u)628 2257 y Fu(1)667 2249 y Fl(\021)d Fq(u)758 2257 y Fu(2)798 2249 y Fl(\021)f(\001)k Fs(\(that)d(is,)j Fq(\030)s Fs(\()p Fq(v)r(;)9 b(w)q Fs(\))15 b Fl(\021)h Fq(s)p Fs(\))i(then)f(return)f(\()p Fq(w)q(;)9 b(P)f Fs(\).)443 2315 y(\(b\))27 b(if)19 b Fq(u)628 2323 y Fu(1)667 2315 y Fl(\021)d(\001)j Fs(and)e Fq(u)899 2323 y Fu(2)939 2315 y Fl(6\021)f(\001)i Fs(\()p Fq(\030)s Fs(\()p Fq(v)r(;)9 b(w)q Fs(\))15 b Fl(\026)g Fq(s)p Fs(\))j(then)f(return)g Fy(split)o Fs(\()p Fq(P)r(;)9 b(w)q(;)g(u)1954 2323 y Fu(2)1979 2315 y Fs(\).)449 2381 y(\(c\))27 b(if)22 b Fq(u)631 2389 y Fu(1)676 2381 y Fl(6\021)e(\001)i Fs(and)f Fq(u)919 2389 y Fu(2)964 2381 y Fl(\021)g(\001)h Fs(\()p Fq(s)e Fl(\026)h Fq(\030)s Fs(\()p Fq(v)r(;)9 b(w)q Fs(\)\))20 b(then)g(c)n(ho)r(ose)g(a)h(v)n (ertex)g Fq(v)1961 2361 y Fm(0)2003 2381 y Fq(=)-34 b Fl(2)21 b Fq(V)12 b Fs(\()p Fq(P)c Fs(\))20 b(and)545 2448 y(return)c Fq(v)743 2428 y Fm(0)777 2448 y Fs(and)691 2533 y Fq(P)735 2510 y Fm(0)765 2533 y Fs(=)f(\()p Fy(u)p Fs(\()p Fq(P)8 b Fs(\))p Fq(;)h(r)r Fs(\()p Fq(P)f Fs(\))p Fq(;)845 2605 y(V)k Fs(\()p Fq(P)c Fs(\))j Fl([)h(f)p Fq(v)1094 2582 y Fm(0)1109 2605 y Fl(g)p Fq(;)845 2678 y Fs(\()p Fq(E)s Fs(\()p Fq(V)f Fs(\))p Fl(nf)p Fs(\()p Fq(v)r(;)e(w)q Fs(\))p Fl(g)p Fs(\))h Fl([)j(f)p Fs(\()p Fq(v)r(;)c(v)1434 2655 y Fm(0)1449 2678 y Fs(\))p Fq(;)g Fs(\()p Fq(v)1546 2655 y Fm(0)1561 2678 y Fq(;)g(w)q Fs(\))p Fl(g)p Fq(;)845 2750 y Fs(\()p Fq(\030)s Fl(nf)p Fs(\(\()p Fq(v)r(;)g(w)q Fs(\))p Fq(;)g(t)p Fs(\))p Fl(g)p Fs(\))g Fl([)j(f)p Fs(\(\()p Fq(v)r(;)d(v)1438 2728 y Fm(0)1453 2750 y Fs(\))p Fq(;)g(u)p Fs(\))p Fq(;)g Fs(\(\()p Fq(v)1650 2728 y Fm(0)1664 2750 y Fq(;)g(w)q Fs(\))p Fq(;)g(u)1807 2758 y Fu(1)1832 2750 y Fs(\))p Fl(g)p Fs(\))p Fq(:)443 2855 y Fs(\(d\))27 b(if)17 b Fq(u)626 2863 y Fu(1)665 2855 y Fl(6\021)f(\001)h Fs(and)e Fq(u)893 2863 y Fu(2)933 2855 y Fl(6\021)h(\001)g Fs(then)g(c)n(ho)r(ose)e(v)n (ertices)h Fq(v)1548 2835 y Fm(0)1564 2855 y Fq(;)9 b(v)1617 2835 y Fm(00)1667 2855 y Fq(=)-34 b Fl(2)15 b Fq(V)e Fs(\()p Fq(P)8 b Fs(\))14 b(and)i(return)f Fq(v)2163 2835 y Fm(0)o(0)2207 2855 y Fs(and)532 2954 y Fq(P)576 2931 y Fm(0)606 2954 y Fs(=)g(\()p Fy(u)p Fs(\()p Fq(P)8 b Fs(\))p Fq(;)h(r)r Fs(\()p Fq(P)f Fs(\))p Fq(;)686 3026 y(V)k Fs(\()p Fq(P)c Fs(\))j Fl([)h(f)p Fq(v)935 3003 y Fm(0)950 3026 y Fq(;)d(v)1003 3003 y Fm(00)1032 3026 y Fl(g)p Fq(;)686 3099 y Fs(\()p Fq(E)s Fs(\()p Fq(V)j Fs(\))p Fl(nf)p Fs(\()p Fq(v)r(;)d(w)q Fs(\))p Fl(g)p Fs(\))g Fl([)k(f)p Fs(\()p Fq(v)r(;)c(v)1275 3076 y Fm(0)1290 3099 y Fs(\))p Fq(;)g Fs(\()p Fq(v)1387 3076 y Fm(0)1402 3099 y Fq(;)g(v)1455 3076 y Fm(00)1483 3099 y Fs(\))p Fq(;)g Fs(\()p Fq(v)1580 3076 y Fm(0)1595 3099 y Fq(;)g(w)q Fs(\))p Fl(g)p Fq(;)686 3172 y Fs(\()p Fq(\030)s Fl(nf)p Fs(\(\()p Fq(v)r(;)g(w)q Fs(\))p Fq(;)g(t)p Fs(\))p Fl(g)p Fs(\))g Fl([)j(f)p Fs(\(\()p Fq(v)r(;)d(v)1279 3149 y Fm(0)1294 3172 y Fs(\))p Fq(;)g(u)p Fs(\))p Fq(;)g Fs(\(\()p Fq(v)1491 3149 y Fm(0)1505 3172 y Fq(;)g(v)1558 3149 y Fm(00)1587 3172 y Fs(\))p Fq(;)g(u)1665 3180 y Fu(2)1690 3172 y Fs(\))p Fq(;)g Fs(\(\()p Fq(v)1809 3149 y Fm(0)1823 3172 y Fq(;)g(w)q Fs(\))p Fq(;)g(u)1966 3180 y Fu(1)1991 3172 y Fs(\))p Fl(g)p Fs(\))p Fq(:)370 3273 y Fs(2.)28 b(Otherwise)18 b(there)f(is)j(no)f(c)n(hild-edge)f(of)i Fq(v)h Fs(lab)r(eled)d(with)h(a)g(substitution)e(that)h(shares)441 3340 y(an)n(y)g(structure)e(with)i Fq(s)p Fs(.)25 b(Return)16 b(\()p Fq(v)r(;)9 b Fy(add)q Fs(\()p Fq(P)r(;)g(v)r(;)g(s)p Fs(\)\).)304 3423 y(Complexit)n(y:)25 b Fq(O)r Fs(\()p Fl(j)p Fq(P)8 b Fl(j)p Fs(\(max)o(\()p Fl(j)p Fq(s)p Fl(j)p Fq(;)h(sub)p 1073 3423 V 20 w(max)p Fs(\()p Fq(P)f Fs(\)\)\))1332 3403 y Fu(2)1354 3423 y Fs(\))18 b(time.)p eop %%Page: 18 20 18 19 bop 304 151 a Fu(18)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)434 1960 y @beginspecial 14 @llx 518 @lly 286 @urx 786 @ury 2720 @rwi @setspecial %%BeginDocument: figures/split.eps %! %%BoundingBox: 14 518 286 786 %%Title: split %%CreationDate: Tue Aug 4 21:47:10 1998 %%Creator: Tgif-3.0-p17 by William Chia-Wei Cheng (william@cs.UCLA.edu) /tgifdict 4 dict def tgifdict begin /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def end %%PageBoundingBox: 14 518 286 786 tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 0 setgray 72 0 mul 72 11.00 mul translate 72 128 div 100.000 mul 100 div dup neg scale gsave /tgiforigctm matrix currentmatrix def % POLY/OPEN-SPLINE 0 setgray gsave newpath 104 156 moveto 104 120 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 256 76 moveto 256 40 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 448 76 moveto 448 36 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 256 312 moveto 256 272 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 448 312 moveto 448 272 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 448 140 moveto 448 208 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 104 156 moveto 104 220 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 104 156 moveto 152 216 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % POLY/OPEN-SPLINE 0 setgray gsave newpath 104 156 moveto 56 216 lineto tgiforigctm setmatrix 1 setlinewidth stroke grestore % TEXT 0 setgray /Courier findfont [10 0 0 -10 0 0] makefont setfont gsave 76 201 moveto (...) show grestore % TEXT 0 setgray /Courier findfont [10 0 0 -10 0 0] makefont setfont gsave 112 201 moveto (...) 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y(in)g Fq(o)p Fs(,)h(then)d(remo)n(v)n(e)i Fq(r)909 1942 y Fp(i)947 1934 y Fs(from)f Fq(D)j Fs(and)d(apply)h Fq(r)1431 1942 y Fp(i)1468 1934 y Fs(to)f(the)g(rest)g(of)h Fq(D)h Fs(un)n(til)g (there)d(are)h(no)441 2000 y(more)f(o)r(ccurrences)e(of)j Fq(r)965 2008 y Fp(i)983 2000 y Fs(.)370 2066 y(3.)28 b(Return)16 b Fq(D)r Fs(.)370 2172 y(Note)21 b(that)h(this)g(algorithm) g(migh)n(t)h(actually)f(increase)f(the)h(size)g(of)h(the)e(dictionary)h (b)n(y)304 2238 y(remo)n(ving)g(a)h(rule)g(that)e(w)n(as)i(not)f(not)g (in)h(the)f(list,)j(but)d(whic)n(h)h(w)n(as)g(used)f(extensiv)n(ely)h (b)n(y)304 2305 y(other)16 b(rules.)25 b(If)18 b(this)g(is)h(the)e (case)g(it)i(migh)n(t)f(b)r(e)g(b)r(etter)d(to)j(mo)r(dify)g(the)g(ab)r (o)n(v)n(e)f(algorithm)h(so)304 2371 y(that)i(rules)h(are)g(only)h (remo)n(v)n(ed)g(if)g(doing)f(so)h(w)n(ould)f(impro)n(v)n(e)i(the)e (dictionary's)f(cost.)34 b(F)-5 b(or)304 2438 y(example,)16 b(if)g(the)f(metric)g(is)h Fq(N)942 2446 y Fu(#)982 2438 y Fs(,)h(rules)e(should)g(only)h(b)r(e)f(remo)n(v)n(ed)g(if)h(they)g(o) r(ccur)d(few)n(er)j(than)304 2504 y(t)n(wice)g(in)h(the)e(rest)h(of)g (the)g(dictionary)-5 b(.)23 b(The)16 b(dictionaries)f(w)n(e)i(will)h (use)e(ha)n(v)n(e)h(no)f(de\014nitional)304 2570 y(dep)r(endencies,)f (so)j(w)n(e)g(will)i(not)d(w)n(orry)i(ab)r(out)d(this)i(problem)f (further.)957 2713 y(5.)31 b Ft(Encoding)20 b(Algorithms)370 2813 y Fs(Once)g(a)g(pattern)e(m)n(ultiset)i(has)g(b)r(een)f (generated,)f(the)h(input)h(terms)f(m)n(ust)i(b)r(e)e(enco)r(ded)304 2879 y(using)c(the)g(patterns,)g(and)g(a)h(dictionary)f(including)g (de\014nitions)g(for)g(all)i(the)e(op)r(erators)e(m)n(ust)304 2946 y(b)r(e)21 b(constructed.)34 b(In)23 b(this)f(section)f(w)n(e)h (will)i(discuss)e(algorithms)f(whic)n(h)i(attempt)d(to)i(\014nd)304 3012 y(go)r(o)r(d)d(enco)r(dings)g(of)i(the)f(input)g(set)h(of)g (terms.)32 b(Finding)21 b(optimal)g(enco)r(dings)e(is)i(probably)304 3079 y(at)f(least)f(NP-hard,)i(but)e(sev)n(eral)h(appro)n(ximate)g (approac)n(hes)e(migh)n(t)i(giv)n(e)h(acceptable,)e(but)304 3145 y(not)e(alw)n(a)n(ys)i(optimal)f(results.)370 3211 y(W)-5 b(e)18 b(will)h(explore)e(four)f(approac)n(hes.)23 b(W)-5 b(e)17 b(will)i(\014rst)e(demonstrate)d(that)j(it)g(is)i(p)r (ossible)d(to)304 3278 y(solv)n(e)21 b(the)e(problem)h(b)n(y)h(brute)e (force,)h(but)g(w)n(e)h(will)g(only)g(b)r(e)f(able)g(to)g(giv)n(e)h(an) f(exp)r(onen)n(tial)304 3344 y(algorithm)h(for)g(doing)h(so.)35 b(Instead,)22 b(acceptable)d(solutions)i(ma)n(y)i(b)r(e)e(found)g(b)n (y)i(using)e(the)304 3411 y Fr(glob)m(al)s(ly)h(b)m(est)k Fs(or)19 b Fr(lo)m(c)m(al)s(ly)j(b)m(est)j Fs(patterns)17 b(\014rst.)27 b(These)18 b(algorithms)g(pro)r(ceed)e(b)n(y)k(rep)r (eatedly)304 3477 y(making)i(pattern)e(replacemen)n(ts)g(that)h(are)g (guaran)n(teed)e(to)j(impro)n(v)n(e)h(the)e(measure)f(of)i(the)p eop %%Page: 27 29 27 28 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(27)304 283 y Fs(input.)48 b(Finally)-5 b(,)30 b(w)n(e)c(will)i (suggest)c(a)i(\\top-do)n(wn")f(algorithm)g(based)g(on)h(the)g(enco)r (ding)304 350 y(strategy)17 b(used)i(in)h(Lemp)r(el-Ziv)f(compression.) 27 b(In)19 b(this)h(pro)r(cedure,)d(the)h(substitution)g(tree)304 416 y(is)e(used)f(as)h(a)g(\014nite)f(automaton)e(in)k(m)n(uc)n(h)f (the)f(same)g(w)n(a)n(y)i(as)f(the)f(Lemp)r(el-Ziv)g(compression)304 483 y(tree)i(is)h(used)g(to)f(scan)h(and)f(enco)r(de)f(the)h(input.)304 604 y(5.1.)32 b Fy(Brute)24 b(F)-5 b(orce.)27 b Fs(W)-5 b(e)20 b(ha)n(v)n(e)h(earlier)e(describ)r(ed)e(ho)n(w)k(all)f(the)f(p)r (ossible)g(enco)r(dings)f(of)i(a)304 671 y(term)g(could)g(b)r(e)g(en)n (umerated.)31 b(The)20 b(brute)g(force)g(algorithm)g(pro)r(ceeds)e(b)n (y)j(generating)e(all)304 737 y(the)d(enco)r(dings)f(of)i(a)g(list)h (of)f(terms,)g(their)f(asso)r(ciated)f(dictionaries,)h(and)g(their)h (costs)f(under)304 804 y(metrics)g Fq(M)24 b Fs(and)16 b Fq(N)25 b Fs(resp)r(ectiv)n(ely)-5 b(.)23 b(It)17 b(returns)f(an)h 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b Fl(\\)j Fs(\006)1846 1305 y Fm(0)1861 1325 y Fq(;)9 b(D)r Fs(\))20 b(and)f(calculate)441 1391 y Fq(siz)r(e)p Fs(\()p Fq(u;)9 b(D)666 1371 y Fm(0)681 1391 y Fs(\))14 b(=)i Fq(M)6 b Fs(\()p Fq(u)p Fs(\))11 b(+)h Fq(N)6 b Fs(\()p Fq(D)1098 1371 y Fm(0)1113 1391 y Fs(\).)370 1458 y(3.)28 b(Return)16 b(the)i(dictionary)f Fq(D)1036 1438 y Fm(0)1069 1458 y Fs(and)h(term)f Fq(u)h Fs(suc)n(h)g(that)f Fq(siz)r(e)p Fs(\()p Fq(u;)9 b(D)1829 1438 y Fm(0)1844 1458 y Fs(\))18 b(is)g(minimized.)370 1564 y(Here)h(w)n(e)h(mak)n(e)h(the)e(simpli\014cation)h(of)g(assuming) g(some)f(straigh)n(tforw)n(ard)f(w)n(a)n(y)j(of)f(en)n(u-)304 1630 y(merating)d(all)j(p)r(ossible)d Fl( )879 1638 y Fp(D)920 1630 y Fs(-deriv)m(ations)h(of)h Fq(t)g Fs(is)g(giv)n(en.)27 b Fq(D)1596 1610 y Fm(\000)p Fu(1)1674 1630 y Fs(is)19 b(decreasing)d(with)j(resp)r(ect)304 1697 y(to)c Fq(M)423 1705 y Fu(#)462 1697 y Fs(,)h(so)g(there)e(can)h(b)r(e)f(at)h(most)g Fl(j)p Fq(t)p Fl(j)h Fs(=)g Fq(M)1246 1705 y Fu(#)1285 1697 y Fs(\()p Fq(t)p Fs(\))f(rules)g(used)f(in)i(an)n(y)g(deriv)m (ation)f(of)g Fq(t)2119 1677 y Fm(0)2150 1697 y Fl( )2205 1705 y Fp(D)2261 1697 y Fq(t)p Fs(.)304 1767 y(There)f(are)h(consequen) n(tly)g(at)g(most)g Fl(j)p Fq(D)r Fl(j)1153 1747 y Fm(j)p Fp(t)p Fm(j)1215 1767 y Fs(suc)n(h)h(deriv)m(ations,)f(if)i Fl(j)p Fq(D)r Fl(j)f Fs(is)g(the)f(n)n(um)n(b)r(er)g(of)h(rules)304 1834 y(in)i(the)g(dictionary)-5 b(.)370 1900 y(This)26 b(algorithm)e(\()p Fy(BF)p Fs(\))i(is)g(correct)d(\(it)i(alw)n(a)n(ys)h (\014nds)f(a)g(solution\))g(and)g(optimal)g(\(the)304 1968 y(solution)15 b(it)h(\014nds)f(is)i(the)e(b)r(est)f(one\).)23 b(It)16 b(can)f(run)h(in)g Fq(O)r Fs(\()p Fl(j)p Fq(D)r Fl(j)1564 1948 y Fm(j)p Fp(t)p Fm(j)1609 1968 y Fs(\))g(time)g(and)f Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)8 b Fs(+)g Fl(j)p Fq(D)r Fl(j)p Fs(\))16 b(space)304 2035 y(\(it)i(needs)f(to)g(remem)n(b)r(er)g (only)h(the)g(b)r(est)e(and)i(the)f(curren)n(t)g(enco)r(ding)f(and)i (dictionary\).)304 2156 y(5.2.)32 b Fy(Lo)r(cal)22 b(Best)g(First.)27 b Fs(The)19 b(lo)r(cal)g(b)r(est-\014rst)d(algorithm)i(attempts)f(to)i (recursiv)n(ely)g(re-)304 2223 y(place)14 b(matc)n(hing)f(patterns)g (that)g(maximize)i(the)f(v)m(alue)g(of)h(a)f(giv)n(en)h(function.)22 b(This)15 b Fr(heuristic)304 2289 y Fs(function)i Fq(f)23 b Fs(:)17 b Fq(P)8 b(at)696 2269 y Fp(Y)696 2305 y Fu(\006)p Fp(X)782 2289 y Fl(\002)k Fk(N)17 b Fl(!)g Fk(N)i Fs(estimates)e(the)h (b)r(ene\014ts)f(resulting)g(from)i(replacing)f(ev)n(ery)304 2356 y(o)r(ccurrence)d(of)k(a)g(pattern)e Fq(p)j Fs(with)f(m)n (ultiplicit)n(y)h Fq(m)g Fs(in)f(the)f(set)h(of)g(terms.)27 b(W)-5 b(e)19 b(will)h(assume)304 2422 y(that)d Fq(f)24 b Fs(is)19 b Fq(O)r Fs(\()p Fq(n)p Fs(\).)304 2528 y Fy(Algorithm)h(5.2.1)h Fs(\()p Fy(lo)r(cal)p 919 2528 20 3 v 21 w(b)r(est)p 1056 2528 V 23 w(\014rst)p Fs(\))p Fy(.)304 2594 y Fs(Input:)i(a)c(term)e Fq(t)e Fl(2)h Fq(T)8 b(er)r(m)886 2602 y Fu(\006)p Fp(X)977 2594 y Fs(and)17 b(a)h(pattern)e(m)n(ultiset)i Fq(S)304 2661 y Fs(Output:)23 b(a)18 b(term)g Fq(u)d Fl(2)g Fq(T)8 b(er)r(m)942 2669 y Fu(\006)p Fm([)p Fu(\006)1036 2658 y Fi(0)1049 2669 y Fp(X)1109 2661 y Fs(and)18 b(dictionary)f Fq(D)304 2727 y Fs(Pro)r(cedure)e Fy(lo)r(cal)p 701 2727 V 22 w(b)r(est)p 839 2727 V 23 w(\014rst)p Fs(\()p Fq(t;)9 b(D)r(;)g(S)s Fs(\):)304 2794 y(Return)16 b Fy(lo)r(cal)p 624 2794 17 3 v 19 w(b)r(est)p 759 2794 V 20 w(\014rst)887 2774 y Fm(0)903 2794 y Fs(\()p Fq(t;)9 b Fy(mak)n(e)p 1117 2794 V 20 w(dictionary)q Fs(\()p Fq(S)s Fs(\))p Fq(;)g(S)s Fs(\).)304 2860 y(Pro)r(cedure)15 b Fy(lo)r(cal)p 701 2860 20 3 v 22 w(b)r(est)p 839 2860 V 23 w(\014rst')p Fs(\()p Fq(t;)9 b(D)r(;)g(S)s Fs(\):)370 2944 y(1.)28 b(Let)19 b Fq(p)h Fs(b)r(e)f(a)h(pattern)e(that)g(maximizes)j Fq(f)6 b Fs(\()p Fq(p;)j(\026)1447 2952 y Fp(S)1479 2944 y Fs(\()p Fq(p)p Fs(\)\))19 b(among)g(all)h(patterns)e(in)i Fq(S)k Fs(that)441 3010 y(matc)n(h)17 b Fq(t)p Fs(.)370 3077 y(2.)28 b(If)18 b Fq(p)h Fs(is)f(non)n(trivial)g(and)g(nonatomic,) f(then)446 3143 y(\(a\))27 b(Let)17 b Fq(s)h Fs(matc)n(h)g Fq(p)g Fs(to)g Fq(t)g Fs(and)g(assume)f Fq(\033)1341 3123 y Fm(0)1356 3143 y Fs(\()p Fy(y)1412 3151 y Fu(1)1436 3143 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b Fy(y)1602 3151 y Fp(n)1632 3143 y Fs(\))15 b Fl( )1724 3151 y Fp(D)1780 3143 y Fq(p)p Fs(.)443 3209 y(\(b\))27 b(Return)485 3310 y Fq(\033)519 3287 y Fm(0)534 3310 y Fs(\()p Fy(lo)r(cal)p 688 3310 17 3 v 19 w(b)r(est)p 823 3310 V 20 w(\014rst)951 3287 y Fm(0)967 3310 y Fs(\([)p Fq(s)p Fs(])p Fy(y)1079 3318 y Fu(1)1104 3310 y Fq(;)9 b(D)r(;)g(S)s Fs(\))p Fq(;)g(:)g(:)g(:)21 b(;)9 b Fy(lo)r(cal)p 1523 3310 V 19 w(b)r(est)p 1658 3310 V 20 w(\014rst)1786 3287 y Fm(0)1802 3310 y Fs(\([)p