%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: confluence.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMTI10 CMMI10 CMSY10 CMBX12 %%+ CMSY8 CMSL10 CMR8 CMMI8 MSAM10 CMR6 CMMI6 CMSS10 CMEX10 CMSY6 CMTI8 %%+ CMTI7 CMR9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -Pamz -Pcmz confluence.dvi -o confluence.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.04.09:1608 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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/IEn 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 IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A 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/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A mul exch 0 get A 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/IEn 256 array N 2 string 0 1 255{IEn S A 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A 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 Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/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 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI7 %!PS-AdobeFont-1.1: CMTI7 1.0 %%CreationDate: 1991 Aug 18 21:07:18 % 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 (CMTI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI7 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 97 /a put dup 98 /b put readonly def /FontBBox{-27 -250 1268 750}readonly def /UniqueXX 5000825 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd80219794832af151fe9d5386d1ed91db54d1f 56dbec55d7c48ed295977e18f23fda49f00cbf6b37dbcbd34dd98770e780450f ab0bb54cb467b1038527da84f47b0cdff9561b343383c7cd96f818e10597fb39 4e91fc05dacdb7b418c8e6362ca4e803b563f227624bc4d8e0e1f76925affaeb 710c3b8220db48814ac08cc6a4d735d85316555bd4ab77a1098370dcb2bd272e fce72e2125b0a783a9b1fe202217eae8eb51e4ee2b9735b997af5b43174c58b4 07c1c7a4b7528d6ddbc7091f1fb8d4a731d78ec08e4c996301846cbeb5673597 69578272b49423899ffa88ed9e3fb5511c3e471150a3e257ba5656d21b8ed57c 4d1f6433eb9e04012c52e0fad62312eb2a88dbff69cde9d39d372dd7934fde7f 906a9c4828c1762dd4e1ae410026c58b56db7df324dad8fc3696323900480ff9 09c9f4b9acbf7d983b9429b25d317613beae8ea83b7626fa7b010aeda862d9d0 9b45c73d841e6d5e36bc8db0151d6dd635442552fa4d942210b738d3b6258846 28e541a7abb75c347ec149ecc27a148cea9cdf7f1e6842ec26e2b9fdac2bd6ae a7bb0a594e0298b7003b2b858a51d0da0ecdb9172942c16eec62c8b4352d64ac ee149e55cd6a72b0e80c67782b42c244b955a3885f87c6b7fa8d83a611e2e10d 5a6fce63f0a6cbeac3684f01f7cb60166a924849aa33626d97a1f7c7d323388b 666e15a2b4a21ee5542aef982c7c88902127b872f9405c9f3ddee6f09361efbb 93e28d89714b02c9263ff96cd423a693ae2834a8f26e926aaab01f70ad759640 dc6d18b0867d828a3c9a3a8758c2b3408100b416b920aee2155e70bcf5c76920 2d54ad46f5af53045e6040862d2cd51f0571f3d440aeb71635a5d3d55570cc88 acde19a70fd94d075e2c186f98d3d9b727268c9bd8ea91b02835c2588c9a8e0b ff41591c7983b09daf2592135992e5292c8ec310c77a16ee5ac90537233216ad 0518323e7d786581a70f987e78a9af042382b674c3727de9151c42a3a9cc3e92 f37e02603f1da72a055de72acafd8e0b0e62391cad5b9a384ef0db1dca3e68b9 a87abccba76bf48cde3ff1da732a7fba92a3db2f1c8a2203f4ff09d043e656e8 fab966fbca7e3eda29b40da77e6b24fd5652bb34ac5197417f68765f8f7e86c2 b639f879bbbb0fefa87b8a295ad8beefd3ffe0432436dc21cd8d72ff42c9877e af259668a7d38ba3d4ca27f96ce57880426ec678ee808e8c50c16b3634b2f361 ec01c509b784510c5f1e2b478499967ecc7dd3e79670287a4859c6778efa8a7e 94a17578c2d5fdb1b1284b9e3175de955857dccb50c6e84853ef6daf16084cc2 e2ae2a3fdbc3ebdb219eda42f5abc1ae10b8b728e34b33cd2062628328f74a2d b6cc3f536341bb709b0d40fa3b80bbc905f04223fd1a084468129d175d2d9020 8e09c350bc048863c1ef76d7bbee743effc978901ef29cfe3e373f5a64a9e3be cc17bd594fadea921e328254f26bdf14846e307919fe6e4d0f9e242044638fc0 40de74e89323b22deb39751156e4938c62b6ab44cfd07342619f614db9f162e5 a8c7a0b36b2311db9d16cd7f4b6e2552064b6432709a45c24c46cc7685c007db d5cb335f55386c26314ffb4687 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 97 /a put dup 98 /b put readonly def /FontBBox{-35 -250 1190 750}readonly def /UniqueXX 5000826 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36f6372968dbd0df8505aef5b22e57f2b9834fef2c2798f9d715aa323 3e09b183f8d92797bbd3a990980982018887612cdc5507a47581f07a1cf99a5c 587021d6b67b0524e780517e7a088a6388097ced01e5a59aa8779c6ad82af704 f6283c0a8b62d57cc5d24220e4d52a7c736389ff9d77df48c0de3d3db6482774 72a5e726bf4d83d29751b70eca9ee9d04a18fb7d011542a2b2737eff052a4eff 2800d8dbc1aeeb7273d9172927c2912b597c90faf4eb15f30201e026c5d06c94 122d69e4c81882d9f10b74f81d8a85a329a5946464c9765706a4e979e3b76913 83ea8ff66e4608e5c6fa0171e0c14bba3efb8fbcd0285e6ca408671bf068e1b9 f1c040e0417a92557f6b43a2b49c772c5d6e1e6fe0315378d71f1b762c987f29 6ada445624744ef29f727436345875bab3b78eff3a86027026983eae46ee5cb6 baefe7a95b00007094774550ad81f1b2c7e3c042a77548631e4baeacfb244b62 80922062c83c0f96d439448951a8f4e8bdf68d60245c60c8a8cf3e340644ff00 efbbcbd9abfe6018af21b3fe10b1381325d2599e22b2c5ad47bd6d5ecf1f55fc 7b382f2bf4889af9fb6beb58dc03087e9de3624e2e171c669acc08ff9a88f879 48749ad40d3c781c14e04ad93b147a89b1f29c82f756d9de7c26ded896aa4918 46e3376cb7cad2c19cd70be96eb4cc3509e812fe2aac63a59c3f5016fe6380ea fa2042e6b070a6a0e9ea6d4fdedbd9147fcc6eedee7b48bcfb29d09c214fd2d1 6297d9220ae3467eeecbaa9efd677e0964f8b737d0db9a5fafc7ecc4ff00bb8c c06359af9c0db731419186f985c3ab3f726a465d094c89610c370b44b90a79d1 8a81b4f83cea9720531ea89b912447206968330307da8e93cc5de1a7d75d4326 4218254a5cbab51cef397b212bccfd41d1a92d92b49f5368325fa9afc182f3d9 6e65f14aeeff0c0fd89ec6c1f3ca0c4202f08f20725423209af093acd826e42e aa59bd8b5c34cac8c89bb3257cf110c5253eaf5dc9c64981d75a556603b436ff 2a6021eea62bfded21fcc65c64027456eca030a1c910f7882659b78d17ac89fb b3c39f0f6663d02bbf572e8d33878d87939fcdda2a8db7debd45dc218e4f3958 b0800b0476e086cff978e4c9315f5134d556a9a027743a29aeec00e691c7798c da40a9300a9043cc5462020df42042fe253034e7f818fcec18850ec7aa87a764 f3cc5681347070889d5c9882ab73d49db8831a3fdf0bd9b5e192252ec57a89e3 7032d39ff018065bb482402d2a23282af074f1aa77da5a95e38663c776278099 7a19df06e571810a5e2ab829b6f9233546dbf8c0c9e928b5e0aa17dd7b5b18f2 8514bf220a3376b433ce88b535dc0541aea6002ddb5474b35ef33b46c33ef091 34c1578127ac19738b16d65204941d8846344971b06f06695de9d6424e990af5 e0cad58c1117d14189cb6bf5270fbadb8186e6d21bec6558a3a5afad7b3908cf 7146de1a18a076b927578666ef66291105e8f8d89da7766644f78ba4fbd3e87f 50f4daaf005972292026a1a5a8c1329efc558adc968507be77148c58f1e520cd d9b44515d7400e 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 1 /periodcentered put dup 2 /multiply put dup 15 /bullet put dup 18 /reflexsubset put dup 32 /arrowleft put dup 33 /arrowright put dup 36 /arrowboth put dup 41 /arrowdblright put dup 42 /arrowdblup put dup 43 /arrowdbldown put dup 50 /element put dup 54 /negationslash put dup 58 /logicalnot put dup 59 /emptyset put dup 82 /R put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301d00e489955b2ce2d9040bb11626f0633ded65afdf8d422c60bce7d6b 678bcd768e8835bf74334debb4eaef0fc0d56c62ddfad531be5ac6ad487d0f2a 19e62d980df7cd38c471dd8697cda3dc3ace663ea1e15c5f4a6c64ea44e374e8 7f4a13780d429346cc92c746619a4f014e813c97cef8d0e05de84d45454fcf74 c9fedace9f7bf703138612da228cf38f7ae47f41219408b9a6fda2f8d2d4c090 f13b0474cc0009a813c8431ff3bee90155fe22e3a7a9b64b85c57f4f3bdf46c8 c4305919e6f042c3a1ce23c52a3207c3b748e034d1bd92cf435213c560b5a971 28b49dc5ec78964f5e81b83cc62949c2239dc62bd43b431ab04d77cf00f4193b d2b7f958dcf8faa04073401360f1f8347448db717c4a19898e2714b0c506a4ee 485e0e3b95a7107df0a2bceda1aed2380ca098978ba65108b519740f768429b7 2a6f9a75a3cd33916f3dfb819f6475378bacb629d765d6d0c42a26b1bec1359a fe95016b668bbdd245962a139eb595a5b5702c01b9766153a1426acc9a142cc3 3e91004e9800461c1be0b6a6b761325b5729fa2ed7e577b7705b668ac6529751 877e0cf62972501353ba2a0e105c295e5988726ffa64c23f0b8870961738378a 644882e662096bb394b6a69eaee3af821bec21c80d9d546d40cc5ac2a83c2699 24073681e03d7b9f70cd2e20431dc6f43b3e311bd544349086c6a34d7d5b7aab 77732eaf08f151df9a93446876ae5307f87f96903ed199b62b1769a7d0575c86 9774b51acd8e47e73701b012a0405053960cc9fb3a80e018ff2768fc3ae19f46 3dc462b5a8ee78a3d30db0f6680595cf70a88ab0856e171825c8d2026af02bca f968d41c91704af2219125ee4c99f10136eb6424b867df98f400e6e9f7c7995c 8b1e75646a53a1a27978cb2ea836ce3d75db8fbc2690aa3a8cd2a4520c93d67e 9b415a70721a0072d2cffb013ccfd96ca51cd42a231e1b84357196d93b99c397 778f3f8068bea09ab432f2c61fd5a5b16c1c80872d00569899f5bf63ecd1b8ab 8abe3de166289e10c8d4f8ddd78ddaf19ec57a20e443954be63f072bb84d04da 0d1ee6e5a4d96baec8b24f8adb9d91142bf1a26ae2b9d226b120939b5220f6ae 6535c96daaf9338b67575d56d2273a5efd39888ef53cba18319aa9fb4d77a3e5 45f03b4bdd3ff965bf458ec2088be2e399ba8029ae832e49b41cdb7e4202d679 3adda71a80f70ab069d527ec9040dc50a5afdf3403504a57057b194338682221 ad3879457a4b0919de37b57b14a0dfbfcdd7f0d40340af3a96bce808139c71c8 cb8124e4587bebd9a76fa4a0582054a16987d5a1d06754a19ed53f064fe2d453 30e8df5b0aac285a2d1d8ebba0b2b48861986de64a31b263fe7fbe77026b9121 3a28575d45db9d5f625ea4b532b55cc95acf33b04ea4c6f4233b5c5b329d7dda a9b079bfac1c20d9b65a68bf255fce7cbda36f2c6f699685d65100e0e2262abf 71c16313f9c11f2709500a5c4400d77878630682cb62b1f6a5c396fc7e20ab70 05208a2ad39986920cac3c7273e72770340b44874d730b0f183310a4d39d9ee7 e7920a3747e94ea60bde916756b3cd5b1839bba6ce26cc0980763b17b727a241 ff077c62effe9133858caf5642c8c5598f7bd141969a86882f082ced65a304c4 63afc7c63a3a99891d62eeef18710d931c16f69e7d6a90bf6938195f7cf0aeb4 56854752f99f4eff60e5c94904767e654a543f86a31fa937989d80a8dd8907c3 e58475723898f1fdced0b4a4cf0c80e78c5e273740e4226f1c5714b9c2439f25 58e4e7b8ba5fd37ba6d8b609564183f210f148c4ce9b39ad1eb4d197fb208299 9418603d6375b63f2a956e03f96a3424b0ef61f106fcd94bc2a0b2b32b041c65 4271b038c6fbfa04be6196cc21853ac4e92382036ebf244729e716aed6c4c196 ada228a6eb188e993fa5886b285e663f17959f85d79ddecefafaeff3d187ae90 8b2b35cc1ea00c412321bb2b27c5eddd0fab8072fd46037355fd3e1250e5c421 0b12dbe9c34337f45e4027e69f4856d40c7d12e654c9ac808e59927bb8b6337c f94823578d0449d43fa420bc802a0771d230362b62c667d3b31904b749ee72fd c55dc3c2dce125257f32927b51b281d4f94495ef30b04c9e62373b1df6703e1b c1fc80d4202acb93850987e51513aade723b395bd36bf3d09ce9e6f228e4c969 3343ba39ed1934fde2fe0e638d9b449f5bfb9a39fe18937ed1dec454d267a8e6 b2ed9e3605d85c943cb89efe9f6e1b598771f400c4cd6a14cd02964c29665d8e a8343eae5a79767a59217178016b93a7fa05d6522fe9d7d80886e64cbc12e6c9 8eb5572f10817078a738154fe7937e60bb8803a148ddf52d718f7eff3b69ef7d 390e1e3222d8c5f27c4006050207280d361643d1e08c7638429871132eaa68df 9cc91eee787d22b46506e716639b10c49f5d885981790a26dd98f7256a8dc304 22be06f46b1a6a4e7ba2400ee668147986701b3566aa5d4b098f3cad801ef867 a53a75262ce8eca474a52c6b9cd39889e13b51a89af555afb8f627068da0931d ecec2633d3733540cb2c84324fc94db7fcda2c364a800147ddf9e58b162474c9 530d4fec112a79b58e99413bf8df9d9cd6b5d0d97e98c26d14dc320da1872ba0 cefcc26a3cd12c6c670d833f660c9a2253bc41214afff859b999ec758217dbab b74ab9a1935df228b0ccd93e71162b2b11c09bd0194b46cba9582ff58719b430 4c2442594ff76b745c3cc8e6f09f75f064057b37b1b33e39d4d991e0c48459f0 42b2460cadb09a01ba959ab91e27fa50f686e0c7ec3ee52733f94a2cac8b5786 b2dfff0ea4fa2334c1dc099747318f7a748919f47cf24859fed77786e9a40412 cdf7396785605cfbefbba4859f9c1a92078bbb4915b29d57b5e473cf93611f6d 6efb8f93f2c8f69c5dd08cc7fc5f8f9403cf87f34f1963a803284aa968bbf6a1 791b66069644a1693f4528bf2a9143618c6afc99119e2a6ca28012981e38e58c bb4be3c42b95d9cbbf485ad91b993a449007dfcf8c830b3c5e5bdd8d397de318 7936bfd367df0ce78e60dafdfe3199544258dbe9d82f3ddf1d32edc17169719f ad0f9f4204309da343fd8e941b84ecb2738a8fa6a963c8685726b37e0c853ed2 af9ccd1554838be0a3aedba3931b11f37057c8a24a64838b515c69dacc81de88 b5492cee43c862737423c1ff5778753b6b7f6a2df23eb7af517dddeed3a1de2d 557f701cfeaeffdb78a4205c24ea14c6cce7a11253f0ec952ac0c88fa4e44dc6 d1ef1beb1181aa494103dd3a6a6c819728dc8fdeaf72df2aaa0b56b20af1eced 8429fcbbb5536ff463368462dd333ece4d048ec62d9ea37329555a0ce7dadd40 1e1d94a563d11708b0784896f55dab921e6356ddc0959f36bfc4da931b6baf97 b8da41224f5e64d166cecfa21e449fc9fcd180a5b4960d3c22f9dafcbfc4df39 638aa891d923f8b719c14d581046350496df505285c5ba1f609f5178cfdd96ec 99551fbe82c71bdc2666b4b3cc5514d3028303f6ce082c93815d996d188832a6 d12a7d029aab2a9428f7f93d89c4f03e9d9c44891bb2c4ff766438e029ed2b0c ee0cb6c59d63a17731e1d7a38373d2d5fcbd1ab3220867e7c1f6ea0a45d6ef3f f57e2e3ed6568ae469a60c4deefd9033eb3f9b75d33b514121e23e1dfaf3eefc db7d574403dfd2d3f1c2f57f8cb031e4e859bdbb18118c82d3ab5bf32e093cf4 f23da056134119f71fa629c176821e60951611dcf4407b2fd518f109389d324a 1c7aed223e7f7b10d203571cf16abb261ab51c2d7eead27634c6e22c19cae030 2523fe32b1d7bdd6e9f6fe9b136e6f413f1aec10cf3fb77381cc04f731c279d9 698d8fa754c047d2e534f4d8e9a2720a7d0de486ffeabb742095b59c19a14ccd ac3b34c8702acce959ec8d0e6057176bd5f38aff426da18b8080ae55509d1571 a1cfe3c5ccf68200f0f4af53d19a2f15815e13ec72e68276064f07369062c7d3 484a3d89bfca0d31da86bb97bb24cf3396ca706669d19984ea1023fd5c47037f 20a3c949b6f443d7efedc29fc54d591dfb49dad21c884b539da979e64d527267 688cb2013b5a5477125f08427b8dbe5ae53031e574afa587a5ff3570a2653818 060b143cd6efe33676bb2c7a84d9d9b57b0dafa9e26b79c4c509f9284876cab3 cc594719c596317fbcdb26bca5bee88623188e8d8054e662785edecfaf5e6425 5971a5b46d5e422a462f49e994a5dc4e6bd2a8c26f49431c4149154737a781a1 ce0309a5de77a992b211e007032bf00fe46350c7644e64b37fd9ad71f7c1662f 84e5502704cfc9b1be6e87926baa98f588d4faf75130276f4c6b30c9175cb85f 0b524ef542c849e1d2c21cf527beafa790f87b215f611c8a5da617329e073ded 1c497bbe05ba39f4ffcf1cec9a9ffb4b8d2c447ab901cbe63b17ad945de786d0 93a0c244c8f199153017631a008be1e9d422584f35c6b60533db2630b84d7099 16cd67b1a902e9f3deec70bf335ad243d8f4d5acc3ad1e99305fdf2c36084f0a bf87b34f8bbec735fb9f16e08c3b1c5f83d6b6bd0b50089286a7759995471b46 8715b26a69434f6ad9d19d3d321fe2554b65937c2fac 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e67a35269b877f161 ead3b9314f0ebb23d1a803608fb8bed47522b2a7814eba409f33973ca88ebb86 148d3d0d91d8056e573e23f17b04c8dfeb3d040cab25f1da3caad77b904b64f1 98dc18b1abbbeb700e3bcad39189c15c31bdc0009f94094d223f303408ab2800 533c624b77a8b576fcb13a492a5aac1035d7766bf5e2ba738c1de82bcb395cb4 344d6f452463521d7a6d8d979625e6900ac2b993de3d6a15cf4473176be41862 e61b3f9e109d500405e6f01936bd4cd77ca687b490df2a2e501e141fbecc7082 86f85fa428a8121af362ef3bee57af 0000000000000000000000000000000000000000000000000000000000000000 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 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 76 /L put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 104 /h put dup 108 /l put dup 109 /m put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a2651e58aec586f836902e8a7d11babd27ed000f14269d6b8774a8e955d 0333ad4d5e8eb3a3c13b2f8be09bd3d3b0c90524bd356ea9d807fcc33e248bc9 ef272d6eb1720212a961ed6ed6b5608cdc2a877539cc04c7465024fa19676193 0d477363659fd7f83cd9df1e11478dd6986b938726f940538e005b155106ff49 3ff4bd4d71f3347cac3efefff4f7754c47c230155348362c04864d2d6dede000 6a1c1fb0b7af9b765f3596bf360baca480cd8fa2a5ef590e6cc5b12f0902609c 5eb42b4127c43c08d3983d60b79b1949c7240250519285fb98b38261097b8b27 7b8dd67b449d30de2ccd0e02d3da3f39cb066fcae198c032b5379c790e6a2b45 839e424e4ab16d2bb9555adc9d00bd84f20047cd21a099fa8d83db67181fe62a 52e1423cec4dd70abba86424c248298633a2617f082821de84d46a50c126d927 6b666958ad160743c3ec08ea7c47f7f2e61d1e072f0a59d793d404e290cbe943 2b4c9f0230df959ceb4c0202cbc53731dddb94247d6e67da8c4cbd59d96d44f5 ecfe2d7c7a351730c565b0838891e4aa0c6dc537c07bbcbf65e327533ba2ef12 dd225ff24131d046f4cb382233ecc1496e9105fa751fbe01b14074374dd0f550 e8ea340c1b2b0927563b5bd9790d242317158aa3062fbbc8bd78a714d9e1979e 9d20957b96cbf676ea1653e8541e6d828910d71fcdf8c7581d25fb2af579ecfb 9bc34fcbc99d1782eb30ebb31e71052f12614622ac6951386236d790d4dacfac eab574a699d420d1393d9cb4968a460bbc48725df587b4461f4fd488bdf8f5dd 59f2bd5f468f5a8e36c361a893b83fbcfcbb97cf110dd793387b912b2755ebde 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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 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 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 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83f4d4be6c2568b5f01f2beefd6f5be299e4291b06e57d60300f382dbf901fcb 3e89d028d2d969d05e40cb3355a5b2d16adb377859926b219ce0d4a6e0f8b3dc 50adc7a2e76493d9cebe63e1cd09192e91cef816c675598c61134fd144d43fa3 3cee667bcc4867eaacb4f5db32d7523a0c500a4876c3ca47c58a6b04f32db381 3b9297b6186bbeca663d6604de8699001a7f7e63171d37710a4e63095fb7ea91 3b08eec0396b9793d9db9af0d88fb4bef17c71eb0a85108adfb7efa11270eecb 0fd127bbfd8c1b22a03502b4994c115e03f303ed39ce0e41d8ad891e833463e6 9f39dfcdea01e1a82afa37af44bdc6c803f615045efd6ef50c5b0b43856413b8 0952761b8d843375392986c1894cb96a6a2757b4a3b866e320b516b408538100 69e862d62d93d0821bd39936f1c9fd7273df3ca7064d8a388309439208dca0a5 9cb33419c98b38d243504059b1543700c5a0dc25aea1e731077400a10447c83a 21ddd734304cfd1432b3933d85201ffddbf5a738056fc52b2c255dc112d37316 23a85e632fcdb1429c171d1de030f59e49798cd51978f1adafddb051442ad4e7 8981afa5228d1560b74919bf01ed5753112f060d119e82b3f9d92813481c03a4 92e6b2db79ef60fb7ac163f4d4ec3a0e96d5ffd9e37666d8dcbe3f4f9eceb71b 959c677d6099ca3daa5060add541d4419a0bb3437ffd1da6b2d92bdb3e60a1e5 b55c5d607cf03d63d2cf200084f63af0974838e865787c9cf517fd5ff661f814 efc1786c0ce34587534748a12c2e81a1704159573955c41b82032bffd79c35be 68b958616eb8b684eb309dd995c840eed518dc9861a2473bf68f15f4b42003d6 50906d0bb0882633714534463f50295dc2f1e90c86e8b6ab419de4c298e9bc15 1d4773931d28d97babbc88df1d8c8483ef9a54c46edc11ef5218903d82282111 a1546fc100c5aae3ac695b8330e9e94dc552ebf789f43ff2a1039a8459973722 b38c3c2867925bbdc8501107109f1e2b8a180d8dccb9b10496fe175862b82f93 10f3d500b7a6d53411ad29fbb436e746c76be20f4e6d12849d8d3d283d3723ea 83ff925d4d3e7944b1d2bf48ec9594d817e31dc028fde4ceb0e4e7d7ed445c34 b471d9477f3a6e3b440f05c5dd2afdfdea069fabf6c00359e6b299fe5e94deee 35987c9e68583c34ba9b79ded441aeaed50552010eef14cbeef1dcc3ad55957d 53dfe2afcdf6d21770704d62ad6dd87313bdcb753babdb9b967264ac73037361 9883abbdb392c943437e552031441558c80f44f487b3b0496de7e60b61c79c8a 80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 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 68 /D put dup 97 /a put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 110 /n put dup 114 /r put dup 115 /s put readonly def /FontBBox{-61 -250 999 759}readonly def /UniqueXX 5000803 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d43275929a6669eca687892476 7868f7357418bea5258a0ce230bd4a90e4c2449c1bb0b09a003fcc1550cc30c0 074be252faae7ebbb4540870d857f3fa917fbc959e50a1005453d38380bb2050 ab7166ae895122c76d0fe468acb4209a6a4732c83c7b3efd4a48f01a0095a944 70612ba2a396dd6f90875a7b48a6806c6d7b781c419e0404c494a7531556efd1 08e97e770410e56fe92b45557101548925cb121fe6c80914681803e6a43f603a 7a5a56dc73ad61f0e07d23b83519d73df4bbb2095c0b43c63ef3839135866235 3fdc68c810d53b36757496e93e0278a3e9ecfb064693408c428ccb882e25cbd8 0a6816f0f43d82cd7e7f65836756fd2bba49cb58c838fedcbd066007a0d05031 7bd08ed1ee6852e941da15d26b694f786dd9aa403dd448eb7459c1f1be275dcf da8d9eea3ce8c4590ffb24f7b35ba9486d675c2c05388d903335b7ca38d54a8b dfb43f0aa7d4c553f962fadf9f7515bd713d93bdb45e1917d291116d3fe3c7b1 7a0303886591e6248ca7862a83feccf080a54fc9780f848d95bae09126f5c52c 88ff1344328249a2bed8726f05890eb9988e69258044b9521bc6a56ba837d06f 28f244e71c374e76aacb5c16a0713c104b38e48a94bcaf268f27a90dc7bcd5f2 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 109 /m put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 9391472f46a58f4d47b46f95c41dbec5e4518446480a4f24096952ed2f1dbc87 45a2d0a1fe65de562d84e47536c41f41ee92e6edbee519141eb65b05f8e0dee0 fcb116be1e9416c3ea3b1ff688a8f78b088424476f33c9d70ace9b088579c0c0 773925848ea8f547252443058fd0bbd044d26b07cf5b95202d134e3bf14b0375 b54204c6e30b47c13a1353cd6c3ac4abdeaec9257fbe54308570be0b5931b3db 51ba6772b74836871c92ff7b0eb1a3277c37c593b7e71b2743826e0d2e7abcac 56de53e49e06c39143ddd1e9035d04eff2cd461c7f82c93d75a85ef06a63feaf bbc2cc04efe98a19110f574a96d5a10d37264bc6ba2bf1c75a25394074884617 7dccc1ca1ca8a7a8f86dfc2ec6e79fec653afdfcc1c36c1cb54a69369837b5b5 d61990fde7c418822b0a0024fb4d4daaa7c42be63aa6fef86c1bc514513b3300 c00fdc0a847861e6a34ae69d046b7367fc16b2187af780c77b24b6e6a7349dea 04f5a879d1274540470becc4f0b3f36cc8223c151b2353fed3e435f96ef8fe67 b2d0f35f93897673546a4f33c9bfa3251835426568b719b9df78cc02a67af08f 