Fq(s)p Fs(])p Fy(y)1914 3318 y Fp(n)1944 3310 y Fq(;)g(D)r(;)g(S)s Fs(\))p Fq(:)370 3411 y Fs(3.)28 b(If)18 b Fq(p)h Fs(is)f(trivial)h(or)f(atomic:)446 3477 y(\(a\))27 b(If)18 b Fq(t)h Fs(is)f(a)g(v)m(ariable)g(or)g(constan)n(t) e(then)h(return)g Fq(t)p eop %%Page: 28 30 28 29 bop 304 151 a Fu(28)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)443 283 y Fs(\(b\))27 b(Otherwise)17 b Fq(t)e Fs(=)g Fq(\033)r Fs(\()p Fq(t)976 291 y Fu(1)1000 283 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b(t)1152 291 y Fp(n)1183 283 y Fs(\).)24 b(Return)563 376 y Fq(\033)r Fs(\()p Fy(lo)r(cal)p 751 376 17 3 v 19 w(b)r(est)p 886 376 V 20 w(\014rst)1014 353 y Fm(0)1030 376 y Fs(\()p Fq(t)1072 384 y Fu(1)1096 376 y Fq(;)9 b(D)r(;)g(S)s Fs(\))p Fq(;)g(:)g(:)g(:)21 b(;)9 b Fy(lo)r(cal)p 1515 376 V 19 w(b)r(est)p 1650 376 V 21 w(\014rst)1779 353 y Fm(0)1794 376 y Fs(\()p Fq(t)1836 384 y Fp(n)1866 376 y Fq(;)g(D)r(;)g(S)s Fs(\))p Fq(:)370 478 y Fs(This)23 b(algorithm)f(\()p Fy(LB)863 486 y Fp(f)892 478 y Fs(\))g(requires)g Fq(O)r Fs(\()p Fl(j)p Fq(S)s Fl(jj)p Fq(t)p Fl(j)1332 457 y Fu(3)1357 478 y Fs(\))h(running)f(time:)33 b(for)22 b(eac)n(h)g(p)r(osition,)h(the)304 544 y(b)r(est)h(pattern)f(m)n(ust)i (b)r(e)g(found)g(and)g(replaced)e(from)i(a)h(set)f(of)g Fl(j)p Fq(S)s Fl(j)i Fs(of)e(them.)46 b(Matc)n(hing)304 610 y(requires)25 b Fl(j)p Fq(t)p Fl(j)i Fs(time)f(and)g(applying)g (the)f(resulting)g(substitution)f(requires)h Fl(j)p Fq(t)p Fl(j)i Fs(time.)49 b(The)304 677 y(results)19 b(are)h(not)h(alw)n(a)n (ys)g(correct)e(or)h(optimal,)i(but)e(ma)n(y)i(pro)r(duce)c(acceptable) g(results)i(for)304 743 y(appropriate)15 b(heuristics)i Fq(f)6 b Fs(.)304 849 y(5.3.)32 b Fy(Global)20 b(Best)h(First.)27 b Fs(The)17 b(global)h(b)r(est-\014rst)d(algorithm)i(sorts)f(the)h (patterns)e(in)j(the)304 915 y(tree)i(in)i(order)e(of)i(decreasing)d (heuristic)h(v)m(alue,)j(giv)n(en)f(some)f(heuristic)f(v)m(alue)i (function)e Fq(f)6 b Fs(.)304 982 y(Then)17 b(it)h(replaces)e(as)h(man) n(y)h(of)g(the)f(o)r(ccurrences)d(as)j(it)h(can)f(of)h(the)e(b)r(est,)h (second)f(b)r(est,)h Fr(etc.)304 1048 y Fs(patterns)f(in)i(decreasing)e (heuristic)h(v)m(alue.)304 1150 y Fy(Algorithm)j(5.3.1)h Fs(\()p Fy(global)p 956 1150 20 3 v 21 w(b)r(est)p 1093 1150 V 23 w(\014rst)p Fs(\))p Fy(.)304 1216 y Fs(Input:)i(a)c(term)e Fq(t)e Fl(2)h Fq(T)8 b(er)r(m)886 1224 y Fu(\006)p Fp(X)977 1216 y Fs(pattern)16 b(m)n(ultiset)i Fq(S)304 1282 y Fs(Output:)23 b(a)18 b(term)g Fq(u)d Fl(2)g Fq(T)8 b(er)r(m)942 1290 y Fu(\006)p Fm([)p Fu(\006)1036 1280 y Fi(0)1049 1290 y Fp(X)1109 1282 y Fs(and)18 b(dictionary)f Fq(D)304 1349 y Fs(Pro)r(cedure)e Fy(global)p 738 1349 V 22 w(b)r(est)p 876 1349 V 23 w(\014rst)p Fs(\()p Fq(t;)9 b(P)f Fs(\):)370 1428 y(1.)28 b(Construct)16 b(dictionary)h Fq(D)f Fs(=)g Fy(mak)n(e)p 1232 1428 17 3 v 20 w(dictionary)p Fs(\()p Fq(S)s Fs(\).)370 1495 y(2.)28 b(If)18 b Fq(S)h Fs(=)c Fl(;)p Fs(,)k(return)d Fq(t)j Fs(and)e Fq(D)r Fs(.)370 1561 y(3.)28 b(Otherwise,)17 b(let)h Fq(p)g Fs(b)r(e)g(a)g(pattern)e (whic)n(h)i(maximizes)g Fq(f)6 b Fs(\()p Fq(p;)j(\026)1730 1569 y Fp(S)1763 1561 y Fs(\()p Fq(p)p Fs(\)\))17 b(o)n(v)n(er)h Fq(S)s Fs(.)370 1628 y(4.)28 b(If)18 b Fq(p)h Fs(is)f(trivial)h(or)f (atomic,)f(go)h(to)g(step)f(6)370 1694 y(5.)28 b(Otherwise,)21 b(if)h Fq(q)g Fl( )878 1702 y Fp(D)938 1694 y Fq(p)p Fs(,)h(recursiv)n(ely)e(replace)e(ev)n(ery)i(o)r(ccurrence)d(of)j Fq(p)g Fs(in)h Fq(t)f Fs(with)g Fq(q)r Fs(.)441 1761 y(\(If)d(o)r(ccurrences)c(o)n(v)n(erlap,)19 b(b)r(oth)d(cannot)h(b)r(e) g(replaced,)g(so)h(c)n(ho)r(ose)f(one.\))370 1827 y(6.)28 b(Remo)n(v)n(e)18 b Fq(p)g Fs(from)g Fq(S)s Fs(.)25 b(Go)19 b(bac)n(k)f(to)g(step)f(2.)370 1928 y(This)f(algorithm)e(\()p Fy(GB)p Fs(\))h(requires)f(at)h(w)n(orst)f Fq(O)r Fs(\()p Fl(j)p Fq(S)s Fl(j)1446 1908 y Fu(2)1471 1928 y Fs(\))h(time)g(to)g (calculate)f(all)h(the)g(heuristic)304 1995 y(v)m(alues,)j Fq(O)r Fs(\()p Fl(j)p Fq(S)s Fl(j)9 b Fs(log)i Fl(j)p Fq(S)s Fl(j)p Fs(\))19 b(time)f(to)g(sort)g(the)g(patterns)e(b)n(y)j (heuristic)e(v)m(alue,)i(and)f(could)g(in)h(the)304 2061 y(w)n(orst)e(case)g(tak)n(e)g Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(jj)p Fq(S)s Fl(j)p Fs(\))i(time)e(to)h(try)f(to)g(replace)g(eac)n(h) g(pattern)f(in)i Fq(S)j Fs(at)c(ev)n(ery)h(lo)r(cation)304 2128 y(in)e Fq(t)p Fs(.)24 b(Th)n(us)17 b(the)e(o)n(v)n(erall)i(time)e (complexit)n(y)i(is)f Fq(O)r Fs(\()p Fl(j)p Fq(S)s Fl(j)p Fs(\()p Fl(j)p Fq(S)s Fl(j)8 b Fs(+)g(log)j Fl(j)p Fq(S)s Fl(j)d Fs(+)g Fl(j)p Fq(t)p Fl(j)p Fs(\))16 b(=)g Fq(O)r Fs(\()p Fl(j)p Fq(S)s Fl(j)2057 2108 y Fu(2)2090 2128 y Fs(+)8 b Fl(j)p Fq(S)s Fl(jj)p Fq(t)p Fl(j)p Fs(\).)304 2194 y(If)25 b(the)g(searc)n(h)f(metho)r(d)f(generates)g Fq(O)r Fs(\(1\))h(or)h Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)p Fs(\))g(patterns)e(then)h(the)h(required)f(times)304 2260 y(are)18 b Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)p Fs(\))h(or)g Fq(O)r Fs(\()p Fl(j)p Fq(t)724 2240 y Fu(2)749 2260 y Fl(j)p Fs(\))g(resp)r(ectiv)n(ely)-5 b(.)27 b(This)19 b(algorithm)f(is)i(not)f(alw)n(a)n(ys)h(correct)d(or)h(optimal,)304 2327 y(but)24 b(ma)n(y)i(pro)r(duce)c(e\016cien)n(t)i(dictionaries)g (and)g(enco)r(dings)f(if)i(the)f(heuristic)g(function)f(is)304 2393 y(w)n(ell-c)n(hosen.)304 2499 y(5.4.)32 b Fy(T)-5 b(op-Do)n(wn.)28 b Fs(The)14 b(follo)n(wing)i(is)f(a)g(v)n(ersion)g(of) g(the)f(Lemp)r(el-Ziv)h(enco)r(ding)e(algorithm:)304 2600 y Fy(Algorithm)20 b(5.4.1)h Fs(\()p Fy(lzenco)r(de)p Fs(\))p Fy(.)304 2667 y Fs(Input:)36 b(a)24 b(string)g(of)g(sym)n(b)r (ols)h(of)f(\000)h(and)f(a)g(ro)r(oted)e(tree)h Fq(T)34 b Fs(=)26 b(\()p Fq(r)n(;)9 b Fs(\000)1824 2647 y Fm(0)1840 2667 y Fq(;)g(E)s Fs(\))25 b(with)f(an)g(edge-)304 2733 y(lab)r(eling)17 b Fq(L)e Fs(:)h Fq(E)j Fl(!)c Fs(\000.)304 2800 y(Output:)23 b(a)18 b(string)f(of)h(sym)n(b)r(ols)h(of)f(\000)12 b Fl([)g Fs(\000)1194 2780 y Fm(0)1209 2800 y Fs(,)19 b(and)e(function)g Fq(M)k Fs(:)16 b(\000)1708 2780 y Fm(0)1739 2800 y Fl(!)f Fq(E)s Fs(.)304 2866 y(Pro)r(cedure)g Fy(lzenco)r(de)p Fs(\()p Fq(\015)s(;)9 b(T)c(;)k(L)p Fs(\):)370 2946 y(1.)28 b(Create)16 b(v)m(ariable)i Fq(x)g Fs(initialized)g(to)g Fq(r)r Fs(.)370 3012 y(2.)28 b(Create)16 b(an)i(empt)n(y)g(mapping)g Fq(M)6 b Fs(.)370 3079 y(3.)28 b(Create)16 b(an)i(empt)n(y)g(string)g Fq(\015)1055 3058 y Fm(0)370 3145 y Fs(4.)28 b(Scan)15 b(the)g(input)h(sequence)e(\()p Fq(\015)1088 3153 y Fu(1)1112 3145 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b(\015)1273 3153 y Fp(n)1303 3145 y Fs(\);)17 b(for)f(eac)n(h)g (1)f Fq(<)g(i)h(<)f(n)i Fs(in)f(order)f(p)r(erform)f(the)441 3211 y(follo)n(wing:)446 3278 y(\(a\))27 b(If)c(v)n(ertex)g Fq(x)h Fs(=)h Fq(v)g Fs(of)e Fq(T)32 b Fs(has)23 b(a)h(c)n(hild-edge)f Fl(f)p Fq(v)r(;)9 b(v)1617 3258 y Fm(0)1632 3278 y Fl(g)24 b Fs(lab)r(eled)e(with)h Fq(\015)2038 3286 y Fp(i)2056 3278 y Fs(,)j(then)c(set)545 3344 y Fq(x)15 b Fs(=)g Fq(v)679 3324 y Fm(0)713 3344 y Fs(and)j(go)f(on)h(to)g(the)f(next)h (sym)n(b)r(ol.)443 3411 y(\(b\))27 b(If)18 b(there)f(is)h(no)g(suc)n(h) g(edge)f(and)h Fq(x)g Fs(=)g Fq(r)i Fs(then)d(set)h Fq(\015)1634 3391 y Fm(0)1664 3411 y Fs(=)e Fq(\015)1755 3391 y Fm(0)1770 3411 y Fq(\015)1799 3419 y Fp(i)449 3477 y Fs(\(c\))27 b(If)13 b(there)f(is)h(no)g(suc)n(h)g(edge)f(and)g Fq(M)6 b Fs(\()p Fq(\033)1313 3457 y Fm(0)1327 3477 y Fs(\))15 b(=)h Fq(x)d Fs(for)f(some)h Fq(\033)1715 3457 y Fm(0)1745 3477 y Fl(2)j Fs(\000)1833 3457 y Fm(0)1861 3477 y Fs(then)c(set)g Fq(\015)2095 3457 y Fm(0)2126 3477 y Fs(=)j Fq(\015)2216 3457 y Fm(0)2232 3477 y Fq(\033)2266 3457 y Fm(0)2281 3477 y Fs(.)p eop %%Page: 29 31 29 30 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(29)443 283 y Fs(\(d\))27 b(Otherwise)16 b(\014nd)i(a)f(new)h(sym)n(b) r(ol)g Fq(\033)1299 263 y Fm(0)1329 283 y Fl(2)d Fs(\000)1416 263 y Fm(0)1450 283 y Fs(and)i(mo)r(dify)g Fq(M)24 b Fs(so)18 b(that)e(it)i(assigns)f Fq(\033)2281 263 y Fm(0)545 350 y Fs(to)g Fq(x)p Fs(,)i(and)f(set)f Fq(\015)904 330 y Fm(0)935 350 y Fs(=)e Fq(\015)1025 330 y Fm(0)1040 350 y Fq(\033)1074 330 y Fm(0)1089 350 y Fs(.)370 416 y(5.)28 b(return)16 b Fq(\015)642 396 y Fm(0)676 416 y Fs(and)i Fq(M)6 b Fs(.)370 528 y(This)17 b(algorithm)f(can)g(b)r(e)g (extended)f(to)h(enco)r(de)f(terms)h(using)g(the)g(patterns)f(in)i(a)g (pattern)304 594 y(substitution)f(tree,)h(as)h(follo)n(ws:)304 705 y Fy(Algorithm)i(5.4.2)h Fs(\()p Fy(top)p 882 705 20 3 v 22 w(do)n(wn)p Fs(\))p Fy(.)304 772 y Fs(Input:)i(a)c(term)e Fq(t)e Fl(2)h Fq(T)8 b(er)r(m)886 780 y Fu(\006)p Fp(X)977 772 y Fs(and)17 b(pattern)f(substitution)g(tree)h Fq(P)8 b Fs(.)304 838 y(Output:)23 b(a)18 b(term)g Fq(u)d Fl(2)g Fq(T)8 b(er)r(m)942 846 y Fu(\006)p Fm([)p Fu(\006)1036 835 y Fi(0)1049 846 y Fp(X)1109 838 y Fs(and)18 b(dictionary)f Fq(D)304 905 y Fs(Pro)r(cedure)e Fy(top)p 664 905 V 23 w(do)n(wn)p Fs(\()p Fq(t;)9 b(P)f Fs(\):)370 993 y(1.)28 b(Create)15 b(a)i(dictionary)f Fq(D)g Fs(=)g Fy(mak)n(e)p 1194 993 17 3 v 20 w(dictionary)p Fs(\()p Fq(P)8 b Fs(\).)23 b(\(W)-5 b(e)17 b(consider)e Fq(P)24 b Fs(to)17 b(represen)n(t)441 1059 y(a)h(pattern)e(set.\))370 1126 y(2.)28 b(If)18 b(\()p Fq(v)r(;)9 b(s)p Fs(\))15 b(=)g Fy(lo)r(okup)p Fs(\()p Fq(P)r(;)9 b(t)p Fs(\))17 b(and)h Fq(p)g Fs(is)h(non)n(trivial) f(and)f(nonatomic)g(then)446 1192 y(\(a\))27 b(Let)17 b Fq(\033)678 1172 y Fm(0)693 1192 y Fs(\()p Fy(y)749 1200 y Fu(1)773 1192 y Fq(;)9 b(:)g(:)g(:)21 b(;)9 b Fy(y)939 1200 y Fp(n)969 1192 y Fs(\))15 b Fl( )1061 1200 y Fp(D)1117 1192 y Fq(p)p Fs(.)25 b(\(There)16 b(m)n(ust)i(b)r(e)g (suc)n(h)f(a)i(rule)e(in)i Fq(D)r Fs(.\))443 1258 y(\(b\))27 b(Set)633 1368 y Fq(u)15 b Fs(=)g Fq(\033)772 1346 y Fm(0)787 1368 y Fs(\()p Fy(elzenco)r(de)q Fs(\([)p Fq(s)p Fs(])p Fy(y)1187 1376 y Fu(1)1212 1368 y Fq(;)9 b(P)f Fs(\))p Fq(;)h(:)g(:)g(:)19 b(;)9 b Fy(elzenco)r(de)r Fs(\([)p Fq(s)p Fs(])p Fy(y)1811 1376 y Fp(n)1841 1368 y Fq(;)g(P)f Fs(\)\))p Fq(:)370 1478 y Fs(3.)28 b(Otherwise,)17 b(if)i Fq(t)c Fs(=)h Fq(\033)r Fs(\()p Fq(t)p Fs(1)p Fq(;)9 b(:)g(:)g(:)19 b(;)9 b(t)1117 1486 y Fp(n)1148 1478 y Fs(\))18 b(then)f(set)711 1589 y Fq(u)e Fs(=)g Fq(\033)r Fs(\()p Fy(elzenco)r(de)p Fs(\()p Fq(t)1179 1597 y Fu(1)1204 1589 y Fq(;)9 b(P)f Fs(\))p Fq(;)h(:)g(:)g(:)20 b(;)9 b Fy(elzenco)r(de)q Fs(\()p Fq(t)1733 1597 y Fp(n)1763 1589 y Fq(;)g(P)f Fs(\)\))p Fq(:)441 1699 y Fs(If)18 b Fq(t)e Fs(=)f Fq(c)j Fs(or)g Fq(t)e Fs(=)f Fq(x)j Fs(for)g(some)f (constan)n(t)g Fq(c)h Fs(or)g(v)m(ariable)g Fq(x)g Fs(then)f(set)g Fq(u)e Fs(=)h Fq(t)p Fs(.)370 1765 y(4.)28 b(Return)16 b Fq(u)j Fs(and)e Fq(D)r Fs(.)370 1876 y(This)g(algorithm)f(\()p Fy(TD)o Fs(\))h(\014nds)f(the)f(term's)h(closest)f(matc)n(hing)h (pattern)f(and)h(substitution)304 1943 y(using)21 b(the)f(substitution) f(tree,)i(enco)r(des)d(that)i(pattern)f(using)i(an)g(op)r(erator)d (from)j(the)f(dic-)304 2009 y(tionary)-5 b(,)21 b(and)f(then)g(pro)r (ceeds)e(recursiv)n(ely)i(on)h(an)n(y)g(subterms.)31 b(If)21 b(enco)r(ding)e(using)h(trivial)304 2075 y(patterns)h(w)n(ere)j (allo)n(w)n(ed,)i(the)e(algorithm)f(w)n(ould)h(nev)n(er)g(terminate)e (when)h(run)h(on)g(terms)304 2142 y(that)c(matc)n(hed)h(only)g(the)g (trivial)h(pattern.)33 b(Therefore,)21 b(when)g(the)g(trivial)h (pattern)d(is)j(the)304 2208 y(b)r(est)15 b(matc)n(h,)i(w)n(e)g(do)g (not)f(try)h(to)f(enco)r(de)f(the)h(term)g(at)g(the)g(ro)r(ot)f(op)r (erator,)g(but)h(instead)g(try)304 2275 y(to)h(enco)r(de)f(its)j (subterms.)370 2341 y(This)f(algorithm)f(can)g(pro)r(duce)e(p)r(o)r(or) h(results)g(b)r(ecause)g(it)h(will)i(replace)e(large,)g(nonrep)r(et-) 304 2408 y(itiv)n(e)h(patterns)d(when)i(replacing)f(smaller,)h(more)g (recurren)n(t)e(patterns)g(w)n(ould)j(b)r(e)e(b)r(etter.)22 b(If,)304 2474 y(for)f(example,)i(the)e(en)n(tire)g(term)g(is)i(presen) n(t)d(in)i(the)f(pattern)f(substitution)g(tree,)i(then)e(the)304 2540 y(top-do)n(wn)g(algorithm)g(will)i(replace)e(it)h(with)g(a)f (single)h(sym)n(b)r(ol,)h(and)f(add)f(an)h(en)n(try)f(in)h(the)304 2607 y(dictionary)f(that)h(de\014nes)f(the)h(sym)n(b)r(ol)i(to)e(b)r(e) g(the)g(en)n(tire)g(term.)35 b(F)-5 b(or)22 b(man)n(y)g(metrics)f(it)h (is)304 2673 y(not)17 b(an)h(impro)n(v)n(emen)n(t)g(to)g(use)g(the)f (dictionary)1171 2783 y Fq(D)f Fs(=)g Fq(u)f Fl(!)g Fq(t)304 2893 y Fs(and)i(enco)r(ding)g Fq(t)h Fs(b)n(y)1203 2980 y Fq(u)d Fl( )1305 2988 y Fp(D)1361 2980 y Fq(t:)304 3079 y Fs(The)g(resulting)g(pair)g(of)h(dictionary)e(and)h(enco)r(ding) f(requires)h(more)g(space)f(than)h(the)g(original)304 3145 y(term)j(did,)i(that)e(is,)j(the)d(enco)r(ded)f(ob)s(ject)h(and)h (dictionary)f(actually)h(ha)n(v)n(e)g(larger)g Fq(M)2149 3153 y Fu(#)2207 3145 y Fs(and)304 3211 y Fq(N)348 3219 y Fu(#)406 3211 y Fs(v)m(alues)f(than)f(the)g(original)h(term)f(did.) 370 3278 y(This)g(algorithm)f(is)h(neither)e(optimal)h(nor)g(correct.) 