64ac11ee06caad0fac78f07ba5d87e9032e30db8ac571f881c6e04ebd3c897ed 97f00b28486b65699f24ac52e2d7d642aad2b7b4a0cb7d85c39882 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 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 66a602a9f142a563fdae3c8d40de6a8aa3248ce46710abf46dc196954cfca135 def696d1ccd10dc0843108c4e457e9e8ac79dbbf6cf1da6a528d981a2c5e1ca8 12d52bcb90169e828b345d262c2ac98fa9ed0de8fc00296dbec62640c5744153 d83debe161084286c3875fe6108d2f7b9fe647f42ca6fb421ce39ee0d2e20e70 3eb4f07b6ce318317ea3b283f36be8d9718c7277ef6e4bf54d50c2cb1ab5cfc4 95d5ad9fd883087971a9c54e450c694d1a5db51f6229f2358519a2ae967b9a80 1cd538e7c452586b53d09497d4acb553ece31063fafddd867ea5790b7c2bec13 59ef4446a7e441fcb2f5ba485741e56d816a1c7f2bc5efc4e8a82a4e26bea0ff 9dd0f73fdee794cc932398df30da430dd57c2d3c636fe151b3ad1d185e5f17ef 1a4490d513bc5b182addcd5ffdf4da6f216464f46a3eb06d2631fb6a13747b29 46a05b1231d19dcc3d1a477f5a62a9a5c6100f601cf5669d61e181aa28928119 bc0cd7f1a44adb54edb0b5c20bcd9119c782e92bf10ab04354b2e1a5b754d953 d6da82587e221450f493f52cd37e8922998c6ba344657e5c6af4c763f850e00d c1be7b5bfe3e23168d1cb029a2a48cb4458088a6e780fa9edd3c1cdeef9055b9 e7cb2ed44a490d8a1ad0091ce31ecb280687f71f57a132386d6fc642e00f95d2 4016dbfc504e7eed78434b96e78f0407ee8abe555c417d05c4be9f5623cd467d 9a8c80621cdf2468c1c349ad0001f84b11f9abff5e4ae7d8cd4a141dea939d14 77db64dd0ad143a8cdc7338b60efe89a8a245a0301fca58a7c51ec0dbb8a992c 80f2ff18cbebdb2f0172631478e55e982292d6486c1d5f81ac379c16532e4c76 ec7d8f72 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 80 /P put dup 98 /b put dup 105 /i put dup 109 /m put dup 110 /n put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 69718c0c5912d3167bb0971546d733e735aa9bcd89f5aa154656031ab3f6fe41 94ec225ad03ab8b67a9da43ab365c4df17c23e79f685c3b6fd5b654e4a6f2603 6cec3a8641b0b57f93eb95f9e238832aeac2061f3153e0f375513c974f1c38f7 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2eb1aff 52c190f50fc981dccb64bb2c1f7a62c216f211c162c2ed1c554ef49fbaf12b31 aa3b2a80fee259dc0a2a4242869ddec7c29e2941792b28bba7f2c6f77ef69638 12b07d6843f33b51e01c8626b26edc34e1e0fffadede5fb468e8cd5de7e5a64c 366967578f415484d122834de4a647541d66989f7d8b25d1712d8d16729bd9a6 4727db53973f043b92ead0e719a448eda8b3621003ff3851db15cd5a7463d77c d95f66caa2ff7cb52ead3c395150ba35beb5eecc35c2cfaf38a214b12238e4fb cdac17a04cf4ca9e0408c08385780c5054793f02ab565f0e5d56a215453c5e54 e77cd064ef5d79005a78fe5cd24f56abd38ae861d0afa0cb017c8c2996f861ad c7ff520c28cd70851d66ea72d92aa2aea4a410f3d92729165916bfec47b8bfee 605388529bb68b98e7f6b0fd6558da2a3ca7b73c4be7e7447c309ebe881fe59b 0a472311dbf491204432c7d57c5747e0480d28e88d18c1834756e3c935cd244f 9b4c660a27d61619101839884d48520c40c7a270e997953d10523c073261b6e2 4d764a6b79cad23e2fe3c604d459fc5cd21588ef4947fd419affc8f4e5034037 f343272998093b3660e3ced46bbfe946b0efeda4f929c0bf06e525a9b5c5cea3 b1567213d522f08649b5c1e09147ebe2192d2c372692d34cdd32b240e1a2b517 0719ed4b610cf1341a10531ee003dbc22f10c3df0e8fd0cfc8f19e84429cb80a 1c884ed1d4f8d311d9887b8d98b2e1c9e4f21d6069325102ee8860ee2cf4c7de dec0a5983732c44cc7b7e03f66686a95f18aa8019a8f59b3ea5b1b52502ef1d5 164468d130e8fab0b3ec1bd4b09d0eef0f0cb4ea1be4a3006ab255800ce30385 13e94562ef83886763136e77169cf294c67c54f0e51b41832721dd293e887a82 feb3e8148ea9855f663da5143c46aa18922302d96064c1edf60a13f3c592c444 9298be006e30542b28b13e09c3a2918828450bcd553d73f114f65701917a7831 3dbe1baf3bffbbc7d86800031721c9b9cf462566f939dddbfcf0a7615e8c01fb 0b42e4f77d7d171c22f8b0c1bf4285d8e743d522f9efdef6074f45d8478b0225 732a433c6cbf9545d296616506f1e77285cc783403f2f07a960cd18d23980702 6fba1a759ad80fe007d466ca04f1c9ef6c4b03f33a46f4e55334187432b1b4f0 5baac0b0359fe3363d80be7f39010eb76e1f169d953a5772bed28021448bf272 e69641335890b7163cd629a9427764f0f98fc4a6202309d4a2fc57787102d275 105b203d59ae1cb8217c3ac51c08c343b070866358b558445d4eac2bcac67246 ecc7a2a03c4f3311e6ead072058b6f4cd3f6820d3f82a19e01609d722e304c64 0bb0869b6080e266e139ec11a1ed90eace796b24b5ff6f29fb5340c170263dd4 58d0c801d0753f6f1c003caf6baa26c082804728c59bdafaaecaf5610047cfa5 6225c1b11d01494c55e1123709d5e68d41ae9666677073f1cb32384bcb4bc66a 253bc013f6b090602129deb61cb4d73106337ce21fdbfe32eed41d6955b0947c 885b12e53ad9beab8c29520fc28e7f963bc842709aa28e1499b324ac96a3e312 aeefddc8ee2213e98a4960a2afd63f494a33e2c71782f264e9708d87dbe6e6c6 72dda85fe914d413e8a80a38ed484c78d1f473a4df2e70010fe21db6125dd234 0a054798f690b7cc00793a749b88047af58171f31cc7804fb6e375b87ab755c6 afe480496510a6c2993208716ee43312fe35055d3abbaf4529272889d293b58b 39b7a3a2064c25ba6c3e46fd0733f78da1b7189650901f55754d19a25ebc45f5 82b379af9f0912250537fd9f7824c1dcd2d1fe2f9440e0c938941ecc48cceaef 65dce6563c7628ef67dc2b259ec91790ba2cdfbff6a43cad2cde6a89658470d9 088c64feb2ddb9307c6afe95c58fc5357bda7b0ed7d9cd10ec051eb368d9b39f 0adcd2ea30b9878dc1a703948139640fcb28b37e32204a5ddc2d29ed776025f2 f3492c4155b76c0f45e36c15af26cbde26db1f972c1d1864e4dfe1c076b08278 07568a5a4384442660fee811d05cdc812c9dd823f4a2072d077baa6356426f2c b6070b17f0207e8907b03ee70b4c402fe0366c1ee1cc79730b74090058f25c3f f16cd60ce142209f9ef4c51f2c03e609491ec52c51b656547a692b177f4e138f 6fe97e1269fb82a14a419d37a969c07813c6e079af03b556c1ed7e2e72ca53bf 63d099ab6d22380b108b6972532530e3c0e3c83aa6c7aed8d42360b68acb2a40 197570b008c8ae0fc44d18458ba23f82c3b1f2a68ae60b557d8be6f0d3aa352f dd38db1876c6c4612b69b66e2a891d8446e0e73e2ce3344ed65f8a48a6c710ce 99849914f849f7cfc9b52cfae6a6515dbc54670ddd3b9c4952f54571c3a6832d 1845b1cb7e7fe916a4c5f5c5cc2e8a88c5ce8a5623b29539488f0d5149757a79 6852898decc253e9b5bd04f5bd6e934b281487dd2682a4b4d7ff3468e86adbea 7c9e159137db2ba6414329da597775019155967f21ae01cbcbb8ec79f5127400 0c5a8cca20a858e318c253684c9fb3164da4adbe741e2c90bcb5712dd569fe32 9dfbbf9cb3a281fcf06be08776d575f1db3276ce1751ed6b9dfda06cb1586d59 72543eabd04edf881c71e58c4f23f154f845fbb2cb8428204ab51d7225eab603 07e89ebd80b5df354ce7a2b1f87c53991b0384113c16d7f5df36e1eef0efbd88 9e8f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e50a08c99d708d7f1b 7f2f2a9fa872f63f820b3b60dc8b518db5d16f8f0036dcfef0bfb148800fcc58 09f4e4b215a147dca09ee07fa968d2375fdacf271ed8980b6f619d95c8a4c44f 0bf94efc5813e10858db2d74719863f85d6e6c87a5cc2f9d56bb8db10a063a72 3013057bce04c467ad6b663f6ef724238de88c1571bdaea9c50dc0921e5f5677 c94176c4d7503ae0f0c58aa7f6de447c1db49698695bc19a13824fee13c360c5 bf1fb37da88a2848e548dd01c20b88096971af2fbada6a21557a9d2196d0295e 3a5cbb4776d8f4a12acbf7c30b99020425ec8f6aceb627f6b3d83210269a856e cfd77c4c2d2bc9ac74434fe31ed256a7f97bc8ac077dcb99cbc52b8d8b9b72c3 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f523acc8529f30413b5b49b1f5ed16c53d742cd54f8d7d3d8bcdbbaf7ae37a91 86a5919a41d454e657df7567be35498382413ab145fe7cd484145e765dbe1f10 3319504c3a7b2a2f91ad5a27e93c5bd1566fe991e076a6383969517f2b3ee497 1021a719c20c60a6f6064a0a08a0e5860dbd9837 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 30 /phi put dup 58 /period put dup 59 /comma put dup 62 /greater put dup 67 /C put dup 80 /P put dup 83 /S put dup 88 /X put dup 89 /Y put dup 97 /a put dup 98 /b put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n 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 /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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7d919c43752a2430e01094dd03b1b355ca2541a56f5f5055dd21c72696de1a09 598216e518464620316925421a399c79428486fe81b67ff5ec1bea64500d30c3 161457774afeaacd19ec20b1bf20b290897e66b4b8cd9ddb54e32629c24f7a24 d06af5547655761f6a0cf0297ed124a69ec83616e0909acbd37f3628789d5169 2ce4fc72220824ac266bf304b8a50c54373d8448b83b96fea7d7aaaa7c2337e4 559534a59449ce6e23ce2b67b1dd546d08477928febf1dd7fed6a8be249110e4 1e645eb2f3d5bfd8f5361aceb39ea9362480c37ec27f0b1c63cfb17bef204ead d0c6a4963eb33015a7a2bcc61a9baa7f0a3bb61f6ac0663998b69f3d41f8db07 f42c19f934ff99ea3193526b96a12139e1b03e3c04fd26ee4cd48676f0bbd513 b3df08ba814d6fa22fe6c69cc1a734f2ea7e3f942dff42ef5cf75f379fe3a390 c64788f1b89a08f97f13daecc38ffe9eb7e827bc9bd484e04fce29ec473a2134 5fd3dda08e93a6783ac6941094452040a1236b31a49819f1bf685f1564fb6bdc fe5e9c5e77a1d8fd95cbcfbe555efc0a05bff798fc3c8f60ed4550e66c6a70d5 7d1956cc44c64b7ea26c87b78ab8cd4b07727e4443a566d516373a09cf4295a4 12e25504f327069f1f1d48cb87c4973468eb18c361f93b4184e9aac87de3af1f 04ee4103f46b5f18b6728ea3e324261307c6aa0604a00ef3c13597166b9e7f79 789808d6032066a22b988797e3ceb6b8c8abebc5994895132f0050470607bbc0 6574755d7244227c4b80ec27255f927e21eceaab8f2aa7da8ae907254659ac93 409c4c5ca269ab9cb1d27b9f21c3708d64299a310c302dffc00c9957f4d24a90 f1d14256d94a760dad4b8b286b944e65b35dafeb2f13063105c889a4c459048d 946e60989b3746261a2b8473853a4ed1be141bb6847605520bb79a22a54d3280 60006ba559d50e68d0f872ff40e376a4247ec17de5b0658d29c420fa207c40f7 819bb852d8c0325d4b7d2c652736774faea27550cd1e3c6710efc13719403402 5cebb92eba04752918d754887bcdf48b71f8a7c541f118542f3da132f868c609 b3db3c3b77c555e6e25fac0e6cc9ae558ebec3a535bd66f57a20bb219ed44a28 afd9c631a2c152d08f530c2bcab16f1c21021a97c24c7aa1b354579774cb6bb7 7b4222319387729ca69faa0fd05d2f990883e34932b92fb3c7a87e3b28d5307e 2b54e8f94f4c28493135b470650fd1807374055e627ca505a56e6462aca65398 2a7fc8b7e998aba913fe7161cb613036a6e4b6572e1c56c734cb5039cf22fa92 e99b178a0e9a6f8f27ccdd74eaea92aac07b82c3e1b00b0c7fd8c5c26040df1a dba0e915003b2ede7ce1f086687ba4744f027280edf1bc9adbc1120c30c0a2d2 fa8976c317e7d365bf33b78717827f7968ec192f647304f12c1f051d268b5729 b2527c34ed8e6dba053ae74566c74db193b568ccdc0f1d13c2cb5925f0269d4d 31e783f59e31642a6fc68c8dd34ad89f452fab1b2c842c62d86e5dbf083c5ba1 aa7a6079e17ad985673e168a1622e21519aa3869211a368fa21eb5675fbedb3a 037eb01c514b7beabef695d608bac3c1d4754d4d886e018d280add022b7415da 2d0f0895cd9a46dc1118cbf5113d8fdb3df1cc2f3e0e32c3eab18a8fd0b92d1c 0d6bd9c5f5ccdcb092c910bf7753b558057119649e5a42f7624542c16ef51a80 658ad6522b2d9b26cb897a6260c1cf48dbe1941f0f2f0bf7992ad41312c89e49 fa3f7b8fcfb543a9f50243fbc8904652d95ee8c914ecb21432d002c1ff8bff39 5e6ce6d84377cf75900932dfe1184a827b5fdc39aa7cf302c40afd2021ef9f4c ca7a60c84c705d60d72b460b87600d482b6b83eadd5ae833ae848f519c7ae5ba f8bd3a9a743ab861cdf05e9621affb71e2f9af887b2d48c3d1ca4530b08dd4a8 56ec3136195eb4bd5ff39a17402efbb41b04f2ed4ec1069ebd5f27f2078e8535 