22 b(It)16 b(can)g(b)r(e)g(implemen)n(ted)g(quic)n(kly:)25 b(it)304 3344 y(can)20 b(run)h(in)g Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(jj)p Fq(P)8 b Fl(j)h Fs(max)q(\()p Fq(sub)p 990 3344 V 18 w(max)p Fs(\()p Fq(P)f Fs(\))p Fq(;)h Fl(j)p Fq(t)p Fl(j)p Fs(\)\))20 b(time,)i(whic)n(h)f(for)g(a)g(\014xed)g Fq(P)28 b Fs(w)n(orks)22 b(out)e(to)304 3411 y Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)420 3391 y Fu(2)445 3411 y Fs(\))d(time.)25 b(In)17 b(fact,)g(the)g(total)g(amoun)n(t)g(of)h(matc) n(hing)e(w)n(ork)i(required)f(is)h(only)g Fl(j)p Fq(t)p Fl(j)h Fs(so)e(the)304 3477 y(total)g(time)h(is)h Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)p Fs(\),)f(as)g(in)h(the)e Fy(ELZ)h Fs(algorithm.)p eop %%Page: 30 32 30 31 bop 304 151 a Fu(30)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)304 283 y Fs(5.5.)32 b Fy(T)-5 b(op-do)n(wn)26 b(brute)f(force.)k Fs(This)21 b(algorithm)g(\()p Fy(TB)p Fs(\))g(is)h(iden)n(tical)f(to)g(the)f (top-do)n(wn)304 350 y(enco)r(ding)d(except)g(that)h(if)i(there)d(is)j (a)f(c)n(hoice)f(b)r(et)n(w)n(een)g(enco)r(ding)f(a)i(term)f(at)g(the)h (ro)r(ot)e(and)304 416 y(lea)n(ving)c(the)f(ro)r(ot)f(alone)i(and)f (enco)r(ding)f(its)i(c)n(hildren,)h(it)f(tries)f(b)r(oth)f(and)i (returns)e(the)h(smaller)304 483 y(enco)r(ding.)370 549 y(This)28 b(enco)r(ding)d(is)j(not)e(the)h(same)g(as)g(the)f(brute)g (force)g(enco)r(ding)g(b)r(ecause)e(it)k(tak)n(es)304 616 y(only)19 b(one)f(term's)g(size)h(in)n(to)g(accoun)n(t)e(at)h(a)h (time,)h(and)e(do)r(es)f(not)i(tak)n(e)f(dictionary)g(size)g(in)n(to) 304 682 y(accoun)n(t.)38 b(Ev)n(en)24 b(if)g(it)f(did,)i(it)f(still)g (migh)n(t)g(miss)g(optimal)f(enco)r(dings)e(b)n(y)j(making)f(lo)r (cally)304 748 y(go)r(o)r(d)16 b(but)i(globally)g(bad)g(c)n(hoices.)370 815 y(T)-5 b(op-do)n(wn)20 b(brute)f(force)g(enco)r(ding)f(is)i(again)g (neither)e(optimal)i(nor)f(correct.)28 b(It)20 b(runs)f(in)304 881 y Fq(O)r Fs(\(2)398 861 y Fm(j)p Fp(t)p Fm(j)442 881 y Fl(j)p Fq(P)8 b Fl(j)h Fs(max)q(\()p Fq(sub)p 736 881 17 3 v 19 w(max)p Fs(\()p Fq(P)f Fs(\))p Fq(;)h Fl(j)p Fq(t)p Fl(j)p Fs(\)\))16 b(time.)25 b(If)18 b Fl(j)p Fq(P)8 b Fl(j)19 b Fs(is)g Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)p Fs(\))f(or)g Fq(O)r Fs(\()p Fl(j)p Fq(t)p Fl(j)1776 861 y Fu(2)1801 881 y Fs(\))g(then)f(this)h(is)h Fq(O)r Fs(\(2)2225 861 y Fm(j)2239 881 y Fq(t)p Fl(j)p Fs(\))304 948 y(time.)j(The)13 b(top-do)n(wn)e(brute)h(force)f(algorithm)h(seems) f(more)h(lik)n(ely)j(to)d(b)r(e)g(useful)g(in)h(enco)r(ding)304 1014 y(small)24 b(terms)e(or)h(parts)g(of)g(terms,)h(b)r(ecause)d(it)j 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y(ho)n(w)14 b(term)g(compression)f(tec)n(hniques)g(could)h(b)r(e)f(extended)g(to)h (mak)n(e)g(it)h(p)r(ossible)e(to)h(optimize)304 782 y(pro)r(ofs.)953 920 y(1.)31 b Ft(Pr)o(oof-Carr)l(ying)22 b(Code)370 1020 y Fr(Pr)m(o)m(of-c)m(arrying)g(c)m(o)m(de)h Fs(\(PCC\))16 b(is)i(a)f(system)g(for)g(mobile)g(co)r(de)f(safet)n(y)h([18].)25 b(It)17 b(pro)n(vides)g(a)304 1086 y(means)f(for)g(securit)n (y-conscious)f Fr(c)m(o)m(de)21 b(c)m(onsumers)h Fs(to)17 b(ensure)e(that)h(co)r(de)f(they)i(receiv)n(e)f(from)304 1152 y Fr(c)m(o)m(de)j(pr)m(o)m(duc)m(ers)k Fs(o)n(v)n(er)16 b(a)f(computer)f(net)n(w)n(ork)h(is)h(safe)f(to)g(execute.)22 b(In)16 b(this)f(system,)h(the)f(co)r(de)304 1219 y(pro)r(ducer)21 b(is)k(resp)r(onsible)d(for)i(pro)n(viding)h(a)f Fr(pr)m(o)m(of)38 b Fs(\(in)24 b(a)g(suitable)g(formal)g(system\))f(that)304 1285 y(the)f(co)r(de)g(adheres)g(to)h(a)g Fr(safety)i(p)m(olicy)p Fs(,)i(made)c(public)g(b)n(y)h(the)f(co)r(de)f(consumers.)38 b(When)304 1352 y(a)25 b(co)r(de)f(consumer)g(wishes)h(to)h(run)f(the)f (co)r(de,)j(it)e(can)g(c)n(hec)n(k)h(the)f(pro)r(of)f(\014rst)h(to)g (decide)304 1418 y(whether)17 b(it)i(is)h(safe.)26 b(Pro)r(of)18 b(c)n(hec)n(king)g(is)i(decidable)e(and)g(can)g(b)r(e)h(implemen)n(ted) f(e\016cien)n(tly)-5 b(,)304 1485 y(while)20 b(program)f(v)n (eri\014cation)h(is)g(undecidable)f(and)h(inheren)n(tly)f(di\016cult.) 31 b(Th)n(us,)22 b(the)d(co)r(de)304 1551 y(pro)r(ducer)c(is)k(resp)r (onsible)e(for)h(doing)g(the)f Fr(har)m(d)26 b Fs(w)n(ork)19 b(of)f(pro)n(ving)g(a)h(program)e(correct,)f(and)304 1617 y(the)21 b(co)r(de)g(consumer)g(c)n(hec)n(k)i(the)f(resulting)f (pro)r(of,)i(whic)n(h)f(is)h Fr(e)m(asy)p Fs(.)39 b(Pro)r(of-carrying) 20 b(co)r(de)304 1684 y(frees)d(the)h(co)r(de)f(consumer)g(from)h(an)g (unpleasan)n(t)f(c)n(hoice)h(b)r(et)n(w)n(een)f(v)n(erifying)i(the)f (program)304 1750 y(on)g(its)g(o)n(wn,)g(or)g(blindly)h(trusting)e(it.) 370 1817 y(Some)j(useful)g(safet)n(y)g(p)r(olicies)g(migh)n(t)h (include)f Fr(memory)j(safety)p Fs(,)g Fr(termination)p Fs(,)h Fr(privacy)p Fs(,)304 1883 y(and)16 b Fr(c)m(orr)m(e)m(ctness)p Fs(.)27 b(Memory)16 b(safet)n(y)g(requires)f(that)h(all)h(memory)f (accesses)e(are)i(to)g(legally)h(al-)304 1949 y(lo)r(cated)d(lo)r (cations.)22 b(T)-5 b(ermination)15 b(requires)g(that)f(the)h(program)g (alw)n(a)n(ys)i(terminates)d(within)304 2016 y(a)23 b(giv)n(en)g(n)n (um)n(b)r(er)g(of)g(steps.)39 b(A)23 b(program)f(pro)n(vides)g(priv)m (acy)h(if)h(it)g(nev)n(er)e(sends)h(informa-)304 2082 y(tion)d(from)h(a)g(lo)r(cal)f(\014le)h(to)g(the)f(net)n(w)n(ork)g (without)g(the)g(user's)g(kno)n(wledge.)33 b(A)21 b(program)e(is)304 2149 y(correct)g(if)j(it)g(alw)n(a)n(ys)g(b)r(eha)n(v)n(es)f(according) f(to)h(some)g(b)r(eha)n(vioral)f(sp)r(eci\014cation,)g(whic)n(h)i(for) 304 2215 y(practical)g(purp)r(oses)g(is)i(usually)g(expressed)e(in)j (some)e(formal)h(system.)41 b(It)24 b(is)g(w)n(ell-kno)n(wn)304 2282 y(that)d(these)f(prop)r(erties)f(are)j(undecidable)e (\(termination,)h(in)i(particular,)f(is)g(precisely)f(the)304 2348 y(Halting)c(Problem\).)370 2414 y(A)k(curren)n(t)e(pro)r (of-carrying)f(co)r(de)h(system)h(pro)n(v)n(es)h(prop)r(erties)d(of)i (programs)f(indirectly)-5 b(,)304 2481 y(b)n(y)21 b(pro)n(ving)g (statemen)n(ts)e(in)i(\014rst-order)e(logic)i(whic)n(h)g(are)f(equiv)m (alen)n(t)g(to)h(program)e(safet)n(y)-5 b(.)304 2547 y(The)19 b(logic)g(used)f(b)n(y)i(this)f(system)f(is)i(represen)n(ted)c (in)k(the)e(Edin)n(burgh)g(Logical)h(F)-5 b(ramew)n(ork)304 2614 y(\(LF\))14 b([13].)24 b(Pro)r(ofs)13 b(are)h(represen)n(ted)f(as) h(pro)r(of-t)n(yp)r(ed)f(terms)h(in)i(LF,)f(and)g(pro)r(of)e(c)n(hec)n (king)i(in)304 2680 y(the)g(PCC)h(logic)h(corresp)r(onds)c(to)i(t)n(yp) r(e)h(c)n(hec)n(king)g(in)h(LF.)g(Because)e(of)h(this)g(corresp)r (ondenc)o(e,)304 2746 y(the)f(pro)r(of)g(c)n(hec)n(k)n(er)h(is)h (simply)g(a)f(LF)g(t)n(yp)r(e)g(c)n(hec)n(k)n(er,)h(whic)n(h)f(only)h (needs)e(to)h(b)r(e)f(implemen)n(ted)304 2813 y(once.)370 2879 y(Pro)r(of)h(represen)n(tations)e(in)j(this)g(system)g(are)f (curren)n(tly)g(three)g(to)h(four)f(times)h(the)g(size)g(of)304 2946 y(the)h(co)r(de)g(they)g(accompan)n(y)-5 b(.)27 b(F)-5 b(or)19 b(the)f(system)h(to)g(b)r(ecome)e(practical,)h(the)g (pro)r(ofs)g(need)g(to)304 3012 y(b)r(e)i(made)h(as)f(small)i(as)f(p)r (ossible,)g(while)h(still)g(b)r(eing)e(e\016cien)n(tly)h(c)n(hec)n(k)m (able.)33 b(One)21 b(p)r(ossible)304 3079 y(approac)n(h)h(to)h (optimizing)g(pro)r(ofs)f(migh)n(t)h(b)r(e)g(to)g(analyze)g(the)g (logic)g(and)g(pro)r(of)f(metho)r(ds)304 3145 y(used)16 b(b)n(y)i(the)e(theorem)f(pro)n(v)n(er)i(to)f(dev)n(elop)h(a)g(set)f (of)h(pro)r(of)f(transformations)e(whic)n(h)j(can)f(b)r(e)304 3211 y(applied)g(to)g(a)h(pro)r(of)f(to)g(reduce)f(its)i(complexit)n(y) -5 b(.)25 b(W)-5 b(e)16 b(will)j(tak)n(e)d(a)h(di\013eren)n(t)f (approac)n(h)f(here,)304 3278 y(whic)n(h)22 b(tak)n(es)f(accoun)n(t)f (only)i(of)g(the)f(syn)n(tactical)f(structure)f(of)j(pro)r(ofs,)f(not)g 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Fy(De\014nition)j(1.1.1.)26 b Fs(A)18 b(\014rst-order)d(statemen)n(t)h Fq(V)c(C)22 b Fs(is)c(a)f Fr(veri\014c)m(ation)22 b(c)m(ondition)j Fs(of)17 b(a)h(pro-)304 1647 y(gram)e Fq(p)g Fs(for)g(a)g(prop)r(ert)n (y)f Fq(P)24 b Fs(in)17 b(some)e(mathematical)g(mo)r(del)h(of)g (computation)e(if)j Fq(V)c(C)20 b Fs(is)d(true)304 1714 y(in)i(the)e(mo)r(del)h(if)h(and)f(only)h(if)g Fq(p)d Fl(2)g Fq(P)8 b Fs(.)25 b(A)18 b Fr(veri\014c)m(ation)23 b(c)m(ondition)g(gener)m(ator)28 b Fq(V)12 b(C)t(Gen)2175 1722 y Fp(P)2230 1714 y Fs(for)304 1780 y(a)20 b(safet)n(y)g(p)r(olicy) h Fq(P)28 b Fs(is)21 b(a)g(\(t)n(ypically)g(computable\))d(function)h (suc)n(h)i(that)e Fq(V)13 b(C)t(Gen)2083 1788 y Fp(P)2119 1780 y Fs(\()p Fq(p)p Fs(\))19 b(is)j(a)304 1847 y(v)n(eri\014cation)17 b(condition)g(of)h Fq(p)g Fs(for)g(the)f(prop)r(ert)n(y)f Fq(P)8 b Fs(.)370 1948 y(Note)17 b(that)g(a)h(program)f(ma)n(y)i(ha)n (v)n(e)f(non)n(trivially)h(di\013eren)n(t)f(v)n(eri\014cation)f (conditions.)304 2050 y Fy(Example)k(1.1.2.)26 b Fs(F)-5 b(or)22 b(the)e(trivial)j(program)d(prop)r(ert)n(y)g(con)n(taining)g (all)j(programs,)e(b)r(oth)304 2117 y Fl(>)f Fs(and)f Fl(:)p Fs(\()p Fl(9)p Fq(x)p Fs(\()p Fl(9)p Fq(y)r Fs(\()p Fl(9)p Fq(z)r Fs(\()p Fl(9)p Fq(n)p Fs(\(\()p Fq(n)14 b Fl(\025)k Fs(3\))f Fl(\033)h Fq(x)1159 2097 y Fp(n)1202 2117 y Fs(+)13 b Fq(y)1287 2097 y Fp(n)1335 2117 y Fs(=)18 b Fq(z)1424 2097 y Fp(n)1455 2117 y Fs(\)\)\)\)\))f(are)i(v)n (eri\014cation)g(conditions)f(for)304 2183 y(an)n(y)g(program)f Fq(p)h Fs(\(if)h(F)-5 b(ermat's)17 b(Last)g(Theorem)g(is)i(true;)e(see) h([25)o(,)h(8]\).)370 2285 y(W)-5 b(e)16 b(prefer)f(v)n(eri\014cation)g (conditions)f(that)h(are)g(easy)h(to)f(pro)n(v)n(e,)i(so)f(v)n (eri\014cation)f(condition)304 2352 y(generators)g(should)i(generate)f (conditions)h(that)g(are)g(as)h(simple)h(as)e(p)r(ossible.)304 2454 y Fy(Example)k(1.1.3.)26 b Fs(F)-5 b(or)18 b(illustration,)g (consider)e(a)j(v)n(ery)f(simple)h(programming)d(language)304 2520 y(for)h(a)i(random-access)c(register)i(mac)n(hine)g(with)i(the)e (follo)n(wing)i(instructions:)441 2600 y Ff(PUT\(R,C\))137 b(;)29 b(put)h(constant)f(value)g(C)g(into)h(R)441 2666 y(MOVE\(R)615 2674 y Fu(1)640 2666 y Ff(,)f(R)727 2674 y Fu(2)752 2666 y Ff(\))g(;)g(move)h(value)f(in)g(R)1304 2674 y Fu(2)1358 2666 y Ff(into)g(R)1532 2674 y Fu(1)441 2733 y Ff(LOAD\(L,R\))108 b(;)29 b(load)h(value)f(from)g(location)h(L)f (to)g(register)h(R)441 2799 y(STORE\(L,R\))79 b(;)29 b(store)h(value)f(from)g(register)h(R)f(to)g(location)h(L)441 2866 y(STOP)253 b(;)29 b(stop)h(the)f(program)304 3012 y Fs(A)24 b(program)f(in)i(this)f(language)f(is)h(a)h(sequence)d(of)j (instructions)d Ff(S)1776 3020 y Fu(1)1801 3012 y Ff(;)29 b Fq(:)9 b(:)g(:)i Ff(;)29 b(S)2020 3020 y Fp(n)2050 3012 y Fs(.)44 b(W)-5 b(e)24 b(will)304 3079 y(assume)18 b(that)g(there)f(is)j(a)f(\014nite)f(set)g(of)h(registers,)f(that)g (memory)h(lo)r(cations)e(and)i(constan)n(ts)304 3145 y(are)h(giv)n(en)i(b)n(y)g(in)n(tegers,)f(and)f(memory)h(lo)r(cations)f (do)h(not)f(o)n(v)n(erlap)i(\()p Fr(i.)g(e.)p Fs(,)h(writing)e(to)f (one)304 3211 y(lo)r(cation)c(nev)n(er)i(a\013ects)f(what)g(is)i(read)e (at)h(other)f(lo)r(cations.\))370 3278 y(Supp)r(ose)f(that)h(w)n(e)i (wish)f(to)g(c)n(hec)n(k)g(that)f(programs)f(only)j(access)d(memory)i (in)h(the)e(range)304 3344 y(from)g(memory)h(address)e Fd(A)28 b Fs(to)17 b Fd(B)10 b Fs(.)19 b(In)e(what)h(follo)n(ws,)g Ff(I)30 b(;)f Fd(S)e Fs(means)18 b(a)f(single)h(instruction)e Ff(I)304 3411 y Fs(heading)h(a)h(sequence)e Fd(S)28 b Fs(of)18 b(instructions.)23 b(A)18 b(v)n(eri\014cation)f(condition)g (generator)e(for)j(this)304 3477 y(v)n(ery)g(simple)h(safet)n(y)f(p)r (olicy)f(w)n(ould)i(b)r(e:)p eop %%Page: 33 35 33 34 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(33)441 283 y Ff(VC)499 291 y Fp(M)548 283 y Ff(\(PUT\(R,C\);)30 b Fd(S)9 b Ff(\))137 b(=)29 b(VC)1188 291 y Fp(M)1238 283 y Ff(\()p Fd(S)9 b Ff(\))441 350 y(VC)499 358 y Fp(M)548 350 y Ff(\(MOVE\(R)751 358 y Fu(1)776 350 y Ff(,)29 b(R)863 358 y Fu(2)888 350 y Ff(\);)h Fd(S)9 b Ff(\))29 b(=)g(VC)1188 358 y Fp(M)1238 350 y Ff(\()p Fd(S)9 b Ff(\))441 416 y(VC)499 424 y Fp(M)548 416 y Ff(\(LOAD\(L,R\);)30 b Fd(S)9 b Ff(\))108 b(=)29 b(\()p Fd(A)39 b Fl(\024)29 b Ff(L)g Fl(^)g Ff(L)g Fl(\024)h Fd(B)9 b Ff(\))29 b Fl(^)g Ff(VC)1774 424 y Fp(M)1824 416 y Ff(\()p Fd(S)9 b Ff(\))441 483 y(VC)499 491 y Fp(M)548 483 y Ff(\(STORE\(L,R\);)30 b Fd(S)10 b Ff(\))78 b(=)29 b(\()p Fd(A)39 b Fl(\024)29 b Ff(L)g Fl(^)g Ff(L)g Fl(\024)h Fd(B)9 b Ff(\))29 b Fl(^)g Ff(VC)1774 491 y Fp(M)1824 483 y Ff(\()p Fd(S)9 b Ff(\))441 549 y(VC)499 557 y Fp(M)548 549 y Ff(\(STOP;)30 b Fd(S)9 b Ff(\))253 b(=)29 b Fl(>)370 694 y Fs(Consider)17 b(the)h(follo)n(wing)g(t)n(w)n(o)h(programs:)441 772 y Ff(P)470 780 y Fu(1)495 772 y Ff(:)441 838 y(PUT\(R)586 846 y Fu(1)611 838 y Ff(,0\);)441 904 y(MOVE\(R)615 912 y Fu(2)640 904 y Ff(,)29 b(R)727 912 y Fu(1)752 904 y Ff(\);)441 971 y(STOP)441 1115 y(P)470 1123 y Fu(2)495 1115 y Ff(:)441 1182 y(LOAD\(100,R)731 1190 y Fu(1)756 1182 y Ff(\);)441 1248 y(LOAD\(101,R)731 1256 y Fu(2)756 1248 y Ff(\);)441 1315 y(STORE\(100,R)760 1323 y Fu(2)785 1315 y Ff(\);)441 1381 y(STORE\(101,R)760 1389 y Fu(1)785 1381 y Ff(\);)441 1447 y(STOP)304 1592 y Fs(The)15 b(\014rst)g(program) g(simply)h(sets)f(t)n(w)n(o)i(registers)d(to)h(zero,)g(so)h(it)g (should)g(alw)n(a)n(ys)g(b)r(e)f(memory-)304 1658 y(safe.)30 b(The)20 b(second)f(program)g(exc)n(hanges)g(the)h(v)m(alues)g(at)g(t)n (w)n(o)h(memory)f(lo)r(cations,)f(so)h(it)h(is)304 1725 y(necessary)16 b(to)i(v)n(erify)h(that)d(the)i(memory)g(accesses)e(are) h(legal.)370 1791 y(If)i(w)n(e)f(calculate)f(the)g(v)n(eri\014cation)g (conditions)g(of)h(the)f(t)n(w)n(o)i(programs)d(w)n(e)j(get)e(the)304 1858 y(follo)n(wing)h(results:)441 1936 y Ff(VC)499 1944 y Fp(M)548 1936 y Ff(\(P)606 1944 y Fu(1)631 1936 y Ff(\))29 b(=)g Fl(>)441 2014 y Ff(VC)499 2022 y Fp(M)548 2014 y Ff(\(P)606 2022 y Fu(2)631 2014 y Ff(\))g(=)g(\()p Fd(A)39 b Fl(\024)29 b Ff(100)g Fl(^)g Ff(100)h Fl(\024)f Fd(B)9 b Ff(\))29 b Fl(^)g Ff(\()p Fd(A)39 b Fl(\024)29 b Ff(101)h Fl(^)f Ff(101)g Fl(\024)g Fd(B)10 b Ff(\))747 2080 y Fl(^)29 b Ff(\()p Fd(A)39 b Fl(\024)29 b Ff(100)g Fl(^)g Ff(100)h Fl(\024)f Fd(B)9 b Ff(\))29 b Fl(^)g Ff(\()p Fd(A)39 b Fl(\024)29 b Ff(101)h Fl(^)e Ff(101)i Fl(\024)f Fd(B)9 b Ff(\))30 b Fl(^)f(>)304 2158 y Fs(As)15 b(w)n(e)g(w)n(ould)h(wish,)h(the)d(v)n(eri\014cation)g(condition)h(for) g(the)f(simple)i(program)e(whic)n(h)h(do)r(es)f(not)304 2225 y(access)20 b(memory)i(is)g(simply)h Fl(>)p Fs(.)37 b(F)-5 b(or)22 b(the)f(more)g(complicated)g(program,)g(there)g(is)h(a)g (more)304 2291 y(complicated)17 b(condition.)27 b(Moreo)n(v)n(er,)19 b(w)n(e)h(can)e(see)h(that)f(there)f(are)i(duplicate)f(goals)h(in)g (the)304 2357 y(v)n(eri\014cation)f(condition,)h(since)g(w)n(e)h(m)n (ust)g(v)n(erify)g(m)n(ultiple)g(accesses)e(to)h(the)g(same)g(memory) 304 2424 y(lo)r(cations.)j(If)c(w)n(e)g(needed)e(to)h(pro)r(duce)e(a)i (compact)f(formal)h(pro)r(of)f(of)h(the)g(second)f(condition)304 2490 y(b)n(y)26 b(hand,)h(w)n(e)f(could)f(consider)f(pro)n(ving)i Ff(\()p Fd(A)39 b Fl(\024)29 b Ff(100)g Fl(^)g Ff(100)h Fl(\024)f Fd(B)9 b Ff(\))26 b Fs(and)f Ff(\()p Fd(A)39 b Fl(\024)29 b Ff(101)g Fl(^)304 2557 y Ff(101)g Fl(\024)g Fd(B)10 b Ff(\))26 b Fs(as)h(lemmas,)i(and)c(citing)h(the)g(lemmas)g(t) n(wice)h(eac)n(h.)48 b(Without)26 b(p)r(erforming)304 2623 y(some)13 b(di\016cult)g(analysis,)h(an)f(automatic)f(theorem)f (pro)n(v)n(er)j(will)g(not)f(recognize)e(or)i(tire)g(of)g(this)304 2689 y(rep)r(etition,)j(and)i(will)h(blindly)g(generate)d(iden)n(tical) i(subpro)r(ofs)d(for)j(eac)n(h)g(iden)n(tical)g(subgoal.)370 2780 y(This)g(example)f(presen)n(ts)f(a)h(misleadingly)h(simple)g(view) g(of)f(the)g(problems)g(of)g(generating)304 2846 y(v)n(eri\014cation)d (conditions,)h(but)g(it)h(do)r(es)e(illustrate)g(the)h(p)r(oin)n(t)g 