08e5f1f6bd5f2a1edc3fd4ae413da6eb1044b3a7d086cda6ec4875a0ecf2f50b 46760dabdb62fdff208ceea8165e4c532432d0d8a8504d915ffa14ac57b9cf68 5aca65af2cf8f0d77d621567b763a9a4c7b7199fe5f89c93df70dfa0c4186e8c 6607c3814e45aa0ecac74baeb8cb07bdeb7d34f7e233cf484f389922623136bc e885050194837026de18983e788c93c7a5ce0cbc4320ef22855d1f0f7fb1a7c5 63dcddb341914898625393566eb062aefded7fe9b07311cb7da4dcfe07c19985 4903c436562a7021a8ea445385491f413601ae173701a22c05020046bdfe831e f5c2423520b399ef221a4805a575d7abfe445840e0eba73e5bbcd1452685b2f7 a9235e8499d8e7d44c26fa76bb1015a6df2c027d2a4249a874f2746e25d79576 f731253da333cca1188809638b1cc8c9b19cd86fd06233864eb9e259064b20a6 f4af8afa8a4719005d30bd8cc79614074e85e4b45a746891ffd40d03458efe1d d8f9d7880f901acd4cc025186d8f3472151ae5fab0bf49c3cd55b538aee67246 7cc4c12932b69f8714550fccec099559e0d139b196493654ac5ddf83671a6496 d18ca2436089260aba5185500e10058da0cb624902dc40ddbf36fdf72ad9ce06 db8110319ec8da1f0659f9d7cf755a47e8cc84d48d3dd450d7cd5bc84d5bb886 6f63b923baf36da6ad29bec5bff5f20c4508b53e8854670cfce17b975f0a78e1 40e22ab1239bd0c8e401b27d2ff880b73401b1df074b648f14860c47620c4768 3ea89d866bd15bd7140a0ae0f2a2c0fc8efedd76a46af6f3b017fa06c01af540 626c6b682ae1b9f6e5c640885c3a3c0fd40e3204909cda606bed5647d17179fb a93cb72d9314019f899671e7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 46 /period put dup 97 /a put dup 101 /e put dup 108 /l put dup 116 /t put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueXX 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3036f2ed7bb65d c6f12e33dd6d45b61272314f98990d08c0f8274e999f4b46af4b5c5da923683c 960face50087915d2044d7211a18e0c22967ef89552662ba990e2dfb9267e09e 354f41e74d44bd80a2d905d2edd60d8829e820c6ab0fdf322964c1ecb6569818 a465d5266d3366d748b412446e602f0cc52f3cad2213ff6ad92c04f6ce8bba94 890fb44eadc41a0c42eda4fe9722e6d4b17a3f6d35cf755d0a1c90d0c2033b16 ba76c456137acee0112e35edfb7206a9cb18dd9274401c70c3039059a583940f 44cc19ffed32940fe8203292e103d2abcda173df85995aee98bf4a1f65ead55f b80bc0c737589fa10cf51999c634e05d80a7dffb61433bbdba17ea9a7b92f763 586105c8277d6a961d092617c64745450b7df4efa70884f6017d9b0331814cf7 02f1140425eb8bf4c269ec042fde895a7f625c4832c1a143a14abeb798e23b59 f88ebd1414a66520435a4c8f5dd7a548a7d6a2fa96f0995bc0fbcf91b1a1eb71 1026a99d2404b2103bb3626978385830a00a02b6aa820cf33785f7aa45e267e6 ebe7d737ded351c1d4e944733feb30a3e6f81c1a7b74ddc9148b46b02a525539 c54a7eb25dec18807557b6ea94c36db0c91121160f61bd37cff40bb2b11aceb7 5f146be38109daeb77803d8f580f9086bc460e5c96f764c1254e36feb77c48a4 419b9a9cb5ec916b73efcc4c5a698909d5c98ca014b7c8e2ebcde195f5ad44f5 75bd2dcebce29979618bee325c726f5d4967cf0e3f7d35058e04dbb7fbcf60f1 436fade30418ba3c7dde0b6aab803140e6aa7c9808ab4d3d31432f5d4cbc007d 32d7541ccb867f6ccba8bd18c3d18524a692ed2ad29db54d7ca84a618052d3af 373cc37b55606fdc057cc538fd07f3b7814629834910740b9082012c053d427d a3987c45f540b90ad471de56ba0d305ad1034b9bfeb56e3fba1b9ae727e50df9 c29f110805cc0ad35bacb0aab1169aa8edc8a6396f48832d699a2fcae2d83fc4 851e61434e98ebb0b9393a66a2cc44d9c95b4a347051452f512ba8267a6ea0ee a442aa5f196e0fb58567824a40f2c267c0d435ecd1926d44b473514d5d2f7fc2 be601e2b2bc96b4c4e624b064ad8ab5df3614863df382f876370ab0f87f33dd7 3714f4a7db3b7b608d2a0bf426ceb5a35514fc726067c1d8221a13151f10cb5b 3ac3253899b3aafb9dd5a84b9a04b0c09a01a8904c973cd7d693dc1a360124e0 bffb6ce6f3613a080b9e27fac4a66c5a9de873ffe0bf284fefafd5cef2906f4e 7003459b60280a89b11f21ae1ce0329a957dcbae939d797968510b4bf9c8089a b600cf6f82ba75cba79898928c13f4132a5cfc8c9eca429228b8a6301384813e 2bec00c3f82a87d9b0932bd6341748ccbe581145333c2c58c49d41cb9aa3bf8a a6c988fea92e0ab70df047a62c7f40975d232b1767e29c6bbd4a6f67ecba79d8 0e5c7b4e19e49972a3366c72c794963e53595d933bb55af5c398d2fb2440d04c 4d419b60c42313e1bf5734c85a47c244d843da94a175bc9f01750e91b6bf5f85 4a3bbd50ccf061c7fa36b99fc3ebfe7376ee7544379f4fea6a20ede5a42da81c 06e4427ef43d192cc3b7861017947debd1d8260026678ee5eefa7497f6759ce7 c493e2125cb4de316869833029d723adc5848168ca5c1da4ef832e1809e22a80 d651a6b9867494f6455e0b1b5566c3ea56c7b4512a32bdeea60ced784d8e75e9 b7820d54139a7bacd59d2161b4d4ad92497e2d097a9f1ed9743aad693f552b28 e514d12408a877701261942d54441c70fe2a9d3a351fe656416cdf1097d1a469 3b067b579de6201713cadb6eeeb7ea23dbc46a6277dbdc51683eeb5c10e15e2e f2335d99e20344a0bc0b5f2e43fafc787d5b149cfa3241971a885a5c3ab051cf 34fc0b3d019121b8349471483a3d600b163cf05f709519305f228e1f1868c5ed 542366ca1a328838d8fc101919dda504fb806d122c6b1cf18599203ee2e3e0ea eacc8455b34be37574534cb87852f1ac722eefc8a64e0509abccf08b4f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 82 /R put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a368f97cc4b552b0cb393586ef85adfbe8b725cd15e9c696e58e1b11 21c368ba02f172045741e4815d8cca8af35469573ec88979a32605484af842c7 c8113b9e8ed45959f6509328d38df9339718d0330e7678d6f8dd1eda9c64e8e9 5386e11244041426e100bb8e8a37f3b65096c3610428646e73f5efa5d3c4e61c 14fdcd7ea9eba748a3a13442e01b14e62bee9269bd5a418b0c16014994ea181c e732ef0c7cb86441f64334e2b2be5ae7675114909aff96971eef01cf53193b05 d2bf8bc8d031eb3d13ed5478eb94ab493beed5ffa631a038f3116a2ae0fd5364 01299390e966530184007284ae5a57ffeca47258e1dd77c988d1fa6e0a175e5d a17039d980d9898c7236bf82b688351b5073f2cf958a787fdd6dc83f7968b61f 3719a9ddd5e871348cc6fedf14380356d2aae820cac2d5487dfbf12633746431 e0e4cc19a2817479e3583c82adcfadf3235f227dfee552cf1040cf934a039e25 782e91b09528248f31dadf3eaa34b7c0fa4e9b93f93fcc38fb3cd9b67b0ef0bf e159cfc9fe2c85cbd05ec7b9a5d5ac25e017d7add5b268d1043af4eedafa63b3 631f699181efa80e605cb2d8f3f604d9c66c13b8c6e4bec37fe92c9a881e10a3 024af384c3fe2884bc55dc6e771d8ec60bd20f3a6208a97005e7e1c189514141 f19775466df796ea09235bcd304e0eaf23f962a8f568a9775952d08886caa286 d63749d9996ae5be11ea6044988f3b980c84022714b5f4e7e5fe89170744db91 2e8155ca935c8a9ae7ca9762c5d441cb305db91fa37b25ccd1496d4dad79e073 dc149b3c299639dbf4101fcb61fd6a3aca47fef5f08fbda390a602cccfbfc1b9 41d911b4982889cf6c48e35e48a2a472a4da6e502a70dc99706563d0f7c9cc71 ecc648e149fd3eb592036c037dd1363e55a89502681d039421a09e2c202699ec 4da7339debd8ae01b95c968fa867fb0b9798a3b092cb60faf8df99d037d5063e 93a3a0a4b053a69f116a29c07ab027bf1c1d615e2b5e6a752fa709c379ac5e97 1f589f36b4200648779374d6c8bcedde3dbc0e3deac2fa375d1c50f69757fa35 065515c89ad87a2d29e5f215aae3f0b54d5016067ea2203397c69406e43e89cf 344642e2dfb9b6365e6f37b54c9cb7a486b1e5bf10eb7b2bce6bb8763abbf129 4f7bda1519058e6a81f9aeb1e51349b8045fa5696a243e033189d739f6f4e889 6a2bd4a27dccd0e0f9e72d499500f2 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 13 /fl 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 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C 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 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 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 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e1d6bfcb64bc459a80d08619772ff89aed1e85a6fcc537e30a748394c3ea9f46 038595958a48e69d9599fd23bd9fdb6b7409213639b35013616f76a7fc3ae7f3 9c080fc3858655bb9ce9a43c17a6a2b1d9b1825cb831ccfa1615c7e8ec812eb4 570f46 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 70 /F put dup 71 /G put dup 73 /I put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 87 /W put dup 97 /a 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put readonly def 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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 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 25 /germandbls put dup 34 /quotedblright 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 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft 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{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 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009b0752a3c36c97d479bdcca03a55813082912f28bafbe097412eeeb33f10dd a8e9420bbee24bb51ca86ad95fc5b8d98addbe32c52ef8e40d04e83ce66b4b96 9e1e17b57e4f1f8e6fc105c786c7d3b38243fa91c5a81bce4de90c2ff78817d5 204f3bff27e4fa0d564f6f206504a867d612c9a43101bd41a2eca8ba932e0053 c0f967609c41c74923a5d0eddc94b121354103e0bcadea2d6a2aee6927dc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 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 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 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 readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204a520640fd7c4d05e35ea5486f a43d0e970787203855daccb1ed0b16325828b13b7c767a1a58c87902f0546ea5 404440fa6526975783c2d7a9572b8e5d5176bac598b8960609f49016aebf3763 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b56cf4076d96ca86c35c2119bb1969116cabe552c1a832a128acd02e7d21a0d3 b66ab30db6acf6fae7393cc8b872ad0fb4a15c29bcbfcfc15ebea5e0ac2eb92e 5ef55448045be4466f5faf6483d105d96e960a70938f048724170e1b55221473 ca5eec415c6348d9ecf568b96e038ae616839b98ae55b866b2d1d78afed64878 df83cfed739e57d95b911fde0e35e9cc280ddbe1b0c055b551bd46f9949a172b 317fb239e069aa46a9208fee59392506750c5241835d891930d12cc95d921c61 4aa2ea7949c5bc9e5a70c776dd628f16d88b44 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % 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 (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 /fl put dup 50 /two put dup 58 /colon put dup 67 /C put dup 72 /H put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 114 /r put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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Fu(for)f(linear)h(terminating)g(TRSs.)39 b(W)-8 b(e)30 b(pro)m(v)m(e)212 5100 y(the)24 b(semi-decidabilit)m(y)h (results)f(for)f(arbitrary)g(\(\014nite\))h(TRSs,)g(except)g(that)g(w)m (e)g(need)g(linearit)m(y)g(for)212 5213 y(the)g(semi-decidabilit)m(y)i (of)e(SCR.)f(W)-8 b(e)25 b(also)f(consider)g(\(non-\)semi-decidabilit)m (y)i(for)e(the)g(prop)s(erties)1920 5462 y(2)p eop %%Page: 3 3 3 2 bop 212 337 a Fu(W)m(CR,)28 b(GCR,)g(and)f(CON)1124 304 y Fq(!)1226 337 y Fu(that)h(do)f(not)h(app)s(ear)f(in)g(an)h Fo(X)35 b Fu(p)s(osition)27 b(of)h(an)g(implication)g Fo(X)33 b Fr(\))25 b Fo(Y)212 450 y Fu(in)30 b(the)h(con\015uence)f (hierarc)m(h)m(y)-8 b(.)353 562 y(The)29 b(pap)s(er)g(is)g(organized)h (in)g(suc)m(h)f(a)g(w)m(a)m(y)i(that)f(it)g(can)g(b)s(e)e(read)i(indep) s(enden)m(tly)e(of)i(the)f(com-)212 675 y(panion)e(pap)s(er)f([4)q(])h (on)g(the)g(termination)h(hierarc)m(h)m(y)-8 b(.)41 b(In)27 b(Section)g(2)h(w)m(e)g(giv)m(e)g(the)g(main)f(de\014nitions)212 788 y(and)37 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b(case)h(of)f(a)g(t)m(w)m(o-letter)i(alphab)s(et) e(\(P)m(ost)h([15)q(]\).)41 b(Matiy)m(asevic)m(h)33 b(and)c (Senizergues)h([12)q(])g(sho)m(w)m(ed)212 3008 y(that)e(PCP)e(is)h (undecidable)g(ev)m(en)h(when)e(restricted)h(to)h(instances)g (consisting)g(of)f(sev)m(en)g(pairs.)40 b(An)212 3121 y(ob)m(vious)f(breadth-\014rst)d(searc)m(h)j(pro)s(cedure)e(yields)h (the)g(semi-decidabilit)m(y)i(of)e(PCP)g(and)f(hence)212 3234 y(the)31 b(complemen)m(t)g(of)g(PCP)e(is)i(not)g(semi-decidable.) 