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(similar)h(asp)r(ects)d(of)i(formal)g(systems)f(\(suc)n(h)h(as)304 3411 y(term)15 b(structure)f(and)i(substitutions\))d(so)j(that)f (researc)n(hers)f(can)i(concen)n(trate)d(on)j(the)f(di\013er-)304 3477 y(ences.)p eop %%Page: 34 36 34 35 bop 304 151 a Fu(34)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)370 283 y Fs(The)d(Edin)n(burgh)f (Logical)g(F)-5 b(ramew)n(ork)17 b(\(LF\)[13)n(])h(is)g(one)e(suc)n(h)h (system.)24 b(LF)17 b(is)g(a)g(simply-)304 350 y(t)n(yp)r(ed)k Fq(\025)p Fs(-calculus)h(with)g(dep)r(enden)n(t)e(t)n(yp)r(es.)36 b(A)22 b(logic)g(consisting)e(of)j(constan)n(ts,)e(function)304 416 y(sym)n(b)r(ols,)f(prop)r(ositions,)d(and)h(pro)r(of)g(rules)h(can) f(b)r(e)g(represen)n(ted)f(b)n(y)i(a)g(LF)h(signature)d(asso-)304 483 y(ciating)h(constan)n(t)f(sym)n(b)r(ols)i(with)g(appropriate)e(t)n (yp)r(es.)27 b(According)17 b(to)h(the)g(Curry-Ho)n(w)n(ard)304 549 y(isomorphism)f(b)r(et)n(w)n(een)g(t)n(yp)r(es)g(and)h(prop)r 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Fr(typ)m(es)p Fs(,)g(and)d Fr(obje)m(cts)24 b Fs(\(or)17 b Fr(terms)5 b Fs(\).)304 1147 y(They)18 b(are)f(de\014ned)g(b)n(y)i(the)e(follo)n (wing)i(BNF)f(grammar:)441 1229 y Fd(Kinds)97 b Ff(K)29 b(::=)g(Type)h(|)f Fs(\005)p Fd(x)9 b Ff(:A.K)441 1295 y Fd(Types)97 b Ff(T)29 b(::=)g(a)g(|)h Fs(\005)p Fd(x)9 b Ff(:A)1112 1303 y Fu(1)1137 1295 y Ff(.A)1195 1303 y Fu(2)1249 1295 y Ff(|)29 b Fq(\025)p Fd(x)10 b Ff(:A)1436 1303 y Fu(1)1461 1295 y Ff(.A)1519 1303 y Fu(2)1573 1295 y Ff(|)29 b(A)g(M)441 1361 y Fd(Objects)39 b Ff(M)29 b(::=)g(c)g(|)h Fd(x)38 b Ff(|)29 b Fq(\025)p Fd(x)10 b Ff(:A.M)30 b(|)f(M)1403 1369 y Fu(1)1457 1361 y Ff(M)1486 1369 y Fu(2)304 1443 y Fs(The)15 b(letters)g Ff(a)h Fs(and)g Ff(c)h Fs(represen)n(t)c(t)n(yp)r(e)j(constan)n(ts)e(and)i(ob)s(ject)e (constan)n(ts)g(resp)r(ectiv)n(ely)-5 b(,)16 b(and)304 1509 y(the)21 b(letter)h Fd(x)32 b Fs(denotes)21 b(an)n(y)i(one)f(of)g (a)h(set)e(of)i(v)m(ariables.)37 b(The)22 b(ob)s(jects)f(and)h(t)n(yp)r (es)g(of)g(the)304 1576 y(form)h Fq(\025)p Fd(x)10 b Ff(:A.M)25 b Fs(or)f Fq(\025)p Fd(x)10 b Ff(:A.B)24 b Fs(represen)n(t)e(functions)g(\(or,)j(for)f(t)n(yp)r(es,)h Fr(typ)m(e)h(families)5 b Fs(\))24 b(whic)n(h)304 1642 y(can)17 b(b)r(e)g(applied)g(to)h(individual)g(ob)s(jects)e(of)i(the)f (sp)r(eci\014ed)f(t)n(yp)r(e)h Ff(A)p Fs(.)h(The)g(\005-t)n(yp)r(es)f (represen)n(t)304 1709 y(dep)r(enden)n(t)h(function)j(t)n(yp)r(es:)30 b(\005)p Fd(x)9 b Ff(:A.B)22 b Fs(denotes)d(the)i(t)n(yp)r(e)g(of)g (functions)f(from)h(ob)s(jects)f(of)304 1775 y(t)n(yp)r(e)e Ff(A)i Fs(to)e(t)n(yp)r(e)g Ff(B)p Fs(,)i(whic)n(h)f(binds)g Fd(x)29 b Fs(in)20 b Ff(B)p Fs(,)f(and)g(\005)p Fd(x)9 b Ff(:A.K)20 b Fs(denotes)d(the)h(kind)h(p)r(opulated)e(b)n(y)304 1842 y(t)n(yp)r(e)h(families)h(from)g(ob)s(jects)e(of)i(t)n(yp)r(e)f Ff(A)i Fs(to)e(t)n(yp)r(es)g(of)h(kind)g Ff(K)p Fs(,)h(whic)n(h)f (binds)f Fd(x)29 b Fs(in)20 b Ff(K)p Fs(.)f(When)304 1908 y Fd(x)26 b Fs(do)r(es)15 b(not)h(app)r(ear)f(free)g(in)i Ff(B)g Fs(or)f Ff(K)p Fs(,)h(w)n(e)g(use)f(the)g(more)g(familiar)h (notation)d Ff(A)29 b Fl(!)h Ff(B)17 b Fs(or)f Ff(A)29 b Fl(!)304 1974 y Ff(K)16 b Fs(to)g(represen)n(t)e(these)h(t)n(yp)r(es) g(and)h(kinds.)24 b(Moreo)n(v)n(er,)16 b(in)h(an)f(application)f Ff(M)29 b(N)17 b Fs(or)f Ff(A)29 b(M)p Fs(,)17 b(of)f(an)304 2041 y(ob)s(ject)g(or)h(a)h(t)n(yp)r(e)f(to)g(another)e(ob)s(ject,)i (the)f(ob)s(ject)g(or)i(t)n(yp)r(e)f Ff(M)h Fs(or)f Ff(A)h Fs(m)n(ust)f(b)r(e)g(a)h(function)e(or)304 2107 y(a)i(t)n(yp)r(e)f (family)-5 b(,)20 b(and)e(ha)n(v)n(e)h(the)e(appropriate)e(t)n(yp)r(e)j (or)g(kind.)370 2174 y(In)g(addition,)g(v)m(ariable)g(t)n(yp)r(es)f (and)h(constan)n(t)e(t)n(yp)r(es)h(and)h(kinds)g(are)g(b)r(ound)e(in)j (LF)304 2240 y Fr(c)m(ontexts)24 b Fs(\000)18 b(and)g Fr(signatur)m(es)24 b Fs(\006)19 b(resp)r(ectiv)n(ely)-5 b(,)17 b(de\014ned)g(as)h(follo)n(ws:)441 2322 y(\000)52 b Ff(::=)29 b Fl(\001)h Ff(|)f Fs(\000)p Ff(,)g Fd(x)9 b Ff(:A)441 2388 y Fs(\006)47 b Ff(::=)29 b Fl(\001)h Ff(|)f Fs(\006)p Ff(,)g(c:A)g(|)g Fs(\006)p Ff(,)g(a:K)304 2470 y Fs(There)17 b(are)g(\014v)n(e)i(main)f(judgmen)n(ts)f(in)h(LF:) 550 2551 y(\006)g(sig)522 b(\006)19 b(is)g(a)f(v)m(alid)g(signature)550 2618 y(\006)d Fl(`)g Fs(\000)j(ctx)412 b(\000)18 b(is)h(a)f(v)m(alid)h (con)n(text)e(under)g(\006)550 2684 y(\000)e Fl(`)634 2692 y Fu(\006)683 2684 y Fq(K)465 b(K)23 b Fs(is)c(a)f(v)m(alid)g (kind)550 2751 y(\000)d Fl(`)634 2759 y Fu(\006)683 2751 y Fq(A)g Fs(:)h Fq(K)377 b Fs(T)n(yp)r(e)18 b Fq(A)g Fs(has)g(kind)h Fq(K)550 2817 y Fs(\000)c Fl(`)634 2825 y Fu(\006)683 2817 y Fq(M)21 b Fs(:)16 b Fq(A)364 b Fs(Ob)s(ject)17 b Fq(M)25 b Fs(has)17 b(t)n(yp)r(e)h Fq(A)304 2899 y Fs(When)g(there)f(is)j(no)e(p)r(ossibilit)n(y)i(for)e(confusion)g(w)n (e)h(will)h(omit)f(the)f(subscript)f(\006)i(in)h(the)e(last)304 2965 y(three)i(judgmen)n(ts.)35 b(W)-5 b(e)22 b(will)i(not)d(go)h(in)n (to)g(the)f(further)f(details)i(of)g(these)e(judgmen)n(ts)h(and)304 3032 y(rules)c(relating)h(them;)f(these)g(can)h(b)r(e)f(found)g(in)i ([13)o(].)304 3135 y Fy(Example)i(1.2.1.)26 b Fs(The)18 b(structure)d(of)j(prop)r(ositions)e(using)i(connectiv)n(es)1165 3232 y(\()p Fl(>)p Fq(;)9 b Fl(^)p Fq(;)g Fl(\033)p Fq(;)g Fl(8)p Fs(\))304 3329 y(can)17 b(b)r(e)h(represen)n(ted)d(b)n(y)k(the)e (follo)n(wing)i(sequence)d(of)i(t)n(yp)r(e)g(declarations:)441 3411 y Ff(\045)29 b(basic)g(types)441 3477 y(i)224 b(:)29 b(Type)-87 b(\045)30 b(individuals)p eop %%Page: 35 37 35 36 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(35)441 283 y Ff(o)224 b(:)29 b(Type)-87 b(\045)30 b(propositions)441 350 y(pf)195 b(:)29 b(o)g Fl(!)h Ff(Type)-230 b(\045)30 b(proof)f(derivations)441 498 y(\045)g(predicate)h(functions)441 564 y(true)137 b(:)29 b(o)441 630 y(and)166 b(:)29 b(o)g Fl(!)h Ff(o)f Fl(!)g Ff(o)441 697 y(imp)166 b(:)29 b(o)g Fl(!)h Ff(o)f Fl(!)g Ff(o)441 763 y(all)166 b(:)29 b(\(i)h Fl(!)f Ff(o\))g Fl(!)h Ff(o)370 912 y Fs(F)-5 b(or)16 b(example,)g(the)f(\014rst-order)e(prop)r(osition)h Fl(8)p Fq(x)p Fs(\(\(\()p Fq(P)8 b Fs(\()p Fq(x)p Fs(\))j Fl(\033)k Fq(Q)p Fs(\()p Fq(x)p Fs(\)\))7 b Fl(^)g Fq(P)h Fs(\()p Fq(x)p Fs(\)\))j Fl(\033)k Fq(Q)p Fs(\()p Fq(x)p Fs(\)\))f(can)304 978 y(b)r(e)g(represen)n(ted)e(in)j(LF)g(b)n(y)i Ff(\(all)29 b Fq(\025)p Fd(x)10 b Ff(:i.\(imp)30 b(\(and)f(\(imp)g(\(P)g Fd(x)10 b Ff(\))29 b(\(Q)g Fd(x)10 b Ff(\)\))29 b(\(P)g Fd(x)10 b Ff(\)\))29 b(\(Q)304 1044 y Fd(x)9 b Ff(\)\)\))p Fs(.)26 b(The)17 b(sym)n(b)r(ols)i Ff(P)f Fs(and)g Ff(Q)g Fs(ha)n(v)n(e)h(t)n(yp)r(e)e Ff(i)29 b Fl(!)h Ff(o)p Fs(.)370 1111 y(The)18 b(follo)n(wing)h(LF)f(constan)n(t)e (declarations)g(sho)n(w)i(ho)n(w)g(the)g(logical)g(structure)d(can)j(b) r(e)304 1177 y(represen)n(ted)d(in)k(LF:)441 1259 y Ff(\045)29 b(proof)g(rules)441 1325 y(true)p 560 1325 18 3 v 21 w(i)87 b(:)29 b(pf)h(true)441 1391 y(and)p 531 1391 V 21 w(i)116 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:o.)p Fs(\005)p Fd(B)f Ff(:o.pf)29 b Fd(A)39 b Fl(!)29 b Ff(pf)h Fd(B)9 b Fl(!)30 b Ff(pf)f(\(and)g Fd(A)39 b(B)9 b Ff(\))441 1458 y(and)p 531 1458 V 21 w(el)87 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:o.)p Fs(\005)p Fd(B)f Ff(:o.pf)29 b(\(and)h Fd(A)38 b(B)10 b Ff(\))29 b Fl(!)g Ff(pf)h Fd(A)441 1524 y Ff(and)p 531 1524 V 21 w(er)87 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:o.)p Fs(\005)p Fd(B)f Ff(:o.pf)29 b(\(and)h Fd(A)38 b(B)10 b Ff(\))29 b Fl(!)g Ff(pf)h Fd(B)441 1591 y Ff(imp)p 531 1591 V 21 w(i)116 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:o.)p Fs(\005)p Fd(B)f Ff(:o.\(pf)29 b Fd(A)39 b Fl(!)29 b Ff(pf)h Fd(B)9 b Ff(\))29 b Fl(!)h Ff(pf)f(\(imp)g Fd(A)39 b(B)10 b Ff(\))441 1657 y(imp)p 531 1657 V 21 w(e)116 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:o.)p Fs(\005)p Fd(B)f Ff(:o.pf)29 b(\(imp)h Fd(A)38 b(B)10 b Ff(\))29 b Fl(!)g Ff(pf)h Fd(A)38 b Fl(!)30 b Ff(pf)f Fd(B)441 1723 y Ff(all)p 531 1723 V 21 w(i)116 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:i)29 b Fl(!)g Ff(o.)p Fs(\005)p Fd(x)9 b Ff(:i.pf)30 b(\()p Fd(A)39 b(x)9 b Ff(\))29 b Fl(!)h Ff(pf)f(\(all)g Fd(A)10 b Ff(\))441 1790 y(all)p 531 1790 V 21 w(e)116 b(:)29 b Fs(\005)p Fd(A)10 b Ff(:i)29 b Fl(!)g Ff(o.)p Fs(\005)p Fd(x)9 b Ff(:i.pf)30 b(\(all)f Fd(A)10 b Ff(\))29 b Fl(!)g Ff(pf)g(\()p Fd(A)39 b(x)10 b Ff(\))370 1960 y Fs(These)27 b(declarations)e(corresp)r(ond)f(to)j(the)f(usual)i(pro)r(of)d(rules)i (for)g(the)g Fl(^)p Fs(,)j Fl(\033)p Fs(,)h(and)c Fl(8)304 2026 y Fs(connectiv)n(es.)21 b(F)-5 b(or)14 b(example,)h(consider)d (the)h(rules)g(for)g Fl(^)p Fs(-in)n(tro)r(duction)f(and)h Fl(\033)p Fs(-in)n(tro)r(duction:)812 2119 y(\000)i Fl(`)g Fq(A)138 b Fs(\000)15 b Fl(`)g Fq(B)p 812 2141 423 3 v 1246 2160 a Fl(^)p Fs(I)900 2188 y(\000)g Fl(`)g Fq(A)d Fl(^)g Fq(B)1368 2119 y Fs(\000)p Fq(;)d(u)15 b Fs(:)h Fq(A)e Fl(`)i Fq(B)p 1368 2141 288 3 v 1666 2161 a Fl(\033)g Fs(I)1745 2137 y Fp(u)1382 2188 y Fs(\000)f Fl(`)g Fq(A)g Fl(\033)g Fq(B)370 2282 y Fs(The)24 b(LF)h(constan)n(t)d Ff(and)p 905 2282 18 3 v 21 w(i)j Fs(tak)n(es)f(4)g(argumen)n(ts:)36 b(t)n(w)n(o)25 b(predicates)d Ff(A)i Fs(and)g Ff(B)p Fs(,)h(and)f(t)n(w)n(o)304 2348 y(pro)r(ofs,)18 b(pro)r(of)k Ff(p)e Fs(of)j Ff(A)d Fs(and)f Ff(q)g Fs(of)24 b Ff(B)p Fs(.)c(The)f(t)n(w)n(o)g(pro)r(of)f(argumen)n(ts)g(represen)n(t)f(the)h (t)n(w)n(o)i(judg-)304 2414 y(men)n(ts)g(that)f(are)g(required)h(in)g (the)g Fl(^)p Fs(-in)n(tro)r(duction)e(rule.)31 b(The)20 b(other)f(t)n(w)n(o)i(argumen)n(ts)d(are)304 2481 y(parameters)d(that)j (\014x)g(precisely)g(whic)n(h)g(v)n(ersion)g(of)g(the)g(rule)g(is)h (used.)24 b(The)18 b(t)n(yp)r(e)g(of)g(a)g(w)n(ell-)304 2547 y(formed)12 b(application)h(of)18 b Ff(and)p 920 2547 V 21 w(i)c Fs(to)f Ff(A)p Fs(,)i Ff(B)p Fs(,)f Ff(p)p Fs(,)h(and)f Ff(q)g Fs(has)f(t)n(yp)r(e)g Ff(pf)29 b(\(and)g(A)g(B\))p Fs(,)15 b(corresp)r(onding)304 2614 y(to)i(the)h(result)f(of)h(the)f Fl(^)p Fs(-in)n(tro)r(duction)f(rule.)370 2680 y(Analogously)-5 b(,)18 b(the)f(constan)n(t)f Ff(imp)p 1103 2680 V 21 w(i)j Fs(tak)n(es)e(three)g(argumen)n(ts:)23 b(t)n(w)n(o)18 b(prop)r(ositions)e Ff(A)i Fs(and)304 2746 y Ff(B)p Fs(,)g(and)g(a)f (function)g Ff(f)h Fs(whic)n(h)g(maps)f(ob)s(jects)f(of)i(t)n(yp)r(e)f Ff(pf)29 b(A)18 b Fs(to)g(ob)s(jects)e(of)h(t)n(yp)r(e)h Ff(pf)29 b(B)p Fs(.)18 b(This)304 2813 y(third)j(argumen)n(t)f (represen)n(ts)f(the)h(single)i(prerequisite)d(judgmen)n(t)i(in)h(the)e Fl(\033)p Fs(-in)n(tro)r(duction)304 2879 y(rule,)j(and)f(again)g(the)f (\014rst)g(t)n(w)n(o)i(argumen)n(ts)e(parameterize)e(the)i(rule.)37 b(The)22 b(result)f(of)h(ap-)304 2946 y(plying)g(the)g(function)e Ff(imp)p 885 2946 V 21 w(i)j Fs(is)f(an)g(ob)s(ject)f(of)h(t)n(yp)r(e)f Ff(pf)29 b(\(imp)h(A)f(B\))p Fs(,)23 b(whic)n(h)f(corresp)r(onds)304 3012 y(to)h(the)h(result)f(of)h(the)f Fl(\033)p Fs(-in)n(tro)r(duction) f(rule.)43 b(In)24 b(natural)f(deduction,)h(w)n(e)g(can)g(only)g(use) 304 3079 y(the)17 b(assumption)g(of)i Fq(A)f Fs(in)h(the)e(pro)r(of)g (of)h Fq(B)s Fs(;)h(the)f(v)m(ariable)g Fq(u)g Fs(is)h(used)f(to)g (enforce)e(this)i(scop)r(e)304 3145 y(restriction.)27 b(In)20 b(the)f(LF)h(represen)n(tation)c(of)k(this)g(pro)r(of)e(rule,)i (w)n(e)g(enforce)e(this)i(restriction)304 3211 y(b)n(y)j(pro)n(viding)f (a)h(function)e(function)g Ff(f)h Fs(from)g(pro)r(ofs)f(of)27 b Ff(A)22 b Fs(to)g(pro)r(ofs)f(of)26 b Ff(B)p Fs(.)e(The)e(b)r(o)r(dy) f(of)304 3278 y(this)c(function)f(is)j(abstracted)14 b(o)n(v)n(er)k(a)g(v)m(ariable)f Fd(x)27 b Fs(of)18 b(t)n(yp)r(e)f Ff(pf)29 b(A)p Fs(,)18 b(whic)n(h)g(only)g(has)f(meaning)304 3344 y(within)h(the)f(function)g(b)r(o)r(dy)-5 b(.)370 3411 y(T)g(o)16 b(represen)n(t)e(the)h(v)n(eri\014cation)f(conditions)h (sho)n(wn)g(ab)r(o)n(v)n(e,)h(w)n(e)g(could)g(in)n(tro)r(duce)d (further)304 3477 y(represen)n(tations)23 b(for)j(the)f(constan)n(ts,)i (function)e(sym)n(b)r(ols,)k(predicates,)e(and)f(pro)r(of)f(rules)p eop %%Page: 36 38 36 37 bop 304 151 a Fu(36)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)304 283 y Fs(needed)f(to)i(express) f(prop)r(erties)e(of)j(natural)f(n)n(um)n(b)r(ers.)33 b(W)-5 b(e)21 b(will)i(instead)d(concen)n(trate)e(on)304 350 y(asp)r(ects)e(of)i(LF)g(that)f(are)g(more)h(relev)m(an)n(t)f(to)g (our)h(discussion)f(of)h(pro)r(of)f(optimization.)370 416 y(If)24 b(w)n(e)g(represen)n(t)d(a)i(logical)h(system)f Fq(L)g Fs(in)h(LF)g(using)f(signature)f(\006,)j(w)n(e)f(can)f(represen) n(t)304 483 y(pro)r(ofs)c(in)i(that)e(system)i(as)f(LF)h(terms)f(whic)n (h)h(w)n(e)g(shall)g(refer)e(to)i(as)f(pro)r(of)f(terms.)32 b(An)20 b Fq(L)p Fs(-)304 549 y(pro)r(of)g Fq(D)k Fs(of)e Fq(L)p Fs(-sen)n(tence)f Fq(p)h Fs(corresp)r(onds)d(to)i(a)i(pro)r(of)d (term)i Fc(p)p Fq(D)r Fc(q)f Fs(in)h(LF)h(of)f(t)n(yp)r(e)f Ff(pf)i Fc(p)p Fq(p)p Fc(q)p Fs(,)304 616 y(where)e Fc(p)p Fl(\001)p Fc(q)i Fs(denotes)d(the)i(LF)g(represen)n(tations)d(of)k Fq(L)p Fs(-pro)r(ofs)e(and)g Fq(L)p Fs(-terms.)37 b(W)-5 b(e)23 b(can)f(also)304 682 y(c)n(hec)n(k)c(alleged)f(pro)r(of)g (represen)n(tations)e(b)n(y)k(v)n(erifying)f Fl(\001)e(`)1535 690 y Fu(\006)1585 682 y Fc(p)p Fq(D)r Fc(q)e Fs(:)h Ff(pf)q Fc(p)p Fq(p)p Fc(q)p Fs(.)304 774 y Fy(Example)21 b(1.2.2.)26 b Fs(W)-5 b(e)21 b(migh)n(t)h(ha)n(v)n(e)f(a)h(pro)r(of)e (in)h(\014rst-order)e(natural)h(deduction)g(of)h(\()p Fq(A)13 b Fl(^)304 840 y Fq(B)s Fs(\))h Fl(\033)i Fs(\()p Fq(B)e Fl(^)f Fq(A)p Fs(\),)k(as)h(follo)n(ws:)p 727 886 438 3 v 1176 894 a Fp(u)727 932 y Fq(u)d Fs(:)h Fq(A)11 b Fl(^)i Fq(B)18 b Fl(`)d Fq(A)d Fl(^)g Fq(B)p 727 955 V 1176 970 a Fl(^)p Fs(E)1250 978 y Fp(R)778 1001 y Fq(u)j Fs(:)h Fq(A)c Fl(^)g Fq(B)18 b Fl(`)d Fq(B)p 1303 886 V 1752 894 a Fp(u)1303 932 y Fq(u)g Fs(:)h Fq(A)c Fl(^)g Fq(B)18 b Fl(`)d Fq(A)d Fl(^)g Fq(B)p 1303 955 V 1752 970 a Fl(^)p Fs(E)1827 978 y Fp(L)1356 1001 y Fq(u)j Fs(:)h Fq(A)c Fl(^)g Fq(B)18 b Fl(`)d Fq(A)p 778 1024 910 3 v 1699 1043 a Fl(^)p Fs(I)1015 1071 y Fq(u)g Fs(:)h Fq(A)11 b Fl(^)i Fq(B)18 b Fl(`)d Fq(B)g Fl(^)d Fq(A)p 974 1093 521 3 v 1505 1113 a Fl(\033)j Fs(I)1583 1088 y Fp(u)989 1140 y Fl(`)g Fs(\()p Fq(A)d Fl(^)g Fq(B)s Fs(\))j Fl(\033)g Fs(\()p Fq(B)g Fl(^)d Fq(A)p 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b(w)n(e)h(migh)n(t)f(wish)g(to)g(view)g(the)f (LF)i(constan)n(ts)c(as)j(functions)f(or)h(op)r(erators)d(on)i(their) 304 616 y(o)n(wn,)g(with)g(appropriate)e(ranks.)370 682 y(In)25 b(the)f(former)g(approac)n(h)f(w)n(e)i(can)g(translate)d(more)j (general)e(pro)r(ofs)g(b)r(ecause)g(in)i(LF,)304 748 y(function)17 b(applications)g(ma)n(y)i(b)r(e)f(partial:)25 b(function)17 b(sym)n(b)r(ols)i(of)f(t)n(yp)r(es)g(that)f(indicate)h (that)304 815 y(they)c(can)h(tak)n(e)g Fq(n)h Fs(argumen)n(ts)e(migh)n (t)h(not)g(alw)n(a)n(ys)h(b)r(e)e(applied)h(to)g(all)h Fq(n)g Fs(argumen)n(ts.)22 b(F)-5 b(or)15 b(the)304 881 y(same)h(reason)f(w)n(e)i(migh)n(t)f(ha)n(v)n(e)h(more)f(opp)r (ortunities)e(to)i(generate)e(useful)i(patterns)e(b)r(ecause)304 948 y(there)j(migh)n(t)h(b)r(e)g(man)n(y)h(o)r(ccurrences)14 b(of)k(partial)g(applications.)24 b(The)18 b(second)f(metho)r(d)g(ma)n (y)304 1014 y(lead)j(to)g(smaller)g(term)f(represen)n(tations)e(than)j (the)f(\014rst)h(metho)r(d.)29 b(If)21 b(the)e(pro)r(ofs)g(w)n(e)h (wish)304 1080 y(to)h(compress)g(do)g(not)g(t)n(ypically)i(ha)n(v)n(e)g (partial)e(function)g(applications,)h(then)f(extra)g(e\013ort)304 1147 y(exp)r(ended)16 b(trying)h(to)h(deal)g(with)g(suc)n(h)g(terms)f (and)g(\014nd)h(useful)g(partial)f(patterns)f(w)n(ould)i(b)r(e)304 1213 y(w)n(asted.)29 b(T)-5 b(ranslating)19 b(pro)r(of)g(rules)h(and)f (other)g(LF)h(function)f(sym)n(b)r(ols)h(as)g(ordinary)f Fq(n)p Fs(-ary)304 1280 y(function)e(sym)n(b)r(ols)h(ma)n(y)h(simplify) g(pattern)d(searc)n(h.)370 1346 y(F)-5 b(or)20 b(simplicit)n(y)-5 b(,)21 b(and)e(b)r(ecause)e(it)j(will)g(result)f(in)g(a)h(faster)d (implemen)n(tation)i(w)n(e)g(will)i(use)304 1413 y(the)i(second)f (metho)r(d)g(to)h(represen)n(t)f(pro)r(of)g(terms.)41 b(The)23 b(pro)r(ofs)f(pro)r(duced)f(b)n(y)k(the)e(PCC)304 1479 y(system)g(are)g(t)n(ypically)i(in)f(a)g(normal)g(form)f(that)g (guaran)n(tees)e(that)i(there)f(are)i(no)f(partial)304 1545 y(function)17 b(applications.)304 1647 y Fy(De\014nition)k(2.1.1.) 