353 3347 y(F)-8 b(or)30 b(preliminaries)f(on)g(rewriting)g(the)g (reader)g(is)g(referred)f(to)h([2)q(,)g(3)q(,)g(10)q(].)40 b(Here)30 b(w)m(e)f(de\014ne)f(the)212 3459 y(prop)s(erties)i(in)h(the) g(con\015uence)g(hierarc)m(h)m(y)g(and)f(recall)i(some)g(de\014nitions) e(of)h(other)g(prop)s(erties)f(of)212 3572 y(TRSs.)40 b(All)31 b(our)f(TRSs)f(are)i(assumed)e(to)i(b)s(e)f(\014nite.)353 3685 y(A)43 b(TRS)e Fr(R)h Fu(is)h(called)g Fs(c)-5 b(on\015uent)52 b Fu(\(or)43 b(Ch)m(urc)m(h-Rosser,)i(CR\))e(if)2735 3652 y Fq(\003)2710 3712 y(R)2774 3685 y Fr( )26 b(\001)f(!)3032 3652 y Fq(\003)3032 3712 y(R)3142 3685 y Fr(\022)45 b(!)3349 3652 y Fq(\003)3349 3712 y(R)3438 3685 y Fr(\001)3518 3652 y Fq(\003)3493 3712 y(R)3557 3685 y Fr( )p Fu(.)212 3798 y(Con\015uence)36 b(is)h(equiv)-5 b(alen)m(t)39 b(to)f(the)f(the)g(prop)s(ert)m(y)f(stating)j(that)e(ev)m(ery)h(t)m(w)m (o)g(con)m(v)m(ertible)h(terms)212 3911 y(ha)m(v)m(e)32 b(a)e(common)h(reduct.)40 b(A)31 b(TRS)e Fr(R)h Fu(is)h(called)g Fs(lo)-5 b(c)g(al)5 b(ly)35 b(c)-5 b(on\015uent)39 b Fu(\(or)31 b(w)m(eakly)h(Ch)m(urc)m(h-Rosser,)212 4024 y(W)m(CR\))40 b(if)603 4038 y Fq(R)667 4024 y Fr( )25 b(\001)h(!)925 4038 y Fq(R)1030 4024 y Fr(\022)40 b(!)1232 3991 y Fq(\003)1232 4051 y(R)1322 4024 y Fr(\001)1401 3991 y Fq(\003)1376 4051 y(R)1441 4024 y Fr( )p Fu(.)68 b(A)39 b(TRS)g Fr(R)h Fu(is)f(called)i Fs(str)-5 b(ongly)43 b(c)-5 b(on\015uent)49 b Fu(\(or)40 b(strongly)212 4137 y(Ch)m(urc)m(h-Rosser,)28 b(SCR\))e(if)1169 4151 y Fq(R)1234 4137 y Fr( )f(\001)g(!)1491 4151 y Fq(R)1581 4137 y Fr(\022)g(!)1768 4104 y Fn(=)1768 4164 y Fq(R)1857 4137 y Fr(\001)1937 4104 y Fq(\003)1912 4164 y(R)1976 4137 y Fr( )p Fu(.)40 b(A)27 b(TRS)f(is)h(said)g(to)h(ha)m(v)m(e)h(the)e Fs(normal)k(form)212 4250 y(pr)-5 b(op)g(erty)40 b Fu(\(NF\))31 b(if)f(ev)m(ery)g(term)f (con)m(v)m(ertible)j(to)e(a)g(normal)f(form)g(rewrites)h(to)g(that)g (normal)f(form.)212 4363 y(The)37 b(normal)g(form)g(prop)s(ert)m(y)g (is)h(equiv)-5 b(alen)m(t)39 b(to)f(the)f(prop)s(ert)m(y)g(stating)i (that)f(ev)m(ery)g(term)g(that)212 4476 y(has)f(a)g(normal)f(form)h(is) f(con\015uen)m(t.)60 b(A)37 b(TRS)f(is)g(said)h(to)h(ha)m(v)m(e)f Fs(unique)h(normal)i(forms)46 b Fu(\(UN\))37 b(if)212 4589 y(di\013eren)m(t)30 b(normal)g(forms)f(are)h(not)g(con)m(v)m (ertible.)43 b(A)30 b(TRS)e(is)i(said)g(to)g(ha)m(v)m(e)h Fs(unique)g(normal)j(forms)212 4701 y(with)g(r)-5 b(esp)g(e)g(ct)35 b(to)f(r)-5 b(e)g(duction)39 b Fu(\(UN)1389 4668 y Fq(!)1464 4701 y Fu(\))32 b(if)f(ev)m(ery)g(term)g(has)g(at)h(most)f(one)g (normal)g(form.)42 b(A)31 b(TRS)f(is)212 4814 y(called)37 b Fs(c)-5 b(onsistent)45 b Fu(\(CON\))36 b(if)g(distinct)g(v)-5 b(ariables)36 b(are)g(not)g(con)m(v)m(ertible.)58 b(A)36 b(TRS)e(is)i(consisten)m(t)212 4927 y(if)f(and)e(only)i(if)g(not)f(all) i(terms)e(are)h(con)m(v)m(ertible.)55 b(A)35 b(TRS)e(is)i(called)h Fs(c)-5 b(onsistent)37 b(with)h(r)-5 b(esp)g(e)g(ct)38 b(to)212 5040 y(r)-5 b(e)g(duction)44 b Fu(\(CON)858 5007 y Fq(!)933 5040 y Fu(\))35 b(if)h(no)f(term)g(rewrites)h(to)g(t)m (w)m(o)h(distinct)f(v)-5 b(ariables.)56 b(All)36 b(prop)s(erties)f(in)g (the)212 5153 y(con\015uence)g(hierarc)m(h)m(y)-8 b(,)38 b(except)e(UN)f(and)g(CON,)g(can)g(b)s(e)g(de\014ned)f(for)h (individual)f(terms)h(in)g(the)1920 5462 y(3)p eop %%Page: 4 4 4 3 bop 212 337 a Fu(ob)m(vious)30 b(w)m(a)m(y)-8 b(.)42 b(F)-8 b(or)30 b(instance,)g(a)g(term)g Fo(t)f Fu(in)g(a)h(TRS)e Fr(R)i Fu(is)f(con\015uen)m(t)h(if)f(and)g(only)g(if)h Fo(s)f Fu(and)g Fo(u)g Fu(ha)m(v)m(e)212 450 y(a)38 b(common)g(reduct)f (whenev)m(er)g Fo(s)1471 417 y Fq(\003)1446 476 y(R)1510 450 y Fr( )g Fo(t)g Fr(!)1799 417 y Fq(\003)1799 476 y(R)1901 450 y Fo(u)p Fu(.)62 b(A)37 b(TRS)g Fr(R)g Fu(is)h(called)g Fs(gr)-5 b(ound)41 b(c)-5 b(on\015uent)47 b Fu(\(or)212 562 y(ground)29 b(Ch)m(urc)m(h-Rosser,)i(GCR\))f(if)g(all)i(its)e (ground)g(terms)g(are)h(con\015uen)m(t.)353 675 y(The)36 b(ab)s(o)m(v)m(e)i(de\014nition)e(of)g(strong)h(con\015uence)f (originates)i(from)e(Huet)h([6)q(])g(and)e(is)i(di\013eren)m(t)212 788 y(from)26 b(the)h(one)g(in)f(Dersho)m(witz)i(and)f(Jouannaud)e ([3].)40 b(They)26 b(call)i(a)f(TRS)f Fr(R)h Fu(strongly)g(con\015uen)m (t)f(if)212 915 y Fq(R)276 901 y Fr( )g(\001)f(!)534 915 y Fq(R)624 901 y Fr(\022)g(!)811 868 y Fn(=)811 928 y Fq(R)900 901 y Fr(\001)960 868 y Fn(=)955 928 y Fq(R)1019 901 y Fr( )p Fu(.)39 b(Klop)25 b([10)q(])g(calls)h(the)g(latter)g(prop) s(ert)m(y)e(sub)s(comm)m(utativit)m(y)j(\(W)m(CR)3518 868 y Fi(6)p Fn(1)3613 901 y Fu(\).)212 1014 y(The)j(de\014nitions)h (of)g(CR)f(and)g(W)m(CR)h(are)g(standard)f([10)q(,)h(2)q(].)42 b(F)-8 b(or)31 b(the)g(de\014nitions)g(of)g(UN,)g(UN)3573 981 y Fq(!)3648 1014 y Fu(,)212 1127 y(and)f(NF)g(w)m(e)h(follo)m(w)g (Klop)f([10)r(].)40 b(The)30 b(notion)h(of)f(CON)g(go)s(es)g(bac)m(k)h (to)g(Sc)m(hmidt-Sc)m(hau\031)g([17)q(];)f(for)212 1240 y(the)h(notion)f(CON)856 1207 y Fq(!)961 1240 y Fu(mo)s(dularit)m(y)g (has)h(b)s(een)e(studied)h([18)q(].)353 1353 y(A)c(rewrite)g(rule)f Fo(l)j Fr(!)d Fo(r)j Fu(is)e(called)h Fs(variable)i(pr)-5 b(eserving)34 b Fu(if)26 b(the)g(sets)g(of)g(v)-5 b(ariables)27 b(in)e Fo(l)j Fu(and)d Fo(r)j Fu(are)212 1466 y(the)h(same.)40 b(W)-8 b(e)29 b(call)h Fo(l)d Fr(!)e Fo(r)31 b Fs(c)-5 b(ol)5 b(lapsing)38 b Fu(if)28 b Fo(r)j Fu(is)d(a)h(v)-5 b(ariable)29 b(and)f Fs(line)-5 b(ar)39 b Fu(if)28 b(b)s(oth)g Fo(l)i Fu(and)d Fo(r)k Fu(are)e(linear)212 1579 y(terms;)40 b(a)d(term)f(is)h(linear)g(if)f(no)h(v)-5 b(ariable)37 b(o)s(ccurs)f(more)h(than)f(once)i(in)e(it.)60 b(A)36 b(TRS)g(is)g(v)-5 b(ariable)212 1692 y(preserving)38 b(\(linear\))h(if)f(all)h(its)f(rewrite)h(rules)e(are)i(so.)64 b(A)38 b(TRS)f(is)h(collapsing)i(if)e(it)h(con)m(tains)g(a)212 1804 y(collapsing)32 b(rewrite)e(rule.)353 1917 y(Next)36 b(w)m(e)f(illustrate)h(ho)m(w)f(w)m(e)g(relate)i(PCP)d(to)h(rewriting.) 55 b(An)34 b(arbitrary)h(PCP)f(instance)h Fo(P)212 2030 y Fu(admits)c(a)f(solution)h(if)g(and)e(only)i(if)f Fh(A)25 b Fr(!)1650 1997 y Fq(\003)1650 2062 y(R)1710 2071 y Fk(0)1745 2062 y Fn(\()p Fm(P)10 b Fn(\))1884 2030 y Fh(B)30 b Fu(for)g(the)g(TRS)439 2368 y Fr(R)516 2382 y Fn(0)556 2368 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)819 2181 y Fg(8)819 2263 y(<)819 2427 y(:)1130 2255 y Fh(A)55 b Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fh(c)p Fu(\)\))137 b(for)30 b(all)h(\()p Fo(\013;)15 b(\014)5 b Fu(\))27 b Fr(2)e Fo(P)945 2368 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s Fu(\))57 b Fr(!)e Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fo(y)s Fu(\)\))117 b(for)30 b(all)h(\()p Fo(\013;)15 b(\014)5 b Fu(\))27 b Fr(2)e Fo(P)941 2480 y Fh(f)6 b Fu(\()p Fo(x;)15 b(x)p Fu(\))57 b Fr(!)e Fh(B)212 2675 y Fu(This)35 b(observ)-5 b(ation)37 b(is)e(quite)h(simple:)52 b Fh(B)35 b Fu(can)h(b)s(e)f(reac)m(hed)i(if) f(and)f(only)h(if)f(a)h(term)g(of)g(the)g(shap)s(e)212 2788 y Fh(f)6 b Fu(\()p Fo(t;)15 b(t)p Fu(\))31 b(can)g(b)s(e)f(reac)m (hed,)h(and)f(this)g(can)h(b)s(e)f(reac)m(hed)h(from)f Fh(A)g Fu(if)g(and)g(only)h(if)f(a)h(PCP)f(solution)g(for)212 2901 y Fo(P)49 b Fu(exists.)59 b(T)-8 b(o)36 b(arriv)m(e)h(at)g (results)f(for)g(linear)g(TRSs)f(and)g(for)h(some)h(tec)m(hnical)h(con) m(v)m(enience)g(the)212 3014 y(basic)31 b(system)f Fr(R)818 3028 y Fn(0)858 3014 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(replaced)h(b)m (y)f(the)h(TRS)e Fr(R)2053 3028 y Fn(1)2093 3014 y Fu(\()p Fo(P)13 b Fu(\))31 b(that)g(consists)g(of)f(the)h(rules)665 3213 y Fh(A)55 b Fr(!)h Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fh(c)p Fu(\)\))481 3326 y Fh(f)h Fu(\()p Fo(x;)15 b(y)s Fu(\))56 b Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fo(y)s Fu(\)\))422 b Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))58 b Fr(!)d Fh(A)481 3439 y(f)6 b Fu(\()p Fo(x;)15 b(y)s Fu(\))56 b Fr(!)g Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))427 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\)\))59 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))481 3551 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s Fu(\))56 b Fr(!)g Fh(A)878 b(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))59 b Fr(!)c Fh(B)212 3751 y Fu(The)32 b(signature)g(of)h Fr(R)980 3765 y Fn(1)1019 3751 y Fu(\()p Fo(P)13 b Fu(\))33 b(consists)g(of)f (the)g(constan)m(ts)i Fh(A)p Fu(,)e Fh(B)p Fu(,)h(and)e Fh(c)p Fu(,)i(binary)f(function)g(sym)m(b)s(ols)212 3864 y Fh(f)42 b Fu(and)35 b Fh(g)q Fu(,)j(and)d(for)g(ev)m(ery)i(sym)m(b)s (ol)e Fo(a)h Fu(from)f(the)h(alphab)s(et)f(\000)h(of)f(the)h(PCP)f (instance)i Fo(P)48 b Fu(a)36 b(unary)212 3977 y(function)21 b(sym)m(b)s(ol)g Fo(a)p Fu(.)38 b(So)21 b(all)h(terms)g(m)m(ust)f(b)s (e)g(constructed)g(from)g(the)h(function)f(sym)m(b)s(ols)g(o)s (ccurring)212 4090 y(in)32 b(the)g(rewrite)g(rules)f(of)i Fr(R)1190 4104 y Fn(1)1229 4090 y Fu(\()p Fo(P)13 b Fu(\).)46 b(This)31 b(con)m(v)m(en)m(tion)j(is)e(also)h(adopted)f(for)g(the)g (TRSs)f(de\014ned)g(in)212 4203 y(later)g(sections.)41 b(W)-8 b(e)31 b(use)e Fh(sans)g(serif)36 b Fu(fon)m(t)30 b(for)f(\014xed)g(function)g(sym)m(b)s(ols)g(lik)m(e)i Fh(A)e Fu(and)g Fh(c)h Fu(but)f(not)g(for)212 4315 y(function)37 b(sym)m(b)s(ols)f(that)i(range)f(o)m(v)m(er)h(elemen)m(ts)h(of)e (certain)g(sets)h(\(lik)m(e)g Fo(a)f Fr(2)e Fu(\000\).)61 b(Moreo)m(v)m(er,)41 b(w)m(e)212 4428 y(drop)29 b(the)i(quan)m (ti\014cation)h(in)e(rule)g(sc)m(hemata)i(b)m(y)e(adopting)h(the)g (follo)m(wing)g(con)m(v)m(en)m(tions:)348 4614 y Fr(\017)46 b Fo(\013)31 b Fu(and)f Fo(\014)35 b Fu(range)c(o)m(v)m(er)h(all)f (pairs)f(\()p Fo(\013;)15 b(\014)5 b Fu(\))32 b(in)e Fo(P)13 b Fu(,)348 4801 y Fr(\017)46 b Fo(a)31 b Fu(ranges)f(o)m(v)m (er)i(all)f(elemen)m(ts)h(of)f(\000.)212 4987 y(So)25 b Fh(A)f Fr(!)i Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fh(c)p Fu(\)\))28 b(stands)c(for)g(the)h Fo(n)f Fu(rules)g Fh(A)h Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)2223 5001 y Fn(1)2263 4987 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)2464 5001 y Fn(1)2506 4987 y Fu(\()p Fh(c)p Fu(\)\),)27 b(.)15 b(.)h(.)f(,)26 b Fh(A)f Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)3204 5001 y Fm(n)3252 4987 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)3453 5001 y Fm(n)3502 4987 y Fu(\()p Fh(c)p Fu(\)\).)353 5100 y(The)24 b(TRS)f Fr(R)818 5114 y Fn(1)858 5100 y Fu(\()p Fo(P)13 b Fu(\))24 b(is)h(the)f(basis)g(for)g(man)m(y)g(of)h(our)e (relativ)m(e)k(undecidabilit)m(y)e(results.)38 b(W)-8 b(e)25 b(sho)m(w)212 5213 y(that)31 b(it)g(shares)f(the)h(desired)e(k)m (ey)j(prop)s(ert)m(y)d(with)h Fr(R)2059 5227 y Fn(0)2099 5213 y Fu(\()p Fo(P)13 b Fu(\).)1920 5462 y(4)p eop %%Page: 5 5 5 4 bop 212 337 a Ff(Lemma)35 b(2.1)46 b Fh(A)25 b Fr(!)939 304 y Fq(\003)939 368 y(R)999 377 y Fk(1)1033 368 y Fn(\()p Fm(P)10 b Fn(\))1172 337 y Fh(B)32 b Fs(if)h(and)g(only)h(if)e Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 486 y(Pr)-5 b(o)g(of)60 b Fu(Supp)s(ose)36 b Fo(\015)42 b Fr(2)37 b Fu(\000)1090 453 y Fn(+)1187 486 y Fu(is)g(a)h(solution)h(for)e Fo(P)13 b Fu(.)62 b(So)38 b Fo(\015)k Fu(=)37 b Fo(\013)2413 500 y Fm(i)2437 509 y Fk(1)2492 486 y Fr(\001)15 b(\001)g(\001)h Fo(\013)2671 500 y Fm(i)2695 508 y Fj(m)2795 486 y Fu(=)37 b Fo(\014)2954 500 y Fm(i)2978 509 y Fk(1)3032 486 y Fr(\001)15 b(\001)g(\001)i Fo(\014)3205 500 y Fm(i)3229 508 y Fj(m)3329 486 y Fu(for)38 b(some)212 599 y Fo(m)25 b Fl(>)g Fu(1)31 b(and)f(1)25 b Fl(6)g Fo(i)863 613 y Fn(1)903 599 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)1136 613 y Fm(m)1228 599 y Fl(6)25 b Fo(n)p Fu(.)40 b(W)-8 b(e)32 b(ha)m(v)m(e)g(the)e(follo)m(wing)i(rewrite)f(sequence)f(in)g Fr(R)3219 613 y Fn(1)3259 599 y Fu(\()p Fo(P)13 b Fu(\):)439 778 y Fh(A)25 b Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)768 792 y Fm(i)792 800 y Fj(m)856 778 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)1057 792 y Fm(i)1081 800 y Fj(m)1146 778 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)1408 740 y Fq(\003)1473 778 y Fh(f)6 b Fu(\()p Fo(\013)1600 792 y Fm(i)1624 801 y Fk(1)1678 778 y Fr(\001)15 b(\001)g(\001)i Fo(\013)1858 792 y Fm(i)1882 800 y Fj(m)1945 778 y Fu(\()p Fh(c)p Fu(\))p Fo(;)e(\014)2146 792 y Fm(i)2170 801 y Fk(1)2225 778 y Fr(\001)g(\001)g(\001)i Fo(\014)2398 792 y Fm(i)2422 800 y Fj(m)2485 778 y Fu(\()p Fh(c)p Fu(\)\))27 b(=)e Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\)\))525 915 y Fr(!)25 b Fh(g)q Fu(\()p Fo(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\)\))30 b Fr(!)1242 878 y Fn(+)1326 915 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))28 b Fr(!)d Fh(B)p Fo(:)212 1095 y Fu(Con)m(v)m(ersely)-8 b(,)47 b(supp)s(ose)41 b(that)i Fh(A)h Fr(!)1464 1062 y Fq(\003)1464 1126 y(R)1524 1135 y Fk(1)1559 1126 y Fn(\()p Fm(P)10 b Fn(\))1718 1095 y Fh(B)p Fu(.)76 b(Bey)m(ond)43 b(the)f(last)h Fh(A)f Fu(o)s(ccurring)g(in)g(this)g(rewrite)212 1207 y(sequence)31 b(it)g(is)f(of)h(the)f(form)439 1387 y Fh(A)25 b Fr(!)g Fh(f)6 b Fu(\()p Fo(\013)768 1401 y Fm(i)792 1409 y Fj(m)856 1387 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)1057 1401 y Fm(i)1081 1409 y Fj(m)1146 1387 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)1408 1349 y Fq(\003)1473 1387 y Fh(f)6 b Fu(\()p Fo(\013)1600 1401 y Fm(i)1624 1410 y Fk(1)1678 1387 y Fr(\001)15 b(\001)g(\001)i Fo(\013)1858 1401 y Fm(i)1882 1409 y Fj(m)1945 1387 y Fu(\()p Fh(c)p Fu(\))p Fo(;)e(\014)2146 1401 y Fm(i)2170 1410 y Fk(1)2225 1387 y Fr(\001)g(\001)g(\001)i Fo(\014)2398 1401 y Fm(i)2422 1409 y Fj(m)2485 1387 y Fu(\()p Fh(c)p Fu(\)\))525 1524 y Fr(!)25 b Fh(g)q Fu(\()p Fo(\013)780 1538 y Fm(i)804 1547 y Fk(1)860 1524 y Fr(\001)15 b(\001)g(\001)h Fo(\013)1039 1538 y Fm(i)1063 1546 y Fj(m)1126 1524 y Fu(\()p Fh(c)p Fu(\))p Fo(;)f(\014)1327 1538 y Fm(i)1351 1547 y Fk(1)1407 1524 y Fr(\001)g(\001)g(\001)h Fo(\014)1579 1538 y Fm(i)1603 1546 y Fj(m)1666 1524 y Fu(\()p Fh(c)p Fu(\)\))27 b Fr(!)1929 1487 y Fq(\003)1993 1524 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))641 1583 y Fg(|)p 682 1583 714 11 v 714 w({z)p 1478 1583 V 714 w(})2257 1524 y Fr(!)25 b Fh(B)212 1746 y Fu(for)32 b(some)h Fo(m)28 b Fl(>)g Fu(1)k(with)g(1)d Fl(6)f Fo(i)1280 1760 y Fn(1)1320 1746 y Fo(;)15 b(:)g(:)g(:)i(;)e(i) 1553 1760 y Fm(m)1648 1746 y Fl(6)28 b Fo(n)p Fu(.)46 b(In)32 b(the)g(underbraced)e(part)j(only)f(rewrite)g(rules)g(of)212 1859 y(the)e(form)f Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\)\))28 b Fr(!)e Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))31 b(are)f(used.)40 b(Hence)30 b Fo(\013)2227 1873 y Fm(i)2251 1882 y Fk(1)2305 1859 y Fr(\001)15 b(\001)g(\001)h Fo(\013)2484 1873 y Fm(i)2508 1881 y Fj(m)2571 1859 y Fu(\()p Fh(c)p Fu(\))27 b(=)e Fo(\014)2855 1873 y Fm(i)2879 1882 y Fk(1)2933 1859 y Fr(\001)15 b(\001)g(\001)h Fo(\014)3105 1873 y Fm(i)3129 1881 y Fj(m)3192 1859 y Fu(\()p Fh(c)p Fu(\),)31 b(giving)g(a)212 1971 y(solution)g(for)f Fo(P)13 b Fu(.)2812 b Fl(\003)353 2159 y Fu(Next)36 b(w)m(e)g(men)m(tion)f(t)m(w)m(o)i (theorems)e(that)h(will)f(b)s(e)f(used)h(a)g(n)m(um)m(b)s(er)f(of)h (times)h(in)e(the)i(sequel.)212 2272 y(The)30 b(\014rst)g(one)g(is)h(a) f(result)h(of)f(Huet)h([6)q(].)212 2459 y Ff(Theorem)k(2.2)46 b Fs(A)31 b(line)-5 b(ar)32 b(TRS)g(is)g(str)-5 b(ongly)33 b(c)-5 b(on\015uent)32 b(if)f(and)i(only)f(if)f(it)g(is)h(str)-5 b(ongly)33 b(close)-5 b(d.)43 b Fl(\003)353 2647 y Fu(Here)34 b(a)f(TRS)f Fr(R)h Fu(is)g(called)i Fs(str)-5 b(ongly)36 b(close)-5 b(d)44 b Fu(if)33 b(b)s(oth)g Fo(s)c Fr(!)2414 2614 y Fn(=)2414 2674 y Fq(R)2508 2647 y Fr(\001)2592 2614 y Fq(\003)2567 2674 y(R)2631 2647 y Fr( )h Fo(t)j Fu(and)f Fo(t)e Fr(!)3151 2614 y Fn(=)3151 2674 y Fq(R)3245 2647 y Fr(\001)3329 2614 y Fq(\003)3304 2674 y(R)3368 2647 y Fr( )g Fo(s)i Fu(for)212 2760 y(ev)m(ery)f(critical)i(pair)d Fr(h)p Fo(s;)15 b(t)p Fr(i)31 b Fu(of)f Fr(R)p Fu(.)353 2872 y(W)-8 b(e)39 b(use)f(the)g(preceding)f(result)h(to)h(relate)g Fr(R)1979 2886 y Fn(1)2018 2872 y Fu(\()p Fo(P)13 b Fu(\))39 b(to)f(some)h(prop)s(erties)e(in)g(the)h(con\015uence)212 2985 y(hierarc)m(h)m(y)-8 b(.)212 3173 y Ff(Lemma)35 b(2.3)46 b Fs(The)33 b(fol)5 b(lowing)33 b(statements)h(ar)-5 b(e)34 b(e)-5 b(quivalent:)320 3340 y(1.)45 b(The)33 b(TRS)g Fr(R)917 3354 y Fn(1)957 3340 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(has)h(the)f(normal)h(form)g(pr)-5 b(op)g(erty.)320 3519 y(2.)45 b(The)33 b(TRS)g Fr(R)917 3533 y Fn(1)957 3519 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(on\015uent.)320 3698 y(3.)45 b(The)33 b(TRS)g Fr(R)917 3712 y Fn(1)957 3698 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(gr)-5 b(ound)33 b(c)-5 b(on\015uent.)320 3877 y(4.)45 b(The)33 b(TRS)g Fr(R)917 3891 y Fn(1)957 3877 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent.)320 4057 y(5.)45 b(The)33 b(PCP)f(instanc)-5 b(e)34 b Fo(P)45 b Fs(admits)34 b(a)f(solution.)212 4243 y(Pr)-5 b(o)g(of)54 b Fu(Since)31 b(con\015uence)h(implies)f(the)g(normal)g(form)g(prop)s (ert)m(y)-8 b(,)32 b(lo)s(cal)g(con\015uence,)g(and)e(ground)212 4356 y(con\015uence,)37 b(according)f(to)g(Lemma)f(2.1)i(it)e (su\016ces)g(to)h(sho)m(w)f(that)h(\(i\))g Fh(A)d Fr(!)2967 4323 y Fq(\003)2967 4387 y(R)3027 4396 y Fk(1)3062 4387 y Fn(\()p Fm(P)10 b Fn(\))3209 4356 y Fh(B)34 b Fu(whenev)m(er)212 4480 y Fr(R)289 4494 y Fn(1)329 4480 y Fu(\()p Fo(P)13 b Fu(\))36 b(has)f(the)h(normal)f(form)g(prop)s(ert)m(y)-8 b(,)37 b(is)f(lo)s(cally)h(con\015uen)m(t,)g(or)f(is)f(ground)g (con\015uen)m(t,)i(and)212 4593 y(\(ii\))h Fr(R)447 4607 y Fn(1)486 4593 y Fu(\()p Fo(P)13 b Fu(\))37 b(is)f(con\015uen)m(t)h (whenev)m(er)f Fh(A)f Fr(!)1751 4560 y Fq(\003)1751 4624 y(R)1811 4633 y Fk(1)1846 4624 y Fn(\()p Fm(P)10 b Fn(\))1994 4593 y Fh(B)p Fu(.)59 b(F)-8 b(or)37 b(\(i\))g(w)m(e)g(note)g(that)g Fh(A)e Fr( )g Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))38 b Fr(!)d Fh(B)212 4718 y Fu(in)h Fr(R)401 4732 y Fn(1)441 4718 y Fu(\()p Fo(P)13 b Fu(\))37 b(with)f Fh(B)g Fu(a)h(normal)f(form,)i(hence)f Fh(A)e Fr(!)2019 4685 y Fq(\003)2019 4749 y(R)2079 4758 y Fk(1)2113 4749 y Fn(\()p Fm(P)10 b Fn(\))2262 4718 y Fh(B)36 b Fu(b)m(y)h (de\014nition)f(of)h(the)f(normal)h(form)212 4844 y(prop)s(ert)m(y)-8 b(,)28 b(lo)s(cal)h(con\015uence,)g(or)f(ground)e(con\015uence.)40 b(F)-8 b(or)29 b(\(ii\))g(w)m(e)f(consider)g(the)f(TRS)g Fr(R)3396 4811 y Fq(0)3396 4868 y Fn(1)3436 4844 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)212 4957 y Fr(R)289 4971 y Fn(1)329 4957 y Fu(\()p Fo(P)13 b Fu(\))24 b Fr([)f(f)p Fh(A)33 b Fr(!)g Fh(B)p Fo(;)15 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s Fu(\))35 b Fr(!)e Fh(B)p Fo(;)15 b Fh(g)q Fu(\()p Fo(x;)g(y)s Fu(\))35 b Fr(!)e Fh(B)p Fr(g)p Fu(.)55 b(Since)36 b Fh(A)c Fr(!)2474 4924 y Fq(\003)2474 4988 y(R)2534 4997 y Fk(1)2569 4988 y Fn(\()p Fm(P)10 b Fn(\))2716 4957 y Fh(B)p Fu(,)36 b(the)g(relations)g Fr(!)3465 4924 y Fq(\003)3465 4988 y(R)3525 4997 y Fk(1)3560 4988 y Fn(\()p Fm(P)10 b Fn(\))212 5083 y Fu(and)28 b Fr(!)478 5050 y Fq(\003)478 5115 y(R)538 5092 y Fe(0)538 5136 y Fk(1)572 5115 y Fn(\()p Fm(P)10 b Fn(\))714 5083 y Fu(coincide.)41 b(The)28 b(TRS)f Fr(R)1570 5050 y Fq(0)1570 5107 y Fn(1)1609 5083 y Fu(\()p Fo(P)13 b Fu(\))29 b(is)f(linear)h(and)e(strongly)i (closed)g(and)e(th)m(us)h(\(strongly\))212 