26 b Fs(The)19 b Fr(r)m(ank)26 b Fq(r)r Fs(\()p Ff(A)p Fs(\))18 b(of)h(t)n(yp)r(e)g Ff(A)g Fs(is)h(the)e(largest)f(n)n(um)n(b) r(er)i(of)g(argumen)n(ts)e(to)304 1713 y(whic)n(h)e(it)g(can)f(b)r(e)f (applied)i(in)g(LF.)g(It)g(can)f(b)r(e)f(recursiv)n(ely)i(de\014ned)e (so)i(that)e(if)i Ff(A)h Fs(=)f(\005)p Fd(x)9 b Ff(:B)2188 1721 y Fu(1)2213 1713 y Ff(.B)2271 1721 y Fu(2)304 1780 y Fs(then)21 b Fq(r)r Fs(\()p Ff(A)o Fs(\))g(=)h Fq(r)r Fs(\()p Ff(B)696 1788 y Fu(2)721 1780 y Fs(\))15 b(+)g(1,)23 b(otherwise)d Fq(r)r Fs(\()p Ff(A)o Fs(\))i(=)g(0.)36 b(The)22 b(rank)g Fq(r)r Fs(\()p Ff(K)o Fs(\))g(of)g(a)g(kind)g Ff(K)g Fs(is)h(de\014ned)304 1846 y(similarly)-5 b(.)370 1913 y(The)25 b(op)r(erator)d(domain)j(\(\006)987 1921 y Fp(LF)1054 1913 y Fq(;)9 b(\036)p Fs(\))25 b(for)g(the)f(LF)h (signature)e(\006)j(and)e(v)m(ariable)h(set)f Fq(X)30 b Fs(is)304 1979 y(de\014ned)16 b(so)i(that)670 2072 y(\006)710 2080 y Fp(LF)792 2072 y Fs(=)e Fl(f)p Ff(Type)p Fl(g)c([)g(f)p 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(the)e(20)i(p)r(ossible)f(com)n(binations)304 1877 y(are)j(giv)n(en)i (here.)31 b(W)-5 b(e)21 b(will)h(giv)n(e)f(results)e(for)h(eac)n(h)g (of)g(the)g(searc)n(h)g(algorithms)f(paired)h(with)304 1944 y(\()p Fy(TD)o Fs(\),)e(and)e(for)h(\()p Fy(ELZ)o Fs(\))g(paired)f(with)h(eac)n(h)f(of)h(the)f(enco)r(ding)f(algorithms.) 23 b(Only)18 b(the)e(small)304 2010 y(and)i(medium)h(pro)r(ofs)f(w)n (ere)h(used)f(with)h(\()p Fy(LP)o Fs(\))g(or)g(\()p Fy(IGI)1509 2018 y Fu(1)1533 2010 y Fy(M)1593 2018 y Fu(1)1619 2010 y Fs(\),)g(and)g(only)g(the)f(small)i(pro)r(ofs)304 2077 y(w)n(ere)d(used)h(with)g(\()p Fy(TB)p Fs(\).)304 2176 y(3.2.)32 b Fy(Exp)r(erimen)n(tal)f(data.)c Fs(T)-5 b(able)27 b(1)g(summarizes)f(the)g(p)r(erformance)e(of)j(the)f(v)m(arious)304 2243 y(algorithm)18 b(com)n(binations)g(on)h(small,)i(medium,)f(and)e (large)h(pro)r(ofs.)26 b(Figures)19 b(5)g(and)g(6)g(plot)304 2309 y(pro)r(of)e(size)h Fr(vs.)h Fs(time)g(and)f(compressed)e(size)i (for)h(di\013eren)n(t)f(searc)n(h)f(tec)n(hniques,)h(and)g(\014gures) 304 2376 y(7)e(and)g(8)g(compare)f(the)g(results)g(for)h(the)g (di\013eren)n(t)f(enco)r(dings.)22 b(The)16 b(searc)n(h)f(summaries)h (also)304 2442 y(include)h(the)h(results)f(for)g(sequen)n(tial)h(\()p Fy(LZ)p Fs(\))g(compression.)p 710 2670 1180 3 v 709 2737 3 67 v 808 2717 a(Metho)r(d)p 1022 2737 V 53 w(Small)p 1213 2737 V 55 w(Medium)p 1466 2737 V 55 w(Large)p 1658 2737 V 54 w(Ov)n(erall)p 1889 2737 V 710 2739 1180 3 v 709 2805 3 67 v 777 2785 a(\()p Fy(LPTD)p Fs(\))p 1022 2805 V 55 w(0.948)p 1213 2805 V 63 w(0.771)p 1466 2805 V 127 w({)p 1658 2805 V 164 w(0.787)p 1889 2805 V 710 2807 1180 3 v 709 2874 3 67 v 778 2854 a(\()p Fy(H1TD)o Fs(\))p 1022 2874 V 55 w(1.047)p 1213 2874 V 63 w(0.848)p 1466 2874 V 127 w(0.631)p 1658 2874 V 65 w(0.708)p 1889 2874 V 710 2876 1180 3 v 709 2943 3 67 v 778 2923 a(\()p Fy(H2TD)o Fs(\))p 1022 2943 V 55 w(0.984)p 1213 2943 V 63 w(0.822)p 1466 2943 V 127 w(0.579)p 1658 2943 V 65 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y(?)626 1261 y(?)618 1260 y(?)606 1269 y(?)615 1272 y(?)606 1269 y(?)609 1265 y(?)587 1273 y(?)-34 b(?)k(?)569 1276 y(?)569 1277 y(?)f(?)563 1278 y(?)e(?)558 1280 y(?)i(?)h(?)1003 1645 y Ft(Figure)22 b(5.)27 b Fs(Searc)n(h)17 b(times)p 625 2740 27 3 v 568 2757 a(0)p 2192 2740 V 625 2548 V 485 2565 a(5000)p 2192 2548 V 625 2356 V 457 2373 a(10000)p 2192 2356 V 625 2163 V 457 2179 a(15000)p 2192 2163 V 625 1971 V 457 1987 a(20000)p 2192 1971 V 625 1779 V 457 1795 a(25000)p 2192 1779 V 625 2740 3 27 v 611 2817 a(0)p 625 1805 V 944 2740 V 249 w(5000)p 944 1805 V 1262 2740 V 193 w(10000)p 1262 1805 V 1581 2740 V 179 w(15000)p 1581 1805 V 1900 2740 V 179 w(20000)p 1900 1805 V 2218 2740 V 178 w(25000)p 2218 1805 V 625 2740 1594 3 v 2218 2740 3 962 v 625 1779 1594 3 v 291 2203 a(comp.)321 2267 y(size)258 2325 y(\()p Fq(S)e Fl(\000)d Fq(\016)r(S)s Fs(\))1329 2873 y(size)17 b(\()p Fq(S)s Fs(\))p 625 2740 3 962 v 654 2285 3 480 v 654 1805 584 3 v 1238 2285 3 480 v 654 2285 584 3 v 970 1850 a 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5 1 v 722 2679 V 723 2678 V 725 2678 V 726 2677 V 727 2676 V 728 2676 V 729 2675 V 730 2674 V 732 2674 V 733 2673 V 721 2681 3 3 v 735 2672 V 737 2670 5 1 v 738 2670 V 740 2669 V 741 2668 V 742 2668 V 743 2667 V 745 2666 V 746 2666 V 747 2665 V 748 2664 V 750 2664 V 737 2672 3 3 v 752 2663 V 754 2661 4 1 v 755 2660 V 756 2660 V 757 2659 V 758 2658 V 759 2658 V 760 2657 V 761 2656 V 762 2656 V 764 2655 V 765 2654 V 766 2654 V 767 2653 V 754 2663 2 3 v 768 2652 V 770 2650 5 1 v 772 2650 V 773 2649 V 774 2648 V 775 2648 V 776 2647 V 777 2646 V 779 2646 V 780 2645 V 781 2644 V 782 2644 V 770 2652 3 3 v 784 2643 V 786 2641 5 1 v 788 2640 V 789 2640 V 790 2639 V 791 2638 V 792 2638 V 793 2637 V 795 2636 V 796 2636 V 797 2635 V 798 2634 V 786 2643 3 3 v 800 2633 V 802 2632 5 1 v 804 2631 V 805 2630 V 806 2630 V 807 2629 V 808 2628 V 809 2628 V 811 2627 V 812 2626 V 813 2626 V 814 2625 V 802 2633 3 3 v 816 2624 V 818 2622 4 1 v 819 2622 V 820 2621 V 821 2620 V 822 2620 V 823 2619 V 824 2618 V 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2565 3 3 v 928 2556 V 930 2554 4 1 v 931 2554 V 932 2553 V 933 2552 V 934 2552 V 935 2551 V 936 2550 V 937 2550 V 939 2549 V 940 2548 V 941 2548 V 942 2547 V 943 2546 V 930 2556 2 3 v 944 2545 V 946 2544 5 1 v 948 2543 V 949 2542 V 950 2542 V 951 2541 V 952 2540 V 953 2540 V 955 2539 V 956 2538 V 957 2538 V 958 2537 V 946 2545 3 3 v 960 2536 V 962 2534 5 1 v 964 2534 V 965 2533 V 966 2532 V 967 2532 V 969 2531 V 970 2530 V 971 2530 V 973 2529 V 974 2528 V 975 2528 V 962 2536 3 3 v 977 2527 V 980 2525 4 1 v 981 2524 V 982 2524 V 983 2523 V 984 2522 V 985 2522 V 986 2521 V 987 2520 V 988 2520 V 989 2519 V 990 2518 V 991 2518 V 992 2517 V 980 2527 2 3 v 994 2516 V 996 2514 5 1 v 997 2514 V 998 2513 V 999 2512 V 1000 2512 V 1002 2511 V 1003 2510 V 1004 2510 V 1005 2509 V 1006 2508 V 1007 2508 V 996 2516 3 3 v 1010 2507 V 1012 2505 5 1 v 1013 2504 V 1014 2504 V 1015 2503 V 1016 2502 V 1018 2502 V 1019 2501 V 1020 2500 V 1021 2500 V 1022 2499 V 1023 2498 V 1012 2507 3 3 v 1026 2497 V 1028 2496 5 1 v 1029 2495 V 1030 2494 V 1031 2494 V 1032 2493 V 1034 2492 V 1035 2492 V 1036 2491 V 1037 2490 V 1038 2490 V 1039 2489 V 1028 2497 3 3 v 1042 2488 V 1044 2486 4 1 v 1045 2486 V 1046 2485 V 1047 2484 V 1048 2484 V 1049 2483 V 1050 2482 V 1051 2482 V 1052 2481 V 1053 2480 V 1054 2480 V 1055 2479 V 1056 2478 V 1044 2488 2 3 v 1058 2477 V 1060 2476 5 1 v 1061 2475 V 1062 2474 V 1063 2474 V 1064 2473 V 1066 2472 V 1067 2472 V 1068 2471 V 1069 2470 V 1070 2470 V 1071 2469 V 1060 2477 3 3 v 1074 2468 V 1076 2466 5 1 v 1077 2466 V 1078 2465 V 1079 2464 V 1080 2464 V 1082 2463 V 1083 2462 V 1084 2462 V 1085 2461 V 1086 2460 V 1087 2460 V 1076 2468 3 3 v 1090 2459 V 1092 2457 5 1 v 1093 2456 V 1094 2456 V 1095 2455 V 1096 2454 V 1098 2454 V 1099 2453 V 1100 2452 V 1101 2452 V 1102 2451 V 1103 2450 V 1092 2459 3 3 v 1106 2449 V 1108 2448 4 1 v 1109 2447 V 1110 2446 V 1111 2446 V 1112 2445 V 1113 2444 V 1114 2444 V 1115 2443 V 1116 2442 V 1117 2442 V 1118 2441 V 1119 2440 V 1120 2440 V 1108 2449 2 3 v 1122 2439 V 1124 2437 5 1 v 1125 2436 V 1126 2436 V 1127 2435 V 1128 2434 V 1130 2434 V 1131 2433 V 1132 2432 V 1133 2432 V 1134 2431 V 1135 2430 V 1124 2439 3 3 v 1138 2429 V 1140 2428 5 1 v 1141 2427 V 1142 2426 V 1143 2426 V 1144 2425 V 1146 2424 V 1147 2424 V 1148 2423 V 1149 2422 V 1150 2422 V 1151 2421 V 1140 2429 3 3 v 1154 2420 V 1156 2418 4 1 v 1157 2418 V 1158 2417 V 1159 2416 V 1160 2416 V 1161 2415 V 1162 2414 V 1163 2414 V 1164 2413 V 1165 2412 V 1166 2412 V 1167 2411 V 1168 2410 V 1156 2420 2 3 v 1170 2409 V 1172 2408 5 1 v 1173 2407 V 1174 2406 V 1175 2406 V 1176 2405 V 1178 2404 V 1179 2404 V 1180 2403 V 1181 2402 V 1182 2402 V 1183 2401 V 1172 2409 3 3 v 1186 2400 V 1188 2398 5 1 v 1189 2398 V 1190 2397 V 1192 2396 V 1193 2396 V 1194 2395 V 1195 2394 V 1197 2394 V 1198 2393 V 1199 2392 V 1200 2392 V 1188 2400 3 3 v 1203 2391 V 1205 2389 5 1 v 1206 2388 V 1207 2388 V 1209 2387 V 1210 2386 V 1211 2386 V 1212 2385 V 1213 2384 V 1214 2384 V 1216 2383 V 1217 2382 V 1205 2391 3 3 v 1219 2381 V 1221 2380 4 1 v 1222 2379 V 1223 2378 V 1224 2378 V 1225 2377 V 1226 2376 V 1227 2376 V 1228 2375 V 1229 2374 V 1230 2374 V 1231 2373 V 1232 2372 V 1233 2372 V 1221 2381 2 3 v 1235 2371 V 1237 2369 5 1 v 1238 2368 V 1239 2368 V 1241 2367 V 1242 2366 V 1243 2366 V 1244 2365 V 1245 2364 V 1246 2364 V 1248 2363 V 1249 2362 V 1237 2371 3 3 v 1251 2361 V 1253 2360 5 1 v 1254 2359 V 1255 2358 V 1257 2358 V 1258 2357 V 1259 2356 V 1260 2356 V 1261 2355 V 1262 2354 V 1264 2354 V 1265 2353 V 1253 2361 3 3 v 1267 2352 V 1269 2350 4 1 v 1270 2350 V 1271 2349 V 1272 2348 V 1273 2348 V 1274 2347 V 1275 2346 V 1276 2346 V 1277 2345 V 1278 2344 V 1279 2344 V 1280 2343 V 1281 2342 V 1269 2352 2 3 v 1283 2341 V 1285 2340 5 1 v 1286 2339 V 1287 2338 V 1289 2338 V 1290 2337 V 1291 2336 V 1292 2336 V 1293 2335 V 1294 2334 V 1296 2334 V 1297 2333 V 1285 2341 3 3 v 1299 2332 V 1301 2330 5 1 v 1302 2330 V 1303 2329 V 1305 2328 V 1306 2328 V 1307 2327 V 1308 2326 V 1309 2326 V 1310 2325 V 1312 2324 V 1313 2324 V 1301 2332 3 3 v 1315 2323 V 1317 2321 5 1 v 1318 2320 V 1319 2320 V 1321 2319 V 1322 2318 V 1323 2318 V 1324 2317 V 1325 2316 V 1326 2316 V 1328 2315 V 1329 2314 V 1317 2323 3 3 v 1331 2313 V 1333 2312 4 1 v 1334 2311 V 1335 2310 V 1336 2310 V 1337 2309 V 1338 2308 V 1339 2308 V 1340 2307 V 1341 2306 V 1342 2306 V 1343 2305 V 1344 2304 V 1345 2304 V 1333 2313 2 3 v 1347 2303 V 1349 2301 5 1 v 1350 2300 V 1351 2300 V 1353 2299 V 1354 2298 V 1355 2298 V 1356 2297 V 1357 2296 V 1358 2296 V 1360 2295 V 1361 2294 V 1349 2303 3 3 v 1363 2293 V 1365 2292 5 1 v 1366 2291 V 1367 2290 V 1369 2290 V 1370 2289 V 1371 2288 V 1372 2288 V 1373 2287 V 1374 2286 V 1376 2286 V 1377 2285 V 1365 2293 3 3 v 1379 2284 V 1381 2282 4 1 v 1382 2282 V 1383 2281 V 1384 2280 V 1385 2280 V 1386 2279 V 1387 2278 V 1388 2278 V 1389 2277 V 1390 2276 V 1391 2276 V 1392 2275 V 1393 2274 V 1381 2284 2 3 v 1395 2273 V 1397 2272 5 1 v 1398 2271 V 1399 2270 V 1401 2270 V 1402 2269 V 1403 2268 V 1404 2268 V 1405 2267 V 1406 2266 V 1408 2266 V 1409 2265 V 1397 2273 3 3 v 1411 2264 V 1413 2262 5 1 v 1414 2262 V 1416 2261 V 1417 2260 V 1418 2260 V 1419 2259 V 1421 2258 V 1422 2258 V 1423 2257 V 1424 2256 V 1426 2256 V 1413 2264 3 3 v 1428 2255 V 1430 2253 5 1 v 1432 2252 V 1433 2252 V 1434 2251 V 1435 2250 V 1436 2250 V 1437 2249 V 1439 2248 V 1440 2248 V 1441 2247 V 1442 2246 V 1430 2255 3 3 v 1444 2245 V 1446 2244 4 1 v 1447 2243 V 1448 2242 V 1449 2242 V 1450 2241 V 1451 2240 V 1452 2240 V 1454 2239 V 1455 2238 V 1456 2238 V 1457 2237 V 1458 2236 V 1459 2236 V 1446 2245 2 3 v 1460 2235 V 1462 2233 5 1 v 1464 2232 V 1465 2232 V 1466 2231 V 1467 2230 V 1468 2230 V 1469 2229 V 1471 2228 V 1472 2228 V 1473 2227 V 1474 2226 V 1462 2235 3 3 v 1476 2225 V 1478 2224 5 1 v 1480 2223 V 1481 2222 V 1482 2222 V 1483 2221 V 1484 2220 V 1485 2220 V 1487 2219 V 1488 2218 V 1489 2218 V 1490 2217 V 1478 2225 3 3 v 1492 2216 V 1494 2214 4 1 v 1495 2214 V 1496 2213 V 1497 2212 V 1498 2212 V 1499 2211 V 1500 2210 V 1502 2210 V 1503 2209 V 1504 2208 V 1505 2208 V 1506 2207 V 1507 2206 V 1494 2216 2 3 v 1508 2205 V 1510 2204 5 1 v 1512 2203 V 1513 2202 V 1514 2202 V 1515 2201 V 1516 2200 V 1517 2200 V 1519 2199 V 1520 2198 V 1521 2198 V 1522 2197 V 1510 2205 3 3 v 1524 2196 V 1526 2194 5 1 v 1528 2194 V 1529 2193 V 1530 2192 V 1531 2192 V 1532 2191 V 1533 2190 V 1535 2190 V 1536 2189 V 1537 2188 V 1538 2188 V 1526 2196 3 3 v 1540 2187 V 1542 2185 5 1 v 1544 2184 V 1545 2184 V 1546 2183 V 1547 2182 V 1548 2182 V 1549 2181 V 1551 2180 V 1552 2180 V 1553 2179 V 1554 2178 V 1542 2187 3 3 v 1556 2177 V 1558 2176 4 1 v 1559 2175 V 1560 2174 V 1561 2174 V 1562 2173 V 1563 2172 V 1564 2172 V 1566 2171 V 1567 2170 V 1568 2170 V 1569 2169 V 1570 2168 V 1571 2168 V 1558 2177 2 3 v 1572 2167 V 1574 2165 5 1 v 1576 2164 V 1577 2164 V 1578 2163 V 1579 2162 V 1580 2162 V 1581 2161 V 1583 2160 V 1584 2160 V 1585 2159 V 1586 2158 V 1574 2167 3 3 v 1588 2157 V 1590 2156 5 1 v 1592 2155 V 1593 2154 V 1594 2154 V 1595 2153 V 1596 2152 V 1597 2152 V 1599 2151 V 1600 2150 V 1601 2150 V 1602 2149 V 1590 2157 3 3 v 1604 2148 V 1606 2146 4 1 v 1607 2146 V 1608 2145 V 1609 2144 V 1610 2144 V 1611 2143 V 1612 2142 V 1614 2142 V 1615 2141 V 1616 2140 V 1617 2140 V 1618 2139 V 1619 2138 V 1606 2148 2 3 v 1620 2137 V 1622 2136 5 1 v 1624 2135 V 1625 2134 V 1626 2134 V 1627 2133 V 1628 2132 V 1629 2132 V 1631 2131 V 1632 2130 V 1633 2130 V 1634 2129 V 1622 2137 3 3 v 1636 2128 V 1638 2126 5 1 v 1640 2126 V 1641 2125 V 1642 2124 V 1644 2124 V 1645 2123 V 1646 2122 V 1647 2122 V 1649 2121 V 1650 2120 V 1651 2120 V 1638 2128 3 3 v 1653 2119 V 1656 2117 5 1 v 1657 2116 V 1658 2116 V 1659 2115 V 1660 2114 V 1662 2114 V 1663 2113 V 1664 2112 V 1665 2112 V 1666 2111 V 1667 2110 V 1656 2119 3 3 v 1670 2109 V 1672 2108 4 1 v 1673 2107 V 1674 2106 V 1675 2106 V 1676 2105 V 1677 2104 V 1678 2104 V 1679 2103 V 1680 2102 V 1681 2102 V 1682 2101 V 1683 2100 V 1684 2100 V 1672 2109 2 3 v 1686 2099 V 1688 2097 5 1 v 1689 2096 V 1690 2096 V 1691 2095 V 1692 2094 V 1694 2094 V 1695 2093 V 1696 2092 V 1697 2092 V 1698 2091 V 1699 2090 V 1688 2099 3 3 v 1702 2089 V 1704 2088 5 1 v 1705 2087 V 1706 2086 V 1707 2086 V 1708 2085 V 1710 2084 V 1711 2084 V 1712 2083 V 1713 2082 V 1714 2082 V 1715 2081 V 1704 2089 3 3 v 1718 2080 V 1720 2078 4 1 v 1721 2078 V 1722 2077 V 1723 2076 V 1724 2076 V 1725 2075 V 1726 2074 V 1727 2074 V 1728 2073 V 1729 2072 V 1730 2072 V 1731 2071 V 1732 2070 V 1720 2080 2 3 v 1734 2069 V 1736 2068 5 1 v 1737 2067 V 1738 2066 V 1739 2066 V 1740 2065 V 1742 2064 V 1743 2064 V 1744 2063 V 1745 2062 V 1746 2062 V 1747 2061 V 1736 2069 3 3 v 1750 2060 V 1752 2058 5 1 v 1753 2058 V 1754 2057 V 1755 2056 V 1756 2056 V 1758 2055 V 1759 2054 V 1760 2054 V 1761 2053 V 1762 2052 V 1763 2052 V 1752 2060 3 3 v 1766 2051 V 1768 2049 5 1 v 1769 2048 V 1770 2048 V 1771 2047 V 1772 2046 V 1774 2046 V 1775 2045 V 1776 2044 V 1777 2044 V 1778 2043 V 1779 2042 V 1768 2051 3 3 v 1782 2041 V 1784 2040 4 1 v 1785 2039 V 1786 2038 V 1787 2038 V 1788 2037 V 1789 2036 V 1790 2036 V 1791 2035 V 1792 2034 V 1793 2034 V 1794 2033 V 1795 2032 V 1796 2032 V 1784 2041 2 3 v 1798 2031 V 1800 2029 5 1 v 1801 2028 V 1802 2028 V 1803 2027 V 1804 2026 V 1806 2026 V 1807 2025 V 1808 2024 V 1809 2024 V 1810 2023 V 1811 2022 V 1800 2031 3 3 v 1814 2021 V 1816 2020 5 1 v 1817 2019 V 1818 2018 V 1819 2018 V 1820 2017 V 1822 2016 V 1823 2016 V 1824 2015 V 1825 2014 V 1826 2014 V 1827 2013 V 1816 2021 3 3 v 1830 2012 V 1832 2010 5 1 v 1833 2010 V 1834 2009 V 1835 2008 V 1836 2008 V 1838 2007 V 1839 2006 V 1840 2006 V 1841 2005 V 1842 2004 V 1843 2004 V 1832 2012 3 3 v 1846 2003 V 1848 2001 4 1 v 1849 2000 V 1850 2000 V 1851 1999 V 1852 1998 V 1853 1998 V 1854 1997 V 1855 1996 V 1856 1996 V 1857 1995 V 1858 1994 V 1859 1994 V 1860 1993 V 1848 2003 2 3 v 1862 1992 V 1864 1990 5 1 v 1865 1990 V 1866 1989 V 1868 1988 V 1869 1988 V 1870 1987 V 1871 1986 V 1873 1986 V 1874 1985 V 1875 1984 V 1876 1984 V 1864 1992 3 3 v 1879 1983 V 1881 1981 5 1 v 1882 1980 V 1883 1980 V 1885 1979 V 1886 1978 V 1887 1978 V 1888 1977 V 1889 1976 V 1890 1976 V 1892 1975 V 1893 1974 V 1881 1983 3 3 v 1895 1973 V 1897 1972 4 1 v 1898 1971 V 1899 1970 V 1900 1970 V 1901 1969 V 1902 1968 V 1903 1968 V 1904 1967 V 1905 1966 V 1906 1966 V 1907 1965 V 1908 1964 V 1909 1964 V 1897 1973 2 3 v 1911 1963 V 1913 1961 5 1 v 1914 1960 V 1915 1960 V 1917 1959 V 1918 1958 V 1919 1958 V 1920 1957 V 1921 1956 V 1922 1956 V 1924 1955 V 1925 1954 V 1913 1963 3 3 v 1927 1953 V 1929 1952 5 1 v 1930 1951 V 1931 1950 V 1933 1950 V 1934 1949 V 1935 1948 V 1936 1948 V 1937 1947 V 1938 1946 V 1940 1946 V 1941 1945 V 1929 1953 3 3 v 1943 1944 V 1945 1942 5 1 v 1946 1942 V 1947 1941 V 1949 1940 V 1950 1940 V 1951 1939 V 1952 1938 V 1953 1938 V 1954 1937 V 1956 1936 V 1957 1936 V 1945 1944 3 3 v 1959 1935 V 1961 1933 4 1 v 1962 1932 V 1963 1932 V 1964 1931 V 1965 1930 V 1966 1930 V 1967 1929 V 1968 1928 V 1969 1928 V 1970 1927 V 1971 1926 V 1972 1926 V 1973 1925 V 1961 1935 2 3 v 1975 1924 V 1977 1922 5 1 v 1978 1922 V 1979 1921 V 1981 1920 V 1982 1920 V 1983 1919 V 1984 1918 V 1985 1918 V 1986 1917 V 1988 1916 V 1989 1916 V 1977 1924 3 3 v 1991 1915 V 1993 1913 5 1 v 1994 1912 V 1995 1912 V 1997 1911 V 1998 1910 V 1999 1910 V 2000 1909 V 2001 1908 V 2002 1908 V 2004 1907 V 2005 1906 V 1993 1915 3 3 v 2007 1905 V 2009 1903 4 1 v 2010 1903 V 2011 1902 V 2012 1902 V 2013 1901 V 2014 1900 V 2015 1900 V 2016 1899 V 2017 1898 V 2018 1898 V 2019 1897 V 2020 1896 V 2021 1896 V 2009 1905 2 3 v 2023 1895 V 2025 1893 5 1 v 2026 1892 V 2027 1892 V 2029 1891 V 2030 1890 V 2031 1890 V 2032 1889 V 2033 1888 V 2034 1888 V 2036 1887 V 2037 1886 V 2025 1895 3 3 v 2039 1885 V 2041 1883 5 1 v 2042 1883 V 2043 1882 V 2045 1882 V 2046 1881 V 2047 1880 V 2048 1880 V 2049 1879 V 2050 1878 V 2052 1878 V 2053 1877 V 2041 1885 3 3 v 2055 1876 V 2057 1874 5 1 v 2058 1874 V 2059 1873 V 2061 1872 V 2062 1872 V 2063 1871 V 2064 1870 V 2065 1870 V 2066 1869 V 2068 1868 V 2069 1868 V 2057 1876 3 3 v 2071 1867 V 2073 1865 4 1 v 2074 1864 V 2075 1864 V 2076 1863 V 2077 1862 V 2078 1862 V 2079 1861 V 2080 1860 V 2081 1860 V 2082 1859 V 2083 1858 V 2084 1858 V 2085 1857 V 2073 1867 2 3 v 2087 1856 V 2089 1854 5 1 v 2090 1854 V 2092 1853 V 2093 1852 V 2094 1852 V 2095 1851 V 2097 1850 V 2098 1850 V 2099 1849 V 2101 1848 V 2102 1848 V 2089 1856 3 3 v 2104 1847 V 2106 1845 5 1 v 2108 1844 V 2109 1844 V 2110 1843 V 2111 1842 V 2112 1842 V 2113 1841 V 2115 1840 V 2116 1840 V 2117 1839 V 2118 1838 V 2106 1847 3 3 v 2120 1837 V 2122 1835 4 1 v 2123 1835 V 2124 1834 V 2125 1834 V 2126 1833 V 2127 1832 V 2129 1832 V 2130 1831 V 2131 1830 V 2132 1830 V 2133 1829 V 2134 1828 V 2135 1828 V 2122 1837 2 3 v 2137 1827 V 2138 1825 5 1 v 2140 1824 V 2141 1824 V 2142 1823 V 2143 1822 V 2144 1822 V 2145 1821 V 2147 1820 V 2148 1820 V 2149 1819 V 2150 1818 V 2138 1827 3 3 v 2152 1817 V 2154 1815 5 1 v 2156 1815 V 2157 1814 V 2158 1814 V 2159 1813 V 2160 1812 V 2161 1812 V 2163 1811 V 2164 1810 V 2165 1810 V 2166 1809 V 2154 1817 3 3 v 2168 1808 V 2170 1806 5 1 v 2172 1806 V 2173 1805 V 2174 1804 V 2175 1804 V 2176 1803 V 2177 1802 V 2179 1802 V 2180 1801 V 2181 1800 V 2182 1800 V 2170 1808 3 3 v 2184 1799 V 2186 1797 4 1 v 2187 1796 V 2188 1796 V 2189 1795 V 2190 1794 V 2192 1794 V 2193 1793 V 2194 1792 V 2195 1792 V 2196 1791 V 2197 1790 V 2198 1790 V 2199 1789 V 2186 1799 2 3 v 2201 1788 V 2202 1786 5 1 v 2204 1786 V 2205 1785 V 2206 1784 V 2207 1784 V 2208 1783 V 2209 1782 V 2211 1782 V 2212 1781 V 2213 1780 V 2214 1780 V 2202 1788 3 3 v 2216 1779 V 970 1910 a(2:1)p 1065 1895 V 1093 1895 V 1120 1895 V 1148 1895 V 1176 1895 V 1203 1895 V 1209 1895 V 625 2740 V 625 2740 V 652 2732 V 678 2724 V 705 2716 V 731 2708 V 758 2700 V 784 2693 V 810 2684 V 837 2677 V 863 2668 V 890 2660 V 916 2652 V 943 2644 V 969 2637 V 996 2628 V 1022 2621 V 1049 2612 V 1075 2604 V 1102 2596 V 1128 2588 V 1155 2580 V 1181 2572 V 1208 2565 V 1234 2557 V 1261 2548 V 1287 2541 V 1313 2532 V 1340 2524 V 1366 2516 V 1393 2508 V 1420 2501 V 1446 2492 V 1472 2485 V 1499 2476 V 1525 2468 V 1552 2460 V 1578 2452 V 1605 2445 V 1631 2437 V 1658 2429 V 1684 2421 V 1711 2412 V 1737 2405 V 1764 2396 V 1790 2389 V 1817 2381 V 1843 2372 V 1870 2365 V 1896 2356 V 1923 2349 V 1949 2340 V 1975 2333 V 2002 2325 V 2028 2317 V 2055 2309 V 2081 2301 V 2108 2293 V 2134 2285 V 2161 2277 V 2187 2269 V 2214 2260 V 2218 2259 V 893 1971 a(LPTD)1115 1970 y Fa(3)1009 2597 y(3)821 2650 y(3)745 2689 y(3)726 2692 y(3)699 2709 y(3)743 2680 y(3)674 2716 y(3)666 2718 y(3)655 2730 y(3)663 2738 y(3)654 2733 y(3)658 2730 y(3)637 2738 y(3)631 2741 y(3)629 2740 y(3)619 2745 y(3)618 2748 y(3)617 2750 y(3)614 2749 y(3)609 2753 y(3)-44 b(3)606 2754 y(3)603 2756 y(3)896 2031 y Fb(H1TD)1116 2027 y Fs(+)2041 2225 y(+)1718 2337 y(+)1413 2417 y(+)1009 2571 y(+)821 2643 y(+)745 2673 y(+)726 2682 y(+)700 2701 y(+)744 2689 y(+)674 2711 y(+)-51 b(+)656 2722 y(+)664 2731 y(+)654 2722 y(+)-39 b(+)637 2733 y(+)632 2737 y(+)629 2735 y(+)620 2741 y(+)618 2743 y(+)617 2746 y(+)-46 b(+)609 2750 y(+)609 2749 y(+)606 2751 y(+)604 2753 y(+)896 2091 y Fb(H2TD)1116 2090 y Fa(2)2042 2288 y(2)1719 2364 y(2)1414 2441 y(2)1010 2581 y(2)822 2645 y(2)746 2678 y(2)727 2690 y(2)700 2708 y(2)744 2677 y(2)675 2716 y(2)667 2717 y(2)656 2728 y(2)664 2736 y(2)655 2733 y(2)659 2730 y(2)638 2736 y(2)632 2741 y(2)630 2740 y(2)620 2745 y(2)619 2748 y(2)618 2750 y(2)615 2749 y(2)610 2753 y(2)-41 b(2)607 2754 y(2)604 2756 y(2)892 2151 y Fb(GL)l(TD)1116 2147 y Fl(\002)1009 2538 y(\002)821 2622 y(\002)745 2677 y(\002)726 2673 y(\002)700 2689 y(\002)744 2675 y(\002)674 2706 y(\002)666 2709 y(\002)656 2721 y(\002)664 2722 y(\002)-53 b(\002)-39 b(\002)637 2727 y(\002)632 2731 y(\002)629 2733 y(\002)620 2741 y(\002)618 2742 y(\002)617 2745 y(\002)614 2743 y(\002)609 2749 y(\002)c(\002)606 2751 y(\002)604 2753 y(\002)862 2211 y Fb(ELZTD)1112 2207 y Fl(4)2038 2318 y(4)1715 2386 y(4)1410 2491 y(4)1006 2610 y(4)818 2647 y(4)742 2694 y(4)723 2689 y(4)696 2699 y(4)740 2695 y(4)671 2714 y(4)-57 b(4)652 2722 y(4)660 2729 y(4)f(4)655 2722 y(4)634 2734 y(4)628 2738 y(4)626 2737 y(4)616 2742 y(4)615 2745 y(4)614 2746 y(4)-52 b(4)606 2750 y(4)j(4)603 2751 y(4)600 2753 y(4)896 2271 y Fb(LZTD)1123 2267 y Fq(?)2050 2296 y(?)1726 2375 y(?)1421 2495 y(?)1018 2607 y(?)830 2643 y(?)753 2697 y(?)734 2684 y(?)707 2696 y(?)751 2693 y(?)682 2713 y(?)674 2712 y(?)663 2727 y(?)671 2732 y(?)662 2729 y(?)666 2720 y(?)645 2735 y(?)639 2737 y(?)637 2736 y(?)627 2741 y(?)626 2744 y(?)625 2745 y(?)-31 b(?)617 2749 y(?)j(?)614 2751 y(?)611 2752 y(?)921 3117 y Ft(Figure)22 b(6.)27 b Fs(Searc)n(h)17 b(e\013ectiv)n(eness)p eop %%Page: 41 43 41 42 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(41)p 566 1268 27 3 v 509 1284 a Fs(0)p 2192 1268 V 566 1140 V 482 1156 a(20)p 2192 1140 V 566 1012 V 482 1028 a(40)p 2192 1012 V 566 884 V 482 900 a(60)p 2192 884 V 566 754 V 482 771 a(80)p 2192 754 V 566 626 V 454 643 a(100)p 2192 626 V 566 498 V 454 515 a(120)p 2192 498 V 566 370 V 454 387 a(140)p 2192 370 V 566 1268 3 27 v 553 1344 a(0)p 566 333 V 897 1268 V 261 w(5000)p 897 333 V 1228 1268 V 205 w(10000)p 1228 333 V 1557 1268 V 189 w(15000)p 1557 333 V 1888 1268 V 191 w(20000)p 1888 333 V 2218 1268 V 190 w(25000)p 2218 333 V 566 1268 1652 3 v 2218 1268 3 962 v 566 306 1652 3 v 269 800 a(time)18 b(\()p Fq(t)p Fs(\))1299 1400 y(size)g(\()p Fq(S)s Fs(\))p 566 1268 3 962 v 596 573 3 240 v 596 333 584 3 v 1180 573 3 240 v 596 573 584 3 v 803 378 a Fb(ELZTD)80 b Fa(3)2035 614 y(3)1701 710 y(3)1383 835 y(3)966 1034 y(3)770 1097 y(3)691 1174 y(3)671 1158 y(3)645 1178 y(3)690 1195 y(3)618 1215 y(3)610 1202 y(3)598 1227 y(3)607 1243 y(3)-54 b(3)601 1229 y(3)579 1250 y(3)573 1257 y(3)570 1254 y(3)561 1265 y(3)561 1269 y(3)-47 b(3)555 1273 y(3)550 1283 y(3)550 1279 y(3)547 1282 y(3)545 1283 y(3)806 438 y Fb(ELZTB)1057 435 y Fs(+)588 344 y(+)574 1023 y(+)573 1107 y(+)576 1079 y(+)570 1107 y(+)572 819 y(+)580 1014 y(+)576 780 y(+)568 1154 y(+)570 1196 y(+)570 1244 y(+)565 1235 y(+)573 1182 y(+)573 1056 y(+)569 1247 y(+)565 1243 y(+)564 1242 y(+)570 886 y(+)569 1200 y(+)570 1040 y(+)572 1232 y(+)d(+)565 1234 y(+)566 1223 y(+)561 1246 y(+)561 1239 y(+)565 1254 y(+)565 1222 y(+)565 1262 y(+)562 1243 y(+)560 1246 y(+)561 1259 y(+)561 1260 y(+)558 1263 y(+)558 1267 y(+)558 1251 y(+)562 1262 y(+)558 1256 y(+)556 1264 y(+)556 1272 y(+)556 1271 y(+)556 1263 y(+)557 1275 y(+)556 1271 y(+)-45 b(+)553 1272 y(+)557 1267 y(+)552 1275 y(+)552 1278 y(+)g(+)550 1280 y(+)550 1276 y(+)550 1279 y(+)550 1276 y(+)549 1278 y(+)550 1279 y(+)550 1276 y(+)549 1280 y(+)550 1278 y(+)549 1280 y(+)i(+)g(+)548 1279 y(+)549 1280 y(+)d(+)j(+)f(+)h(+)811 498 y Fb(ELZLB)81 b Fa(2)2036 430 y(2)1702 697 y(2)1384 895 y(2)967 1051 y(2)771 1106 y(2)692 1189 y(2)672 1166 y(2)646 1173 y(2)691 1186 y(2)619 1211 y(2)611 1189 y(2)599 1221 y(2)608 1238 y(2)598 1219 y(2)-37 b(2)580 1235 y(2)574 1237 y(2)571 1231 y(2)562 1250 y(2)562 1243 y(2)559 1263 y(2)556 1258 y(2)551 1269 y(2)551 1267 y(2)548 1277 y(2)546 1283 y(2)803 558 y Fb(ELZGB)1057 555 y Fl(\002)2036 511 y(\002)1701 331 y(\002)966 848 y(\002)770 1172 y(\002)692 1196 y(\002)672 1216 y(\002)645 1215 y(\002)690 1218 y(\002)618 1235 y(\002)610 1246 y(\002)598 1247 y(\002)608 1243 y(\002)597 1254 y(\002)601 1243 y(\002)580 1260 y(\002)573 1259 y(\002)570 1263 y(\002)561 1266 y(\002)561 1268 y(\002)-46 b(\002)556 1274 y(\002)550 1280 y(\002)550 1278 y(\002)h(\002)545 1280 y(\002)969 1645 y Ft(Figure)22 b(7.)27 b Fs(Enco)r(ding)16 b(times)p 625 2751 27 3 v 568 2768 a(0)p 2192 2751 V 625 2559 V 485 2576 a(5000)p 2192 2559 V 625 2367 V 457 2384 a(10000)p 2192 2367 V 625 2173 V 457 2190 a(15000)p 2192 2173 V 625 1981 V 457 1998 a(20000)p 2192 1981 V 625 1789 V 457 1806 a(25000)p 2192 1789 V 625 2751 3 27 v 611 2828 a(0)p 625 1816 V 944 2751 V 249 w(5000)p 944 1816 V 1262 2751 V 193 w(10000)p 1262 1816 V 1581 2751 V 179 w(15000)p 1581 1816 V 1900 2751 V 179 w(20000)p 1900 1816 V 2218 2751 V 178 w(25000)p 2218 1816 V 625 2751 1594 3 v 2218 2751 3 962 v 625 1789 1594 3 v 291 2213 a(comp.)321 2278 y(size)258 2336 y(\()p Fq(S)f Fl(\000)d Fq(\016)r(S)s Fs(\))1329 2884 y(size)17 b(\()p Fq(S)s Fs(\))p 625 2751 3 962 v 654 2176 3 360 v 654 1816 584 3 v 1238 2176 3 360 v 654 2176 584 3 v 970 1860 a Fb(1:1)p 1065 1845 144 3 v 625 2751 3 3 v 625 2749 5 1 v 626 2748 V 627 2748 V 629 2747 V 630 2746 V 631 2746 V 632 2745 V 633 2744 V 634 2744 V 636 2743 V 637 2742 V 625 2751 3 3 v 639 2742 V 641 2740 4 1 v 642 2739 V 643 2738 V 644 2738 V 645 2737 V 646 2736 V 647 2736 V 648 2735 V 649 2734 V 650 2734 V 651 2733 V 652 2732 V 653 2732 V 641 2742 2 3 v 655 2731 V 657 2729 5 1 v 658 2728 V 659 2728 V 661 2727 V 662 2726 V 663 2726 V 664 2725 V 665 2724 V 666 2724 V 668 2723 V 669 2722 V 657 2731 3 3 v 671 2722 V 673 2720 5 1 v 674 2719 V 675 2718 V 677 2718 V 678 2717 V 679 2716 V 680 2716 V 681 2715 V 682 2714 V 684 2714 V 685 2713 V 673 2722 3 3 v 687 2712 