5213 y(con\015uen)m(t)j(b)m (y)f(Theorem)g(2.2.)42 b(Hence)31 b Fr(R)1643 5227 y Fn(1)1682 5213 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)g(con\015uen)m(t)f(to) s(o.)1113 b Fl(\003)1920 5462 y Fu(5)p eop %%Page: 6 6 6 5 bop 353 337 a Fu(The)20 b(second)h(theorem)f(states)i(that)f(pro)m (ving)g(con\015uence)f(is)h(equiv)-5 b(alen)m(t)21 b(to)h(pro)m(ving)e (con\015uence)212 450 y(of)j(the)f(w)m(ell-t)m(yp)s(ed)i(terms)e (according)i(to)f(an)m(y)g(man)m(y-sorted)g(t)m(yp)s(e)g(discipline)f (whic)m(h)g(is)h(compatible)212 562 y(with)30 b(the)h(rewrite)g(system) g(under)e(consideration.)42 b(This)30 b(is)h(kno)m(wn)f(as)h(the)g Fs(p)-5 b(ersistenc)g(e)38 b Fu(\([20)r(]\))31 b(of)212 675 y(con\015uence.)212 888 y Ff(Theorem)k(2.4)46 b Fs(Con\015uenc)-5 b(e)33 b(is)g(a)g(p)-5 b(ersistent)34 b(pr)-5 b(op)g(erty.)1364 b Fl(\003)353 1100 y Fu(F)-8 b(or)28 b(instance,)g(to)g(pro)m(v)m(e)f (con\015uence)g(of)g(TRS)f Fr(R)2061 1114 y Fn(0)2100 1100 y Fu(\()p Fo(P)13 b Fu(\))27 b(ab)s(o)m(v)m(e)h(it)g(su\016ces)e (to)i(pro)m(v)m(e)f(con\015uence)212 1213 y(of)g(ev)m(ery)g(w)m(ell-t)m (yp)s(ed)h(term)e(according)i(to)f(the)g(man)m(y-sorted)g(t)m(yp)s(e)f (declarations:)41 b Fh(A)p Fo(;)15 b Fh(B)10 b Fu(:)30 b(1,)e Fh(f)16 b Fu(:)31 b(2)13 b Fr(\002)212 1326 y Fu(2)26 b Fr(!)f Fu(1,)j Fh(c)10 b Fu(:)31 b(2,)e(and)d Fo(a)10 b Fu(:)31 b(2)26 b Fr(!)f Fu(2)i(for)g(ev)m(ery)h Fo(a)f Fu(of)g(the)g(alphab)s(et)g(of)h Fo(P)13 b Fu(.)39 b(Note)29 b(that)e(these)h(declarations)212 1439 y(are)37 b(compatible)h(with)e Fr(R)1131 1453 y Fn(0)1171 1439 y Fu(\()p Fo(P)13 b Fu(\))37 b(since)g(for)f(ev)m(ery)h(rewrite)g(rule) f Fo(l)i Fr(!)d Fo(r)k Fu(b)s(oth)d(terms)h Fo(l)h Fu(and)e Fo(r)j Fu(are)212 1552 y(w)m(ell-t)m(yp)s(ed)g(\(b)m(y)f(assigning)g(t) m(yp)s(e)g(2)g(to)h(eac)m(h)g(v)-5 b(ariable\))39 b(and)e(ha)m(v)m(e)i (the)f(same)g(t)m(yp)s(e)g(1.)64 b(So)37 b(in)h(a)212 1665 y(con\015uence)30 b(pro)s(of)g(w)m(e)h(do)f(not)h(ha)m(v)m(e)g(to) h(consider)e(ill-t)m(yp)s(ed)h(terms)f(lik)m(e)i Fh(f)6 b Fu(\()p Fh(f)g Fu(\()p Fh(c)p Fo(;)15 b Fh(A)p Fu(\))p Fo(;)g Fh(B)p Fu(\).)353 1778 y(P)m(ersistence)24 b(is)e(closely)i (related)f(to)g(mo)s(dularit)m(y)-8 b(.)38 b(A)22 b(prop)s(ert)m(y)g (of)g(TRSs)f(is)i(said)f(to)h(b)s(e)e(mo)s(dular)212 1891 y(if)38 b(the)f(union)g(of)h(t)m(w)m(o)h(TRSs)d(with)h(the)h(prop) s(ert)m(y)f(and)g(disjoin)m(t)h(signatures)f(has)h(the)f(prop)s(ert)m (y)-8 b(.)212 2004 y(V)g(an)38 b(de)g(P)m(ol)h([14)q(])f(sho)m(w)m(ed)g (that)g(a)g(comp)s(onen)m(t)g(closed)h(prop)s(ert)m(y)e(is)h(p)s (ersisten)m(t)g(if)f(and)g(only)h(if)212 2117 y(it)d(is)f(mo)s(dular)e (for)i(man)m(y-sorted)h(TRSs.)50 b(\(A)34 b(prop)s(ert)m(y)f Fo(P)47 b Fu(of)34 b(TRSs)f(is)h(comp)s(onen)m(t)g(closed)h(if)f(a)212 2230 y(TRS)28 b Fr(R)h Fu(satis\014es)h Fo(P)42 b Fu(if)29 b(and)g(only)g(if)h(ev)m(ery)g(equiv)-5 b(alence)30 b(class)g(of)f Fr($)2625 2197 y Fq(\003)2625 2256 y(R)2719 2230 y Fu(has)g(the)g(prop) s(ert)m(y)g Fo(P)13 b Fu(.\))40 b(So)212 2342 y(pro)m(ving)27 b(p)s(ersistence)g(amoun)m(ts)g(to)h(t)m(yp)s(e-c)m(hec)m(king)h(mo)s (dularit)m(y)e(pro)s(ofs.)38 b(F)-8 b(or)28 b(the)f(sp)s(ecial)g(case)h (of)212 2455 y(con\015uence)i(this)h(is)f(done)g(in)g([1)q(])h(for)f (the)g(mo)s(dularit)m(y)h(pro)s(of)e(in)h([11)r(].)353 2568 y(T)-8 b(o)31 b(pro)m(v)m(e)g(some)f(of)g(the)g(relativ)m(e)j (undecidabilit)m(y)d(results)g(for)g(prop)s(erties)f(dealing)i(with)f (con-)212 2681 y(v)m(ersion)23 b(\(UN)h(and)e(CON\),)h(w)m(e)g(need)f (to)i(relate)g(solv)-5 b(abilit)m(y)25 b(of)e Fo(P)35 b Fu(to)24 b(the)f(existence)h(of)f(a)g Fs(c)-5 b(onversion)212 2794 y Fu(b)s(et)m(w)m(een)37 b Fh(A)g Fu(and)f Fh(B)p Fu(.)59 b(TRS)36 b Fr(R)1292 2808 y Fn(1)1331 2794 y Fu(\()p Fo(P)13 b Fu(\))37 b(is)g(not)g(suitable)g(for)g(this)f(purp)s (ose)f(b)s(ecause)i(in)f Fr(R)3329 2808 y Fn(1)3369 2794 y Fu(\()p Fo(P)13 b Fu(\))37 b(the)212 2907 y(terms)45 b Fh(A)g Fu(and)f Fh(B)g Fu(ma)m(y)i(b)s(e)e(con)m(v)m(ertible)j(ev)m (en)f(if)f Fo(P)58 b Fu(admits)45 b(no)g(solution.)85 b(F)-8 b(or)46 b(instance,)k(in)212 3020 y Fr(R)289 3034 y Fn(1)329 3020 y Fu(\()p Fr(f)p Fu(\(100)p Fo(;)15 b Fu(10\))p Fo(;)g Fu(\(10)p Fo(;)g Fu(1\))q Fr(g)p Fu(\))37 b(w)m(e)31 b(ha)m(v)m(e)g Fh(A)25 b Fr(!)g Fh(f)6 b Fu(\(100\()p Fh(c)p Fu(\))p Fo(;)15 b Fu(10\()p Fh(c)p Fu(\)\))31 b Fr( )25 b Fh(f)6 b Fu(\(0\()p Fh(c)p Fu(\))p Fo(;)15 b Fu(0\()p Fh(c)p Fu(\)\))30 b Fr(!)3003 2987 y Fq(\003)3068 3020 y Fh(B)p Fu(.)353 3133 y(Let)h Fr(R)593 3147 y Fn(2)633 3133 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)973 3100 y Fn(1)973 3157 y(2)1012 3133 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(R)1332 3100 y Fn(2)1332 3157 y(2)1371 3133 y Fu(\()p Fo(P)13 b Fu(\))31 b(with)f Fr(R)1827 3100 y Fn(1)1827 3157 y(2)1867 3133 y Fu(\()p Fo(P)13 b Fu(\))31 b(consisting)g(of)f(the)h(rules)610 3336 y Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g Fh(c)p Fu(\))34 b Fr(!)c Fh(A)481 3449 y(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(i)p Fu(\()p Fo(w)r Fu(\)\))35 b Fr(!)30 b Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(w)r Fu(\))212 3658 y(and)30 b Fr(R)466 3625 y Fn(2)466 3682 y(2)505 3658 y Fu(\()p Fo(P)13 b Fu(\))31 b(consisting)h(of)e(the)h(rules)481 3861 y Fh(f)6 b Fu(\()p Fo(\013)608 3875 y Fm(i)637 3861 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)850 3875 y Fm(i)879 3861 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(i)p Fu(\()p Fo(z)t Fu(\))p Fo(;)g(w)r Fu(\))34 b Fr(!)c Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t(;)g(i)p Fu(\()p Fo(w)r Fu(\)\))186 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))815 3974 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\))p Fo(;)g Fh(c)p Fu(\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))435 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))35 b Fr(!)30 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(z)t Fu(\))2467 4087 y Fh(g)q Fu(\()p Fh(c)p Fo(;)g Fh(c)p Fo(;)g Fh(c)p Fu(\))34 b Fr(!)c Fh(B)212 4289 y Fu(where)g Fo(i)g Fu(ranges)h(o)m(v)m (er)g(1,)g(.)15 b(.)h(.)f(,)31 b Fo(n)p Fu(.)40 b(Note)31 b(that)g Fr(R)1905 4303 y Fn(2)1945 4289 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(terminating)h(for)f(all)h(PCP)e(instances)i Fo(P)13 b Fu(.)212 4501 y Ff(Lemma)35 b(2.5)46 b Fh(A)25 b Fr($)939 4468 y Fq(\003)939 4532 y(R)999 4541 y Fk(2)1033 4532 y Fn(\()p Fm(P)10 b Fn(\))1172 4501 y Fh(B)32 b Fs(if)h(and)g(only)h(if)e Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 4634 y(Pr)-5 b(o)g(of)63 b Fu(First)40 b(supp)s(ose)f(that)i Fo(P)53 b Fu(admits)41 b(a)g(solution)f Fo(\015)47 b Fu(=)42 b Fo(\013)2399 4648 y Fm(i)2423 4657 y Fk(1)2477 4634 y Fr(\001)15 b(\001)g(\001)h Fo(\013)2656 4648 y Fm(i)2680 4656 y Fj(m)2785 4634 y Fu(=)42 b Fo(\014)2949 4648 y Fm(i)2973 4657 y Fk(1)3027 4634 y Fr(\001)15 b(\001)g(\001)h Fo(\014)3199 4648 y Fm(i)3223 4656 y Fj(m)3327 4634 y Fu(for)40 b(some)212 4746 y Fo(m)25 b Fl(>)g Fu(1)i(and)g(1)e Fl(6)g Fo(i)856 4760 y Fn(1)896 4746 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)1129 4760 y Fm(m)1221 4746 y Fl(6)25 b Fo(n)p Fu(.)39 b(Then)26 b(w)m(e)i(ha)m(v)m(e)g(the)f(follo)m(wing)h(con)m(v)m(ersion)g(b)s(et)m (w)m(een)g Fh(A)e Fu(and)g Fh(B)p Fu(:)439 4951 y Fh(A)f Fr( )g Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g Fh(c)p Fu(\))1054 4913 y Fq(\003)1094 4951 y Fr( )25 b Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(i)1550 4965 y Fm(m)1635 4951 y Fr(\001)g(\001)g (\001)h Fo(i)1787 4965 y Fn(1)1827 4951 y Fu(\()p Fh(c)p Fu(\)\))1998 4913 y Fq(\003)2038 4951 y Fr( )25 b Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(i)2658 4965 y Fn(1)2717 4951 y Fr(\001)g(\001)g(\001)h Fo(i)2869 4965 y Fm(m)2936 4951 y Fu(\()p Fh(c)p Fu(\))p Fo(;)f Fh(c)p Fu(\))525 5089 y Fr(!)25 b Fh(g)q Fu(\()p Fo(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(i)1157 5103 y Fn(1)1217 5089 y Fr(\001)g(\001)g(\001)h Fo(i)1369 5103 y Fm(m)1436 5089 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)1698 5051 y Fq(\003)1763 5089 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)2035 5103 y Fn(1)2093 5089 y Fr(\001)g(\001)g(\001)h Fo(i)2245 5103 y Fm(m)2312 5089 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)2574 5051 y Fq(\003)2639 5089 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fu(\))29 b Fr(!)c Fh(B)p Fo(:)1920 5462 y Fu(6)p eop %%Page: 7 7 7 6 bop 212 337 a Fu(Next)34 b(supp)s(ose)d(that)j Fh(A)e Fu(and)h Fh(B)f Fu(are)i(con)m(v)m(ertible.)50 b(Because)35 b Fr(R)2435 351 y Fn(2)2474 337 y Fu(\()p Fo(P)13 b Fu(\))34 b(is)f(v)-5 b(ariable)33 b(preserving)g(and)212 450 y(non-collapsing,) 46 b(it)d(follo)m(ws)g(that)g(all)f(steps)g(in)g(a)g(con)m(v)m(ersion)i (b)s(et)m(w)m(een)e Fh(A)g Fu(and)f Fh(B)h Fu(tak)m(e)h(place)212 562 y(at)g(the)f(ro)s(ot)h(p)s(osition.)75 b(It)43 b(is)f(also)h(easy)f (to)h(see)g(that)g(an)m(y)f(term)g(of)g(the)h(form)e Fh(f)6 b Fu(\()p Fo(t)3260 576 y Fn(1)3300 562 y Fo(;)15 b(t)3373 576 y Fn(2)3413 562 y Fo(;)g(t)3486 576 y Fn(3)3525 562 y Fo(;)g(t)3598 576 y Fn(4)3638 562 y Fu(\))212 675 y(in)42 b(a)g(con)m(v)m(ersion)h(b)s(et)m(w)m(een)g Fh(A)f Fu(and)f Fh(B)g Fu(satis\014es)i Fo(t)2007 689 y Fn(4)2091 675 y Fu(=)h Fo(i)2237 689 y Fm(m)2319 675 y Fr(\001)15 b(\001)g(\001)h Fo(i)2471 689 y Fn(1)2511 675 y Fu(\()p Fh(c)p Fu(\))43 b(with)f(1)j Fl(6)f Fo(i)3119 689 y Fn(1)3159 675 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)3392 689 y Fm(m)3503 675 y Fl(6)45 b Fo(n)212 788 y Fu(for)f(some)h Fo(m)j Fl(>)g Fu(0.)83 b(F)-8 b(urthermore,)48 b(there)d(exists)f(a)h(con)m(v) m(ersion)h(in)e Fr(R)2832 755 y Fn(1)2832 813 y(2)2871 788 y Fu(\()p Fo(P)13 b Fu(\))45 b(b)s(et)m(w)m(een)g Fh(A)f Fu(and)212 901 y Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(i)552 915 y Fm(m)637 901 y Fr(\001)g(\001)g(\001)h Fo(i)789 915 y Fn(1)829 901 y Fu(\()p Fh(c)p Fu(\)\))34 b(for)e(all)h Fo(m)28 b Fl(>)h Fu(0)j(and)g(1)d Fl(6)f Fo(i)1945 915 y Fn(1)1985 901 y Fo(;)15 b(:)g(:)g(:)h(;)f(i)2217 915 y Fm(m)2313 