V 689 2710 5 1 v 690 2710 V 691 2709 V 693 2708 V 694 2708 V 695 2707 V 696 2706 V 697 2706 V 698 2705 V 700 2704 V 701 2704 V 689 2712 3 3 v 703 2703 V 705 2701 4 1 v 706 2700 V 707 2700 V 708 2699 V 709 2698 V 710 2698 V 711 2697 V 712 2696 V 713 2696 V 714 2695 V 715 2694 V 716 2694 V 717 2693 V 705 2703 2 3 v 719 2692 V 721 2690 5 1 v 722 2690 V 723 2689 V 725 2688 V 726 2688 V 727 2687 V 728 2686 V 729 2686 V 730 2685 V 732 2684 V 733 2684 V 721 2692 3 3 v 735 2683 V 737 2681 5 1 v 738 2680 V 740 2680 V 741 2679 V 742 2678 V 743 2678 V 745 2677 V 746 2676 V 747 2676 V 748 2675 V 750 2674 V 737 2683 3 3 v 752 2674 V 754 2672 4 1 v 755 2671 V 756 2670 V 757 2670 V 758 2669 V 759 2668 V 760 2668 V 761 2667 V 762 2666 V 764 2666 V 765 2665 V 766 2664 V 767 2664 V 754 2674 2 3 v 768 2663 V 770 2661 5 1 v 772 2660 V 773 2660 V 774 2659 V 775 2658 V 776 2658 V 777 2657 V 779 2656 V 780 2656 V 781 2655 V 782 2654 V 770 2663 3 3 v 784 2654 V 786 2652 5 1 v 788 2651 V 789 2650 V 790 2650 V 791 2649 V 792 2648 V 793 2648 V 795 2647 V 796 2646 V 797 2646 V 798 2645 V 786 2654 3 3 v 800 2644 V 802 2642 5 1 v 804 2642 V 805 2641 V 806 2640 V 807 2640 V 808 2639 V 809 2638 V 811 2638 V 812 2637 V 813 2636 V 814 2636 V 802 2644 3 3 v 816 2635 V 818 2633 4 1 v 819 2632 V 820 2632 V 821 2631 V 822 2630 V 823 2630 V 824 2629 V 825 2628 V 827 2628 V 828 2627 V 829 2626 V 830 2626 V 831 2625 V 818 2635 2 3 v 832 2624 V 834 2622 5 1 v 836 2622 V 837 2621 V 838 2620 V 839 2620 V 840 2619 V 841 2618 V 843 2618 V 844 2617 V 845 2616 V 846 2616 V 834 2624 3 3 v 848 2615 V 850 2613 5 1 v 852 2612 V 853 2612 V 854 2611 V 855 2610 V 856 2610 V 857 2609 V 859 2608 V 860 2608 V 861 2607 V 862 2606 V 850 2615 3 3 v 864 2606 V 866 2604 4 1 v 867 2603 V 868 2602 V 869 2602 V 870 2601 V 871 2600 V 872 2600 V 873 2599 V 875 2598 V 876 2598 V 877 2597 V 878 2596 V 879 2596 V 866 2606 2 3 v 880 2595 V 882 2593 5 1 v 884 2592 V 885 2592 V 886 2591 V 887 2590 V 888 2590 V 889 2589 V 891 2588 V 892 2588 V 893 2587 V 894 2586 V 882 2595 3 3 v 896 2586 V 898 2584 5 1 v 900 2583 V 901 2582 V 902 2582 V 903 2581 V 904 2580 V 905 2580 V 907 2579 V 908 2578 V 909 2578 V 910 2577 V 898 2586 3 3 v 912 2576 V 914 2574 5 1 v 916 2574 V 917 2573 V 918 2572 V 919 2572 V 920 2571 V 921 2570 V 923 2570 V 924 2569 V 925 2568 V 926 2568 V 914 2576 3 3 v 928 2567 V 930 2565 4 1 v 931 2564 V 932 2564 V 933 2563 V 934 2562 V 935 2562 V 936 2561 V 937 2560 V 939 2560 V 940 2559 V 941 2558 V 942 2558 V 943 2557 V 930 2567 2 3 v 944 2556 V 946 2554 5 1 v 948 2554 V 949 2553 V 950 2552 V 951 2552 V 952 2551 V 953 2550 V 955 2550 V 956 2549 V 957 2548 V 958 2548 V 946 2556 3 3 v 960 2547 V 962 2545 5 1 v 964 2544 V 965 2544 V 966 2543 V 967 2542 V 969 2542 V 970 2541 V 971 2540 V 973 2540 V 974 2539 V 975 2538 V 962 2547 3 3 v 977 2538 V 980 2536 4 1 v 981 2535 V 982 2534 V 983 2534 V 984 2533 V 985 2532 V 986 2532 V 987 2531 V 988 2530 V 989 2530 V 990 2529 V 991 2528 V 992 2528 V 980 2538 2 3 v 994 2527 V 996 2525 5 1 v 997 2524 V 998 2524 V 999 2523 V 1000 2522 V 1002 2522 V 1003 2521 V 1004 2520 V 1005 2520 V 1006 2519 V 1007 2518 V 996 2527 3 3 v 1010 2518 V 1012 2516 5 1 v 1013 2515 V 1014 2514 V 1015 2514 V 1016 2513 V 1018 2512 V 1019 2512 V 1020 2511 V 1021 2510 V 1022 2510 V 1023 2509 V 1012 2518 3 3 v 1026 2508 V 1028 2506 5 1 v 1029 2506 V 1030 2505 V 1031 2504 V 1032 2504 V 1034 2503 V 1035 2502 V 1036 2502 V 1037 2501 V 1038 2500 V 1039 2500 V 1028 2508 3 3 v 1042 2499 V 1044 2497 4 1 v 1045 2496 V 1046 2496 V 1047 2495 V 1048 2494 V 1049 2494 V 1050 2493 V 1051 2492 V 1052 2492 V 1053 2491 V 1054 2490 V 1055 2490 V 1056 2489 V 1044 2499 2 3 v 1058 2488 V 1060 2486 5 1 v 1061 2486 V 1062 2485 V 1063 2484 V 1064 2484 V 1066 2483 V 1067 2482 V 1068 2482 V 1069 2481 V 1070 2480 V 1071 2480 V 1060 2488 3 3 v 1074 2479 V 1076 2477 5 1 v 1077 2476 V 1078 2476 V 1079 2475 V 1080 2474 V 1082 2474 V 1083 2473 V 1084 2472 V 1085 2472 V 1086 2471 V 1087 2470 V 1076 2479 3 3 v 1090 2470 V 1092 2468 5 1 v 1093 2467 V 1094 2466 V 1095 2466 V 1096 2465 V 1098 2464 V 1099 2464 V 1100 2463 V 1101 2462 V 1102 2462 V 1103 2461 V 1092 2470 3 3 v 1106 2460 V 1108 2458 4 1 v 1109 2458 V 1110 2457 V 1111 2456 V 1112 2456 V 1113 2455 V 1114 2454 V 1115 2454 V 1116 2453 V 1117 2452 V 1118 2452 V 1119 2451 V 1120 2450 V 1108 2460 2 3 v 1122 2450 V 1124 2448 5 1 v 1125 2447 V 1126 2446 V 1127 2446 V 1128 2445 V 1130 2444 V 1131 2444 V 1132 2443 V 1133 2442 V 1134 2442 V 1135 2441 V 1124 2450 3 3 v 1138 2440 V 1140 2438 5 1 v 1141 2438 V 1142 2437 V 1143 2436 V 1144 2436 V 1146 2435 V 1147 2434 V 1148 2434 V 1149 2433 V 1150 2432 V 1151 2432 V 1140 2440 3 3 v 1154 2431 V 1156 2429 4 1 v 1157 2428 V 1158 2428 V 1159 2427 V 1160 2426 V 1161 2426 V 1162 2425 V 1163 2424 V 1164 2424 V 1165 2423 V 1166 2422 V 1167 2422 V 1168 2421 V 1156 2431 2 3 v 1170 2420 V 1172 2418 5 1 v 1173 2418 V 1174 2417 V 1175 2416 V 1176 2416 V 1178 2415 V 1179 2414 V 1180 2414 V 1181 2413 V 1182 2412 V 1183 2412 V 1172 2420 3 3 v 1186 2411 V 1188 2409 5 1 v 1189 2408 V 1190 2408 V 1192 2407 V 1193 2406 V 1194 2406 V 1195 2405 V 1197 2404 V 1198 2404 V 1199 2403 V 1200 2402 V 1188 2411 3 3 v 1203 2402 V 1205 2400 5 1 v 1206 2399 V 1207 2398 V 1209 2398 V 1210 2397 V 1211 2396 V 1212 2396 V 1213 2395 V 1214 2394 V 1216 2394 V 1217 2393 V 1205 2402 3 3 v 1219 2392 V 1221 2390 4 1 v 1222 2390 V 1223 2389 V 1224 2388 V 1225 2388 V 1226 2387 V 1227 2386 V 1228 2386 V 1229 2385 V 1230 2384 V 1231 2384 V 1232 2383 V 1233 2382 V 1221 2392 2 3 v 1235 2381 V 1237 2380 5 1 v 1238 2379 V 1239 2378 V 1241 2378 V 1242 2377 V 1243 2376 V 1244 2376 V 1245 2375 V 1246 2374 V 1248 2374 V 1249 2373 V 1237 2381 3 3 v 1251 2372 V 1253 2370 5 1 v 1254 2370 V 1255 2369 V 1257 2368 V 1258 2368 V 1259 2367 V 1260 2366 V 1261 2366 V 1262 2365 V 1264 2364 V 1265 2364 V 1253 2372 3 3 v 1267 2363 V 1269 2361 4 1 v 1270 2360 V 1271 2360 V 1272 2359 V 1273 2358 V 1274 2358 V 1275 2357 V 1276 2356 V 1277 2356 V 1278 2355 V 1279 2354 V 1280 2354 V 1281 2353 V 1269 2363 2 3 v 1283 2352 V 1285 2350 5 1 v 1286 2350 V 1287 2349 V 1289 2348 V 1290 2348 V 1291 2347 V 1292 2346 V 1293 2346 V 1294 2345 V 1296 2344 V 1297 2344 V 1285 2352 3 3 v 1299 2343 V 1301 2341 5 1 v 1302 2340 V 1303 2340 V 1305 2339 V 1306 2338 V 1307 2338 V 1308 2337 V 1309 2336 V 1310 2336 V 1312 2335 V 1313 2334 V 1301 2343 3 3 v 1315 2333 V 1317 2332 5 1 v 1318 2331 V 1319 2330 V 1321 2330 V 1322 2329 V 1323 2328 V 1324 2328 V 1325 2327 V 1326 2326 V 1328 2326 V 1329 2325 V 1317 2333 3 3 v 1331 2324 V 1333 2322 4 1 v 1334 2322 V 1335 2321 V 1336 2320 V 1337 2320 V 1338 2319 V 1339 2318 V 1340 2318 V 1341 2317 V 1342 2316 V 1343 2316 V 1344 2315 V 1345 2314 V 1333 2324 2 3 v 1347 2313 V 1349 2312 5 1 v 1350 2311 V 1351 2310 V 1353 2310 V 1354 2309 V 1355 2308 V 1356 2308 V 1357 2307 V 1358 2306 V 1360 2306 V 1361 2305 V 1349 2313 3 3 v 1363 2304 V 1365 2302 5 1 v 1366 2302 V 1367 2301 V 1369 2300 V 1370 2300 V 1371 2299 V 1372 2298 V 1373 2298 V 1374 2297 V 1376 2296 V 1377 2296 V 1365 2304 3 3 v 1379 2295 V 1381 2293 4 1 v 1382 2292 V 1383 2292 V 1384 2291 V 1385 2290 V 1386 2290 V 1387 2289 V 1388 2288 V 1389 2288 V 1390 2287 V 1391 2286 V 1392 2286 V 1393 2285 V 1381 2295 2 3 v 1395 2284 V 1397 2282 5 1 v 1398 2282 V 1399 2281 V 1401 2280 V 1402 2280 V 1403 2279 V 1404 2278 V 1405 2278 V 1406 2277 V 1408 2276 V 1409 2276 V 1397 2284 3 3 v 1411 2275 V 1413 2273 5 1 v 1414 2272 V 1416 2272 V 1417 2271 V 1418 2270 V 1419 2270 V 1421 2269 V 1422 2268 V 1423 2268 V 1424 2267 V 1426 2266 V 1413 2275 3 3 v 1428 2265 V 1430 2264 5 1 v 1432 2263 V 1433 2262 V 1434 2262 V 1435 2261 V 1436 2260 V 1437 2260 V 1439 2259 V 1440 2258 V 1441 2258 V 1442 2257 V 1430 2265 3 3 v 1444 2256 V 1446 2254 4 1 v 1447 2254 V 1448 2253 V 1449 2252 V 1450 2252 V 1451 2251 V 1452 2250 V 1454 2250 V 1455 2249 V 1456 2248 V 1457 2248 V 1458 2247 V 1459 2246 V 1446 2256 2 3 v 1460 2245 V 1462 2244 5 1 v 1464 2243 V 1465 2242 V 1466 2242 V 1467 2241 V 1468 2240 V 1469 2240 V 1471 2239 V 1472 2238 V 1473 2238 V 1474 2237 V 1462 2245 3 3 v 1476 2236 V 1478 2234 5 1 v 1480 2234 V 1481 2233 V 1482 2232 V 1483 2232 V 1484 2231 V 1485 2230 V 1487 2230 V 1488 2229 V 1489 2228 V 1490 2228 V 1478 2236 3 3 v 1492 2227 V 1494 2225 4 1 v 1495 2224 V 1496 2224 V 1497 2223 V 1498 2222 V 1499 2222 V 1500 2221 V 1502 2220 V 1503 2220 V 1504 2219 V 1505 2218 V 1506 2218 V 1507 2217 V 1494 2227 2 3 v 1508 2216 V 1510 2214 5 1 v 1512 2214 V 1513 2213 V 1514 2212 V 1515 2212 V 1516 2211 V 1517 2210 V 1519 2210 V 1520 2209 V 1521 2208 V 1522 2208 V 1510 2216 3 3 v 1524 2207 V 1526 2205 5 1 v 1528 2204 V 1529 2204 V 1530 2203 V 1531 2202 V 1532 2202 V 1533 2201 V 1535 2200 V 1536 2200 V 1537 2199 V 1538 2198 V 1526 2207 3 3 v 1540 2197 V 1542 2196 5 1 v 1544 2195 V 1545 2194 V 1546 2194 V 1547 2193 V 1548 2192 V 1549 2192 V 1551 2191 V 1552 2190 V 1553 2190 V 1554 2189 V 1542 2197 3 3 v 1556 2188 V 1558 2186 4 1 v 1559 2186 V 1560 2185 V 1561 2184 V 1562 2184 V 1563 2183 V 1564 2182 V 1566 2182 V 1567 2181 V 1568 2180 V 1569 2180 V 1570 2179 V 1571 2178 V 1558 2188 2 3 v 1572 2177 V 1574 2176 5 1 v 1576 2175 V 1577 2174 V 1578 2174 V 1579 2173 V 1580 2172 V 1581 2172 V 1583 2171 V 1584 2170 V 1585 2170 V 1586 2169 V 1574 2177 3 3 v 1588 2168 V 1590 2166 5 1 v 1592 2166 V 1593 2165 V 1594 2164 V 1595 2164 V 1596 2163 V 1597 2162 V 1599 2162 V 1600 2161 V 1601 2160 V 1602 2160 V 1590 2168 3 3 v 1604 2159 V 1606 2157 4 1 v 1607 2156 V 1608 2156 V 1609 2155 V 1610 2154 V 1611 2154 V 1612 2153 V 1614 2152 V 1615 2152 V 1616 2151 V 1617 2150 V 1618 2150 V 1619 2149 V 1606 2159 2 3 v 1620 2148 V 1622 2146 5 1 v 1624 2146 V 1625 2145 V 1626 2144 V 1627 2144 V 1628 2143 V 1629 2142 V 1631 2142 V 1632 2141 V 1633 2140 V 1634 2140 V 1622 2148 3 3 v 1636 2139 V 1638 2137 5 1 v 1640 2136 V 1641 2136 V 1642 2135 V 1644 2134 V 1645 2134 V 1646 2133 V 1647 2132 V 1649 2132 V 1650 2131 V 1651 2130 V 1638 2139 3 3 v 1653 2129 V 1656 2128 5 1 v 1657 2127 V 1658 2126 V 1659 2126 V 1660 2125 V 1662 2124 V 1663 2124 V 1664 2123 V 1665 2122 V 1666 2122 V 1667 2121 V 1656 2129 3 3 v 1670 2120 V 1672 2118 4 1 v 1673 2118 V 1674 2117 V 1675 2116 V 1676 2116 V 1677 2115 V 1678 2114 V 1679 2114 V 1680 2113 V 1681 2112 V 1682 2112 V 1683 2111 V 1684 2110 V 1672 2120 2 3 v 1686 2109 V 1688 2108 5 1 v 1689 2107 V 1690 2106 V 1691 2106 V 1692 2105 V 1694 2104 V 1695 2104 V 1696 2103 V 1697 2102 V 1698 2102 V 1699 2101 V 1688 2109 3 3 v 1702 2100 V 1704 2098 5 1 v 1705 2098 V 1706 2097 V 1707 2096 V 1708 2096 V 1710 2095 V 1711 2094 V 1712 2094 V 1713 2093 V 1714 2092 V 1715 2092 V 1704 2100 3 3 v 1718 2091 V 1720 2089 4 1 v 1721 2088 V 1722 2088 V 1723 2087 V 1724 2086 V 1725 2086 V 1726 2085 V 1727 2084 V 1728 2084 V 1729 2083 V 1730 2082 V 1731 2082 V 1732 2081 V 1720 2091 2 3 v 1734 2080 V 1736 2078 5 1 v 1737 2078 V 1738 2077 V 1739 2076 V 1740 2076 V 1742 2075 V 1743 2074 V 1744 2074 V 1745 2073 V 1746 2072 V 1747 2072 V 1736 2080 3 3 v 1750 2071 V 1752 2069 5 1 v 1753 2068 V 1754 2068 V 1755 2067 V 1756 2066 V 1758 2066 V 1759 2065 V 1760 2064 V 1761 2064 V 1762 2063 V 1763 2062 V 1752 2071 3 3 v 1766 2061 V 1768 2060 5 1 v 1769 2059 V 1770 2058 V 1771 2058 V 1772 2057 V 1774 2056 V 1775 2056 V 1776 2055 V 1777 2054 V 1778 2054 V 1779 2053 V 1768 2061 3 3 v 1782 2052 V 1784 2050 4 1 v 1785 2050 V 1786 2049 V 1787 2048 V 1788 2048 V 1789 2047 V 1790 2046 V 1791 2046 V 1792 2045 V 1793 2044 V 1794 2044 V 1795 2043 V 1796 2042 V 1784 2052 2 3 v 1798 2041 V 1800 2040 5 1 v 1801 2039 V 1802 2038 V 1803 2038 V 1804 2037 V 1806 2036 V 1807 2036 V 1808 2035 V 1809 2034 V 1810 2034 V 1811 2033 V 1800 2041 3 3 v 1814 2032 V 1816 2030 5 1 v 1817 2030 V 1818 2029 V 1819 2028 V 1820 2028 V 1822 2027 V 1823 2026 V 1824 2026 V 1825 2025 V 1826 2024 V 1827 2024 V 1816 2032 3 3 v 1830 2023 V 1832 2021 5 1 v 1833 2020 V 1834 2020 V 1835 2019 V 1836 2018 V 1838 2018 V 1839 2017 V 1840 2016 V 1841 2016 V 1842 2015 V 1843 2014 V 1832 2023 3 3 v 1846 2013 V 1848 2012 4 1 v 1849 2011 V 1850 2010 V 1851 2010 V 1852 2009 V 1853 2008 V 1854 2008 V 1855 2007 V 1856 2006 V 1857 2006 V 1858 2005 V 1859 2004 V 1860 2004 V 1848 2013 2 3 v 1862 2003 V 1864 2001 5 1 v 1865 2000 V 1866 2000 V 1868 1999 V 1869 1998 V 1870 1998 V 1871 1997 V 1873 1996 V 1874 1996 V 1875 1995 V 1876 1994 V 1864 2003 3 3 v 1879 1993 V 1881 1992 5 1 v 1882 1991 V 1883 1990 V 1885 1990 V 1886 1989 V 1887 1988 V 1888 1988 V 1889 1987 V 1890 1986 V 1892 1986 V 1893 1985 V 1881 1993 3 3 v 1895 1984 V 1897 1982 4 1 v 1898 1982 V 1899 1981 V 1900 1980 V 1901 1980 V 1902 1979 V 1903 1978 V 1904 1978 V 1905 1977 V 1906 1976 V 1907 1976 V 1908 1975 V 1909 1974 V 1897 1984 2 3 v 1911 1973 V 1913 1972 5 1 v 1914 1971 V 1915 1970 V 1917 1970 V 1918 1969 V 1919 1968 V 1920 1968 V 1921 1967 V 1922 1966 V 1924 1966 V 1925 1965 V 1913 1973 3 3 v 1927 1964 V 1929 1962 5 1 v 1930 1962 V 1931 1961 V 1933 1960 V 1934 1960 V 1935 1959 V 1936 1958 V 1937 1958 V 1938 1957 V 1940 1956 V 1941 1956 V 1929 1964 3 3 v 1943 1955 V 1945 1953 5 1 v 1946 1952 V 1947 1952 V 1949 1951 V 1950 1950 V 1951 1950 V 1952 1949 V 1953 1948 V 1954 1948 V 1956 1947 V 1957 1946 V 1945 1955 3 3 v 1959 1945 V 1961 1944 4 1 v 1962 1943 V 1963 1942 V 1964 1942 V 1965 1941 V 1966 1940 V 1967 1940 V 1968 1939 V 1969 1938 V 1970 1938 V 1971 1937 V 1972 1936 V 1973 1936 V 1961 1945 2 3 v 1975 1935 V 1977 1933 5 1 v 1978 1932 V 1979 1932 V 1981 1931 V 1982 1930 V 1983 1930 V 1984 1929 V 1985 1928 V 1986 1928 V 1988 1927 V 1989 1926 V 1977 1935 3 3 v 1991 1925 V 1993 1924 5 1 v 1994 1923 V 1995 1922 V 1997 1922 V 1998 1921 V 1999 1920 V 2000 1920 V 2001 1919 V 2002 1918 V 2004 1918 V 2005 1917 V 1993 1925 3 3 v 2007 1916 V 2009 1914 4 1 v 2010 1914 V 2011 1913 V 2012 1912 V 2013 1912 V 2014 1911 V 2015 1910 V 2016 1910 V 2017 1909 V 2018 1908 V 2019 1908 V 2020 1907 V 2021 1906 V 2009 1916 2 3 v 2023 1905 V 2025 1904 5 1 v 2026 1903 V 2027 1902 V 2029 1902 V 2030 1901 V 2031 1900 V 2032 1900 V 2033 1899 V 2034 1898 V 2036 1898 V 2037 1897 V 2025 1905 3 3 v 2039 1896 V 2041 1894 5 1 v 2042 1894 V 2043 1893 V 2045 1892 V 2046 1892 V 2047 1891 V 2048 1890 V 2049 1890 V 2050 1889 V 2052 1888 V 2053 1888 V 2041 1896 3 3 v 2055 1887 V 2057 1885 5 1 v 2058 1884 V 2059 1884 V 2061 1883 V 2062 1882 V 2063 1882 V 2064 1881 V 2065 1880 V 2066 1880 V 2068 1879 V 2069 1878 V 2057 1887 3 3 v 2071 1877 V 2073 1876 4 1 v 2074 1875 V 2075 1874 V 2076 1874 V 2077 1873 V 2078 1872 V 2079 1872 V 2080 1871 V 2081 1870 V 2082 1870 V 2083 1869 V 2084 1868 V 2085 1868 V 2073 1877 2 3 v 2087 1867 V 2089 1865 5 1 v 2090 1864 V 2092 1864 V 2093 1863 V 2094 1862 V 2095 1862 V 2097 1861 V 2098 1860 V 2099 1860 V 2101 1859 V 2102 1858 V 2089 1867 3 3 v 2104 1857 V 2106 1856 5 1 v 2108 1855 V 2109 1854 V 2110 1854 V 2111 1853 V 2112 1852 V 2113 1852 V 2115 1851 V 2116 1850 V 2117 1850 V 2118 1849 V 2106 1857 3 3 v 2120 1848 V 2122 1846 4 1 v 2123 1846 V 2124 1845 V 2125 1844 V 2126 1844 V 2127 1843 V 2129 1842 V 2130 1842 V 2131 1841 V 2132 1840 V 2133 1840 V 2134 1839 V 2135 1838 V 2122 1848 2 3 v 2137 1837 V 2138 1836 5 1 v 2140 1835 V 2141 1834 V 2142 1834 V 2143 1833 V 2144 1832 V 2145 1832 V 2147 1831 V 2148 1830 V 2149 1830 V 2150 1829 V 2138 1837 3 3 v 2152 1828 V 2154 1826 5 1 v 2156 1826 V 2157 1825 V 2158 1824 V 2159 1824 V 2160 1823 V 2161 1822 V 2163 1822 V 2164 1821 V 2165 1820 V 2166 1820 V 2154 1828 3 3 v 2168 1819 V 2170 1817 5 1 v 2172 1816 V 2173 1816 V 2174 1815 V 2175 1814 V 2176 1814 V 2177 1813 V 2179 1812 V 2180 1812 V 2181 1811 V 2182 1810 V 2170 1819 3 3 v 2184 1809 V 2186 1808 4 1 v 2187 1807 V 2188 1806 V 2189 1806 V 2190 1805 V 2192 1804 V 2193 1804 V 2194 1803 V 2195 1802 V 2196 1802 V 2197 1801 V 2198 1800 V 2199 1800 V 2186 1809 2 3 v 2201 1799 V 2202 1797 5 1 v 2204 1796 V 2205 1796 V 2206 1795 V 2207 1794 V 2208 1794 V 2209 1793 V 2211 1792 V 2212 1792 V 2213 1791 V 2214 1790 V 2202 1799 3 3 v 2216 1789 V 970 1920 a(2:1)p 1065 1905 V 1093 1905 V 1120 1905 V 1148 1905 V 1176 1905 V 1203 1905 V 1209 1905 V 625 2751 V 625 2751 V 652 2743 V 678 2735 V 705 2727 V 731 2719 V 758 2711 V 784 2703 V 810 2695 V 837 2687 V 863 2679 V 890 2671 V 916 2662 V 943 2655 V 969 2647 V 996 2639 V 1022 2631 V 1049 2623 V 1075 2615 V 1102 2607 V 1128 2599 V 1155 2591 V 1181 2583 V 1208 2575 V 1234 2568 V 1261 2559 V 1287 2551 V 1313 2543 V 1340 2535 V 1366 2527 V 1393 2519 V 1420 2512 V 1446 2503 V 1472 2495 V 1499 2487 V 1525 2479 V 1552 2471 V 1578 2463 V 1605 2455 V 1631 2447 V 1658 2440 V 1684 2432 V 1711 2423 V 1737 2416 V 1764 2407 V 1790 2399 V 1817 2392 V 1843 2383 V 1870 2376 V 1896 2367 V 1923 2360 V 1949 2351 V 1975 2343 V 2002 2335 V 2028 2327 V 2055 2320 V 2081 2311 V 2108 2304 V 2134 2296 V 2161 2287 V 2187 2280 V 2214 2271 V 2218 2269 V 862 1981 a(ELZTD)79 b Fa(3)2041 2332 y(3)1718 2400 y(3)1413 2505 y(3)1009 2624 y(3)821 2661 y(3)745 2708 y(3)726 2702 y(3)699 2713 y(3)743 2709 y(3)674 2728 y(3)-52 b(3)655 2736 y(3)663 2742 y(3)f(3)658 2736 y(3)637 2748 y(3)631 2752 y(3)629 2750 y(3)619 2756 y(3)618 2758 y(3)617 2760 y(3)-47 b(3)609 2764 y(3)j(3)606 2765 y(3)603 2766 y(3)864 2041 y Fb(ELZTB)1116 2038 y Fs(+)644 2744 y(+)632 2746 y(+)h(+)-31 b(+)633 2745 y(+)629 2748 y(+)-43 b(+)637 2746 y(+)c(+)626 2749 y(+)629 2748 y(+)628 2752 y(+)624 2750 y(+)632 2749 y(+)i(+)626 2752 y(+)e(+)j(+)628 2750 y(+)h(+)629 2748 y(+)629 2749 y(+)624 2753 y(+)622 2750 y(+)624 2752 y(+)618 2754 y(+)620 2753 y(+)624 2752 y(+)g(+)624 2753 y(+)621 2754 y(+)d(+)618 2756 y(+)j(+)f(+)h(+)g(+)621 2753 y(+)617 2757 y(+)d(+)613 2758 y(+)614 2757 y(+)613 2758 y(+)616 2757 y(+)h(+)613 2758 y(+)f(+)616 2757 y(+)610 2760 y(+)j(+)f(+)609 2761 y(+)h(+)609 2760 y(+)g(+)608 2761 y(+)g(+)h(+)e(+)h(+)g(+)g(+)606 2762 y(+)g(+)608 2761 y(+)605 2762 y(+)605 2764 y(+)f(+)h(+)869 2101 y Fb(ELZLB)81 b Fa(2)2042 2216 y(2)1719 2317 y(2)1414 2489 y(2)1010 2610 y(2)822 2658 y(2)746 2708 y(2)727 2700 y(2)700 2710 y(2)744 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b(expressed)f(in)j(LF)g(or)f(LF)1086 2384 y Fp(i)1123 2376 y Fs(are)f(one)h(example)g(in)h(whic)n(h)g (optimization)e(migh)n(t)h(b)r(e)304 2442 y(p)r(ossible.)k(In)16 b(this)f(case,)g(the)f(meaning)h(whic)n(h)g(m)n(ust)h(b)r(e)e(preserv)n (ed)g(is)h(the)g(t)n(yp)r(e)f(of)i(the)e(pro)r(of)304 2508 y(terms)k(rather)f(than)h(the)h(normal)f(form.)28 b(A)19 b(LF)g(pro)r(of)e(optimization)h(algorithm)h(could)f(use)304 2575 y Fq(\025)p Fs(-abstraction)d(to)i(abstract)d(out)j(rep)r(eated)d (patterns,)h(reducing)h(pro)r(of)g(size.)24 b(The)17 b(resulting)304 2641 y(LF)d(terms)g(could)f(still)j(b)r(e)d(c)n(hec)n (k)m(able)h(as)g(pro)r(ofs)f(with)h(the)g(same)f(t)n(yp)r(es)h(as)g (the)g(original)g(pro)r(of)304 2708 y(terms,)k(but)f(require)h(less)g (storage)e(space)h(or)h(c)n(hec)n(king)g(time.)25 b(W)-5 b(e)19 b(shall)f(see)g(in)h(this)f(section)304 2774 y(ho)n(w)h(suc)n(h) f(optimizations)f(can)h(b)r(e)g(made)g(and)h(ho)n(w)f(term)g (compression)f(migh)n(t)i(b)r(e)f(used)g(to)304 2841 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Fs(,)16 b(where)e Fq(P)1368 3324 y Fm(0)1398 3344 y Fs(is)i(iden)n(tical)f(to)f Fq(P)23 b Fs(except)13 b(that)h(ev)n(ery)h(o)r(c-)304 3411 y(currence)h(of)23 b Ff(and)p 684 3411 V 21 w(el)d Fs(has)e(b)r(een)g(replaced)f(b)n(y)j Fd(x)10 b Fs(.)27 b(Since)18 b(the)g(t)n(w)n(o)i(terms)e(are)g Fq(\014)s Fs(-equiv)m(alen)n(t,)304 3477 y(they)f(ha)n(v)n(e)i(the)e (same)h(t)n(yp)r(e,)g(and)f(so)h(the)g(pro)r(ofs)e(they)i(represen)n(t) e(are)h(equiv)m(alen)n(t.)p eop %%Page: 44 46 44 45 bop 304 151 a Fu(44)230 b(JAMES)18 b(R.)h(CHENEY)272 b(CARNEGIE)18 b(MELLON)i(UNIVERSITY)370 283 y Fs(Moreo)n(v)n(er,)c(w)n (e)h(can)f Fq(\025)p Fs(-abstract)e(o)n(v)n(er)j Fr(arbitr)m(ary)24 b Fs(pro)r(of)15 b(rules,)i(to)f(obtain)f(functions)g(from)304 350 y(arbitrary)20 b(\(ev)n(en)g(in)n(v)m(alid\))j(pro)r(of)c(rules)i (to)g(pro)r(ofs.)33 b(W)-5 b(e)21 b(could)g(nev)n(er)g(misuse)g(this)h (abilit)n(y)304 416 y(to)d(pro)r(duce)f(represen)n(tations)f(of)j 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b(general,)g(if)h(a)e(pro)r (of)g(con)n(tains)f(man)n(y)j(rep)r(eated)c(patterns)g(of)j(pro)r(of)e (rules,)j(w)n(e)f(can)304 948 y(p)r(erform)16 b(the)i Fq(\025)p Fs(-abstraction)e(pro)r(cedure)f(outlined)j(ab)r(o)n(v)n(e)g (to)f(pro)n(vide)h(a)h(represen)n(tation)c(of)304 1014 y(the)22 b(pro)r(of)f(whic)n(h)i(is)g(less)g(complex.)37 b(Eac)n(h)23 b(rep)r(etition)d(of)j(the)e(pattern)g(is)i(replaced)e(b)n (y)j(a)304 1080 y(single)g(v)m(ariable)g(sym)n(b)r(ol,)j(and)d(the)g (en)n(tire)g(pattern)e(only)j(needs)e(to)h(app)r(ear)f(once,)i(when)304 1147 y(the)17 b(parameterized)e(pro)r(of)i(is)h(applied)g(to)g(it,)g (so)g(if)h(there)e(are)g(man)n(y)i(rep)r(etitions)c(of)j(a)g(large)304 1213 y(pattern,)24 b(the)g(new)h(equiv)m(alen)n(t)f(pro)r(of)f(will)j (b)r(e)e(smaller.)45 b(When)24 b(this)g(transformation)e(is)304 1280 y(p)r(erformed,)d(w)n(e)h(\014rst)g(replace)f(o)r(ccurrences)d(of) 21 b(patterns)d(in)j(a)f(term)g(with)g(v)m(ariables,)h(then)304 1346 y Fq(\025)p Fs(-abstract)h(o)n(v)n(er)i Fq(x)h Fs(:)h Fq(\034)6 b Fs(,)27 b(where)22 b Fq(\034)31 b Fs(is)25 b(the)e(t)n(yp)r(e)g(represen)n(ting)f(the)h(new)h(pro)r(of)f(rule,)i (and)304 1413 y(\014nally)19 b(apply)h(the)e(resulting)g(function)g(to) g(a)i(deriv)m(ation)e(of)h(the)f(pro)r(of)g(rule)h(\(that)e(is,)j(a)g (LF)304 1479 y(term)d(of)h(t)n(yp)r(e)g Fq(\034)6 b Fs(.\))370 1545 y(As)22 b(in)h(pro)r(of)d(compression,)i(it)h(w)n(ould)f(b)r(e)g (useful)f(to)h(ha)n(v)n(e)h(heuristic)e(estimates)f(of)i(the)304 1612 y(v)m(alue)g(of)h(a)g(pattern,)f(or)g(the)g(impro)n(v)n(emen)n(t)h (in)g(pro)r(of)f(size)g(resulting)g(from)g(using)h(it.)38 b(T)-5 b(o)304 1678 y(de\014ne)17 b(one)g(suc)n(h)h(heuristic,)g(w)n(e) h(will)g(mak)n(e)g(a)f(reasonable)f(assumption)f(that)i(v)m(ariables)f (are)304 1745 y(assigned)e(a)i(\014xed)g(cost)e(\(sa)n(y)-5 b(,)18 b(1\).)24 b(If)17 b(the)f(n)n(um)n(b)r(er)g(of)h(sym)n(b)r(ols)g (a)f(pattern)f(replaces)g(is)j Fq(S)s Fs(,)f(the)304 1811 y(size)f(of)g(its)g(t)n(yp)r(e)g(is)g Fq(T)8 b Fs(,)18 b(the)d(size)h(of)g(its)g(deriv)m(ation)f(is)i Fq(D)r Fs(,)f(and)g(the)g(frequency)e(of)i(the)g(pattern)304 1877 y(in)24 b(the)e(pro)r(of)g(is)i Fq(F)8 b Fs(,)25 b(then)d(the)h(impro)n(v)n(emen)n(t)g(in)h(pro)r(of)e(measure)g(giv)n (en)i(b)n(y)g(replacing)e Fq(F)304 1944 y Fs(o)r(ccurrences)14 b(of)k(the)f(pattern)f(is)1050 2044 y Fq(F)8 b Fs(\()p Fq(S)14 b Fl(\000)f Fs(1\))e Fl(\000)h Fs(\()p Fq(D)i Fs(+)e Fq(T)c Fs(\))p Fq(:)-1246 b Fs(\(2\))304 2144 y(W)-5 b(e)13 b(sa)n(v)n(e)g Fq(S)5 b Fl(\000)r Fs(1)14 b(sym)n(b)r(ols)f(for)f(eac)n(h)h(o)r(ccurrence)c(b)r(ecause)i(w)n(e)i (remo)n(v)n(e)g Fq(S)j Fs(sym)n(b)r(ols)e(and)e(replace)304 2211 y(them)20 b(with)g(one)g(v)m(ariable)h(sym)n(b)r(ol,)h(with)e (cost)g(1.)32 b(W)-5 b(e)21 b(will)h(alw)n(a)n(ys)g(b)r(e)e(able)g(to)g (construct)304 2277 y(the)c(deriv)m(ation)g(from)g(the)g(pattern)f(b)n (y)j Fq(\025)p Fs(-abstracting)d(o)n(v)n(er)i(the)f(free)g(pattern)f(v) 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y(giv)n(en)k(here)g(are)f(not)h(v)n(ery)g(e\016cien)n(t,)i(and)e (further)f(study)g(is)i(needed)e(to)g(\014nd)h(impro)n(v)n(ed)304 1080 y(v)n(ersions)18 b(or)f(other,)g(b)r(etter)f(approac)n(hes.)988 1205 y(3.)31 b Ft(A)n(ckno)o(wledgements)370 1305 y Fs(The)25 b(author)f(wishes)i(to)f(thank)f(P)n(eter)h(Lee,)i(George)d(Necula,)j (F)-5 b(rank)26 b(Pfenning,)g(and)304 1371 y(P)n(eter)19 b(Andrews)g(for)h(their)g(helpful)g(commen)n(ts)g(and)g(advice)g(in)h (conducting)d(this)i(researc)n(h)304 1437 y(and)d(writing)h(this)g (thesis.)1139 1562 y Ft(References)327 1651 y Fx([1])23 b(John)15 b(Co)q(c)o(k)o(e.)e(Global)h(common)g(sub)q(expression)f (elimination.)i Fw(SIGPLAN)h(Notic)n(es)p Fx(,)e(5\(7\):20{24,)e(July) 400 1706 y(1970.)i Fw(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)i(a)g(Symp)n (osium)g(on)g(Compiler)g(Optimization)p Fx(.)327 1761 y([2])23 b(Arth)o(ur)f(L.)f(Delc)o(her)g(and)f(Simon)h(Kasif.)e (E\016cien)o(t)i(parallel)f(term)h(matc)o(hing)g(and)f(an)o (ti-uni\014cation.)400 1817 y Fw(Journal)d(of)g(A)o(utomate)n(d)g(R)n (e)n(asoning)p Fx(,)d(9\(3\):391{406,)e(Decem)o(b)q(er)k(1992.)327 1872 y([3])23 b(N.)14 b(Dersho)o(witz)e(and)h(J.)g(P)l(.)g(Jouannaud.)f (Rewrite)h(systems.)f(In)h(J.)g(v)m(an)g(Leeu)o(w)o(en,)g(editor,)h Fw(Handb)n(o)n(ok)g(of)400 1927 y(The)n(or)n(etic)n(al)19 b(Computer)g(Scienc)n(e)e(\(V)l(ol.)k(B:)e(F)l(ormal)i(Mo)n(dels)e(and) h(Semantics\))p Fx(,)f(Amsterdam,)g(1990.)400 1983 y(North-Holland.)d (T)l(o)f(app)q(ear.)327 2038 y([4])23 b(Nac)o(h)o(um)13 b(Dersho)o(witz.)e(A)i(taste)f(of)h(rewriting.)g(T)l(ec)o(hnical)f(rep) q(ort)h(93-5,)f(Leibnitz)h(Cen)o(ter)g(for)g(Researc)o(h)400 2093 y(in)j(Computer)h(Science,)e(Hebrew)h(Univ)o(ersit)o(y)l(,)g (Jerusalem,)f(Israel,)h(1993.)327 2149 y([5])23 b(Nac)o(h)o(um)17 b(Dersho)o(witz)f(and)h(Shm)o(uel)g(Zaks.)f(P)o(atterns)g(in)h(trees.)g Fw(Ninth)h(Col)r(lo)n(quium)g(on)g(T)l(r)n(e)n(es)g(in)f(A)o(l-)400 2204 y(gebr)n(a)g(and)g(Pr)n(o)n(gr)n(amming)p Fx(,)d(pages)h(93{102,)e 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Fx(,)f(42,)g(1995.)327 2702 y([9])23 b(M.)e(F)l(ranz.)f(Adaptiv)o(e)g (compression)f(of)h(syn)o(tax)f(trees)h(and)g(iterativ)o(e)g(dynamic)g (co)q(de)f(optimization:)400 2758 y(Tw)o(o)26 b(basic)f(tec)o (hnologies)f(for)i(mobile)g(ob)s(ject)f(systems.)g Fw(L)n(e)n(ctur)n(e) f(Notes)j(in)f(Computer)g(Scienc)n(e)p Fx(,)400 2813 y(1222:263{??,)13 b(1997.)304 2868 y([10])22 b(F)l(erenc)16 b(G)o(\023)-22 b(ecseg)13 b(and)j(Magn)o(us)e(Stein)o(b)o(y)l(.)i Fw(T)l(r)n(e)n(e)g(A)o(utomata)p Fx(.)g(Ak)m(ad)o(\023)-22 b(emiai)15 b(Kiad\023)-24 b(o,)14 b(Budap)q(est,)i(1984.)304 2924 y([11])22 b(P)o(eter)g(Graf.)f(Substitution)i(tree)e(indexing.)g (In)h(Jieh)f(Hsiang,)h(editor,)h Fw(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)i (the)h(6th)f(In-)400 2979 y(ternational)17 b(Confer)n(enc)n(e)e(on)i(R) n(ewriting)d(T)l(e)n(chniques)h(and)j(Applic)n(ations)f(\(R)l(T)l (A-95\))p Fx(,)g(v)o(olume)e(914)f(of)400 3034 y Fw(LNCS)p Fx(,)i(pages)e(117{131,)f(Kaiserslautern,)h(German)o(y)l(,)i(April)g 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(basis)h(for)g(computer)h(programming.)f Fw(Communic)n(ations)g(of)h (the)400 3477 y(A)o(CM)p Fx(,)16 b(26\(1\):53{56,)c(Jan)o(uary)j(1983.) p eop %%Page: 49 51 49 50 bop 895 151 a Fu(FIRST-ORDER)20 b(TERM)e(COMPRESSION)547 b(49)304 283 y Fx([16])22 b(Da)o(vid)10 b(A.)i(Hu\013man.)f(A)g(metho)q (d)h(for)f(the)g(construction)g(of)g(minim)o(um-redundancy)g(co)q(des.) f Fw(Pr)n(o)n(c)n(e)n(e)n(dings)400 339 y(of)17 b(the)g(IRE)p Fx(,)f(40\(9\):1098{1101)o(,)c(Septem)o(b)q(er)k(1952.)304 394 y([17])22 b(G.)f(M.)g(Kup)q(er,)i(K.)e(W.)g(McAlo)q(on,)h(K.)f(V.)g (P)o(alem,)g(and)g(K.)g(J.)g(P)o(erry)l(.)g(An)g(e\016cien)o(t)g (algorithm)g(for)400 450 y(an)o(ti-uni\014cation.)16 b(T)l(ec)o(hnical)f(Rep)q(ort)g(R)o(C)g(14065,)f(IBM)i(Y)l(orkto)o(wn)f (Heigh)o(ts,)g(F)l(ebruary)h(1988.)304 505 y([18])22 b(George)15 b(C.)g(Necula.)g(Pro)q(of-carrying)g(co)q(de.)g(In)g Fw(Confer)n(enc)n(e)g(R)n(e)n(c)n(or)n(d)g(of)h(POPL)h('97:)22 b(The)16 b(24th)i(A)o(CM)400 560 y(SIGPLAN-SIGA)o(CT)i(Symp)n(osium)f (on)h(Principles)g(of)f(Pr)n(o)n(gr)n(amming)f(L)n(anguages)p Fx(,)h(pages)f(106{119,)400 616 y(P)o(aris,)d(F)l(rance,)h(15{17)d(Jan) o(uary)i(1997.)304 671 y([19])22 b(George)e(C.)h(Necula)g(and)f(P)o (eter)h(Lee.)g(E\016cien)o(t)g(represen)o(tation)g(and)f(v)m(alidation) g(of)h(logical)e(pro)q(ofs.)400 726 y(T)l(ec)o(hnical)d(Rep)q(ort)f (CS-TR)g(97-172,)e(Carnegie)j(Mellon)f(Univ)o(ersit)o(y)l(,)g(1997.)304 782 y([20])22 b(M.)c(S.)f(P)o(aterson)f(and)h(M.)g(N.)h(W)l(egman.)f (Linear)g(uni\014cation.)g(In)h Fw(Confer)n(enc)n(e)e(R)n(e)n(c)n(or)n (d)h(of)h(the)g(Eighth)400 837 y(A)o(nnual)g(A)o(CM)h(Symp)n(osium)e (on)i(The)n(ory)g(of)f(Computing)p Fx(,)f(pages)f(181{186,)f(Hershey)l (,)i(P)o(ennsylv)m(ania,)400 892 y(3{5)e(Ma)o(y)g(1976.)304 948 y([21])22 b(F)l(rank)16 b(Pfenning.)h(Elf:)22 b(A)17 b(language)d(for)j(logic)e(de\014nition)i(and)f(v)o(eri\014ed)g (metaprogramming.)g(In)g Fw(Pr)n(o-)400 1003 y(c)n(e)n(e)n(dings,)d(F)l 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