901 y Fl(6)29 b Fo(n)p Fu(.)46 b(W)-8 b(e)33 b(claim)g(that)g(there)g (exists)g(a)212 1014 y(con)m(v)m(ersion)439 1212 y Fh(A)25 b Fr($)616 1174 y Fq(\003)616 1244 y(R)676 1221 y Fk(1)676 1265 y(2)711 1244 y Fn(\()p Fm(P)10 b Fn(\))850 1212 y Fo(t)25 b Fr($)999 1174 y Fq(\003)999 1244 y(R)1059 1221 y Fk(2)1059 1265 y(2)1094 1244 y Fn(\()p Fm(P)10 b Fn(\))1232 1212 y Fh(B)212 1426 y Fu(for)38 b(some)g(term)h Fo(t)f Fu(=)f Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(i)1339 1440 y Fm(m)1424 1426 y Fr(\001)g(\001)g (\001)h Fo(i)1576 1440 y Fn(1)1616 1426 y Fu(\()p Fh(c)p Fu(\)\).)66 b(This)37 b(easily)i(follo)m(ws)g(from)f(the)g(preceding)g (obser-)212 1539 y(v)-5 b(ations)45 b(b)m(y)f(considering)g(the)g(last) h(application)h(of)e(a)g(rewrite)h(rule)e(of)i Fr(R)2925 1506 y Fn(1)2925 1564 y(2)2964 1539 y Fu(\()p Fo(P)13 b Fu(\))45 b(in)e(a)i(shortest)212 1652 y(con)m(v)m(ersion)30 b(b)s(et)m(w)m(een)f Fh(A)f Fu(and)g Fh(B)p Fu(.)39 b(Let)29 b Fr(R)1629 1619 y Fn(3)1629 1677 y(2)1669 1652 y Fu(\()p Fo(P)13 b Fu(\))29 b(b)s(e)e(the)i(TRS)f(obtained)g(from)g Fr(R)2988 1619 y Fn(2)2988 1677 y(2)3028 1652 y Fu(\()p Fo(P)13 b Fu(\))29 b(b)m(y)f(orien)m(ting)212 1765 y(all)439 1963 y Fh(f)6 b Fu(\()p Fo(\013)566 1977 y Fm(i)595 1963 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)808 1977 y Fm(i)838 1963 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(i)p Fu(\()p Fo(z)t Fu(\))p Fo(;)g(w)r Fu(\))29 b Fr(!)c Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t(;)g(i)p Fu(\()p Fo(w)r Fu(\)\))212 2161 y(rules)21 b(from)h(righ)m(t)g(to)h(left.)38 b(Clearly)-8 b(,)25 b Fr(R)1537 2128 y Fn(2)1537 2185 y(2)1577 2161 y Fu(\()p Fo(P)13 b Fu(\))22 b(and)f Fr(R)1985 2128 y Fn(3)1985 2185 y(2)2025 2161 y Fu(\()p Fo(P)13 b Fu(\))22 b(generate)h(the)f(same)h(con)m(v)m(ersion)g(relation.)212 2274 y(The)37 b(crucial)i(observ)-5 b(ation)38 b(is)f(that)i Fr(R)1577 2241 y Fn(1)1577 2298 y(2)1616 2274 y Fu(\()p Fo(P)13 b Fu(\))38 b(and)f Fr(R)2056 2241 y Fn(3)2056 2298 y(2)2096 2274 y Fu(\()p Fo(P)13 b Fu(\))38 b(are)g(orthogonal)h (and)e(th)m(us)g(con\015uen)m(t.)212 2387 y(Because)29 b Fh(A)d Fu(and)h Fh(B)g Fu(are)g(normal)g(forms,)h(the)f(ab)s(o)m(v)m (e)h(con)m(v)m(ersion)h(b)s(et)m(w)m(een)f Fh(A)e Fu(and)h Fh(B)f Fu(has)h(the)h(form)439 2584 y Fh(A)698 2547 y Fq(\003)529 2617 y(R)589 2594 y Fk(1)589 2638 y(2)624 2617 y Fn(\()p Fm(P)10 b Fn(\))738 2584 y Fr( )25 b Fo(t)g Fr(!)1003 2547 y Fq(\003)1003 2616 y(R)1063 2594 y Fk(3)1063 2638 y(2)1098 2616 y Fn(\()p Fm(P)10 b Fn(\))1237 2584 y Fh(B)p Fo(:)212 2794 y Fu(It)31 b(follo)m(ws)g(that)439 2992 y Fh(A)25 b Fr( )g Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g Fh(c)p Fu(\))1054 2954 y Fq(\003)1094 2992 y Fr( )25 b Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fo(;)g(i)1550 3006 y Fm(m)1635 2992 y Fr(\001)g(\001)g(\001)h Fo(i)1787 3006 y Fn(1)1827 2992 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)2089 2954 y Fq(\003)2154 2992 y Fh(f)6 b Fu(\()p Fo(\015)2270 3006 y Fn(1)2310 2992 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)2507 3006 y Fn(2)2548 2992 y Fu(\()p Fh(c)p Fu(\))p Fo(;)g(i)2729 3006 y Fn(1)2785 2992 y Fr(\001)g(\001)g(\001)i Fo(i)2938 3006 y Fm(m)3005 2992 y Fu(\()p Fh(c)p Fu(\))p Fo(;)e Fh(c)p Fu(\))525 3130 y Fr(!)25 b Fh(g)q Fu(\()p Fo(\015)769 3144 y Fn(1)810 3130 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)1007 3144 y Fn(2)1048 3130 y Fu(\()p Fh(c)p Fu(\))p Fo(;)g(i)1229 3144 y Fn(1)1285 3130 y Fr(\001)g(\001)g(\001)i Fo(i)1438 3144 y Fm(m)1505 3130 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)1767 3092 y Fq(\003)1832 3130 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)2104 3144 y Fn(1)2161 3130 y Fr(\001)g(\001)g(\001)i Fo(i)2314 3144 y Fm(m)2381 3130 y Fu(\()p Fh(c)p Fu(\)\))26 b Fr(!)2643 3092 y Fq(\003)2708 3130 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g Fh(c)p Fu(\))29 b Fr(!)c Fh(B)212 3327 y Fu(with)38 b Fo(\015)474 3341 y Fn(1)551 3327 y Fu(=)f Fo(\013)717 3341 y Fm(i)741 3350 y Fk(1)795 3327 y Fr(\001)15 b(\001)g(\001)i Fo(\013)975 3341 y Fm(i)999 3349 y Fj(m)1099 3327 y Fu(and)37 b Fo(\015)1330 3341 y Fn(2)1408 3327 y Fu(=)g Fo(\014)1567 3341 y Fm(i)1591 3350 y Fk(1)1645 3327 y Fr(\001)15 b(\001)g(\001)i Fo(\014)1818 3341 y Fm(i)1842 3349 y Fj(m)1905 3327 y Fu(.)63 b(The)37 b(step)h Fh(f)6 b Fu(\()p Fr(\001)15 b(\001)g(\001)i Fu(\))38 b Fr(!)f Fh(g)q Fu(\()p Fr(\001)15 b(\001)g(\001)j Fu(\))38 b(is)g(p)s(ossible)f(only)212 3440 y(if)d Fo(m)d Fl(>)g Fu(1)j(and)g(the)g(sequence)g(from)g Fh(g)q Fu(\()p Fo(\015)1659 3454 y Fn(1)1699 3440 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)1896 3454 y Fn(2)1937 3440 y Fu(\()p Fh(c)p Fu(\))p Fo(;)g(i)2118 3454 y Fn(1)2175 3440 y Fr(\001)g(\001)g(\001)h Fo(i)2327 3454 y Fm(m)2394 3440 y Fu(\()p Fh(c)p Fu(\)\))35 b(to)g Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)2961 3454 y Fn(1)3018 3440 y Fr(\001)g(\001)g(\001)i Fo(i)3171 3454 y Fm(m)3237 3440 y Fu(\()p Fh(c)p Fu(\)\))36 b(en)m(tails)212 3553 y(that)31 b Fo(\015)456 3567 y Fn(1)521 3553 y Fu(=)25 b Fo(\015)664 3567 y Fn(2)703 3553 y Fu(.)41 b(W)-8 b(e)32 b(conclude)e(that)h Fo(P)44 b Fu(admits)30 b(a)h(solution.)1293 b Fl(\003)212 3839 y Ft(3)135 b(SCR)45 b Fd(\))f Ft(CR)212 4041 y Fu(Let)31 b Fr(R)452 4055 y Fn(3)491 4041 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)831 4055 y Fn(1)871 4041 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(f)p Fh(B)25 b Fr(!)g Fh(C)p Fo(;)15 b Fh(C)26 b Fr(!)f Fh(A)p Fr(g)p Fu(.)212 4248 y Ff(Lemma)35 b(3.1)46 b Fs(The)33 b(TRS)g Fr(R)1240 4262 y Fn(3)1279 4248 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(for)f(every)f(PCP)g(instanc)-5 b(e)34 b Fo(P)13 b Fs(.)212 4380 y(Pr)-5 b(o)g(of)60 b Fu(One)36 b(easily)i(c)m(hec)m(ks)h(that)e(the)h(linear)f(TRS)f Fr(R)2148 4347 y Fq(0)2148 4404 y Fn(3)2188 4380 y Fu(\()p Fo(P)13 b Fu(\))37 b(=)f Fr(R)2550 4394 y Fn(3)2589 4380 y Fu(\()p Fo(P)13 b Fu(\))26 b Fr([)e(f)p Fh(B)36 b Fr(!)h Fh(A)p Fr(g)g Fu(is)g(strongly)212 4493 y(closed)46 b(hence)g (\(strongly\))g(con\015uen)m(t)g(b)m(y)f(Theorem)g(2.2.)87 b(Since)45 b(the)h(relations)g Fr(!)3273 4460 y Fq(\003)3273 4524 y(R)3333 4533 y Fk(3)3368 4524 y Fn(\()p Fm(P)10 b Fn(\))3527 4493 y Fu(and)212 4617 y Fr(!)303 4584 y Fq(\003)303 4650 y(R)363 4627 y Fe(0)363 4671 y Fk(3)398 4650 y Fn(\()p Fm(P)g Fn(\))542 4617 y Fu(coincide,)31 b Fr(R)992 4631 y Fn(3)1032 4617 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(con\015uen)m(t.)1922 b Fl(\003)212 4840 y Ff(Lemma)35 b(3.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 4854 y Fn(3)1279 4840 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(str)-5 b(ongly)34 b(c)-5 b(on\015uent)34 b(if)e(and)i(only)f (if)g Fo(P)45 b Fs(admits)34 b(a)f(solution.)212 4973 y(Pr)-5 b(o)g(of)56 b Fu(In)32 b(a)h(shortest)g(rewrite)g(sequence)g (from)g Fh(A)f Fu(to)i Fh(B)e Fu(the)h(rules)f Fh(B)d Fr(!)h Fh(C)j Fu(and)f Fh(C)d Fr(!)g Fh(A)k Fu(are)g(not)212 5086 y(used.)45 b(Hence)33 b Fh(A)28 b Fr(!)912 5053 y Fq(\003)912 5117 y(R)972 5126 y Fk(3)1006 5117 y Fn(\()p Fm(P)10 b Fn(\))1148 5086 y Fh(B)32 b Fu(if)g(and)f(only)h(if)g Fh(A)c Fr(!)1970 5053 y Fq(\003)1970 5117 y(R)2030 5126 y Fk(1)2065 5117 y Fn(\()p Fm(P)10 b Fn(\))2206 5086 y Fh(B)p Fu(.)46 b(According)32 b(to)h(Lemma)f(2.1)i(w)m(e)e(ha)m(v)m (e)212 5210 y(to)g(sho)m(w)e(that)i Fr(R)824 5224 y Fn(3)863 5210 y Fu(\()p Fo(P)13 b Fu(\))32 b(is)f(strongly)g(con\015uen)m(t)g (if)g(and)f(only)h(if)g Fh(A)25 b Fr(!)2594 5177 y Fq(\003)2594 5242 y(R)2654 5251 y Fk(3)2689 5242 y Fn(\()p Fm(P)10 b Fn(\))2829 5210 y Fh(B)p Fu(.)42 b(In)30 b Fr(R)3148 5224 y Fn(3)3187 5210 y Fu(\()p Fo(P)13 b Fu(\))32 b(w)m(e)f(ha)m(v)m (e)1920 5462 y(7)p eop %%Page: 8 8 8 7 bop 212 337 a Fh(B)25 b Fr( )g Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))28 b Fr(!)d Fh(A)p Fu(.)39 b(If)27 b Fr(R)1084 351 y Fn(3)1123 337 y Fu(\()p Fo(P)13 b Fu(\))28 b(is)f(strongly)g(con\015uen)m(t)g(then)g Fh(B)e Fr(!)2497 304 y Fn(=)2581 337 y Fr(\001)2632 304 y Fq(\003)2671 337 y Fr( )g Fh(A)p Fu(,)j(so)f(either)g Fh(B)3349 304 y Fq(\003)3389 337 y Fr( )e Fh(A)h Fu(or)212 450 y Fh(B)h Fr(!)g Fh(C)503 417 y Fq(\003)542 450 y Fr( )g Fh(A)p Fu(.)44 b(Since)31 b(an)m(y)h(reduction)g(sequence)f(from)g Fh(A)g Fu(to)i Fh(C)e Fu(m)m(ust)g(pass)h(through)e Fh(B)p Fu(,)i(in)f(b)s(oth)212 562 y(cases)g(w)m(e)g(ha)m(v)m(e)h(the)e (desired)g Fh(A)25 b Fr(!)1426 529 y Fq(\003)1491 562 y Fh(B)p Fu(.)353 675 y(Con)m(v)m(ersely)-8 b(,)47 b(if)41 b Fh(A)j Fr(!)1136 642 y Fq(\003)1220 675 y Fh(B)d Fu(then)h(one)g (easily)h(c)m(hec)m(ks)g(that)f Fr(R)2558 689 y Fn(3)2598 675 y Fu(\()p Fo(P)13 b Fu(\))42 b(is)g(strongly)g(closed)h(and)212 788 y(therefore)31 b(strongly)g(con\015uen)m(t)f(b)m(y)g(Theorem)g (2.2.)1616 b Fl(\003)212 998 y Ff(Corollary)35 b(3.3)46 b Fs(Str)-5 b(ong)29 b(c)-5 b(on\015uenc)g(e)29 b(is)f(not)h(c)-5 b(o-semi-de)g(cidable)28 b(for)h(line)-5 b(ar)29 b(c)-5 b(on\015uent)29 b(TRSs.)41 b Fl(\003)353 1208 y Fu(In)30 b(the)g(pro)s(ofs)g(of)g(semi-decidabilit)m(y)i(in)e(this)h(pap)s(er)e (w)m(e)i(mak)m(e)g(use)f(of)g(the)h(observ)-5 b(ation)31 b(that)212 1321 y(the)24 b(set)h(of)f(con)m(v)m(ersions)h(is)f (recursiv)m(ely)h(en)m(umerable.)38 b(This)23 b(observ)-5 b(ation)25 b(is)f(an)g(easy)h(consequence)212 1434 y(of)45 b(the)g(fact)h(that)g(there)f(are)g(only)g(coun)m(table)h(in\014nitely) f(man)m(y)g(terms)g(\(b)s(ecause)g(the)h(set)f(of)212 1547 y(v)-5 b(ariables)31 b(is)g(assumed)e(to)i(b)s(e)f(coun)m(tably)h (in\014nite\))g(and)e(con)m(v)m(ersions.)212 1757 y Ff(Theorem)35 b(3.4)46 b Fs(Str)-5 b(ong)34 b(c)-5 b(on\015uenc)g(e)33 b(is)g(semi-de)-5 b(cidable)33 b(for)h(line)-5 b(ar)33 b(TRSs.)212 1890 y(Pr)-5 b(o)g(of)52 b Fu(Let)29 b Fr(R)f Fu(b)s(e)g(an)h(arbitrary)f(\014nite)g(linear)h(TRS.)f(According)i(to)f (Theorem)f(2.2)i Fr(R)e Fu(is)h(strongly)212 2003 y(con\015uen)m(t)34 b(if)g(and)g(only)g(if)g Fo(s)d Fr(!)1330 1970 y Fn(=)1330 2030 y Fq(R)1426 2003 y Fr(\001)1511 1970 y Fq(\003)1486 2030 y(R)1551 2003 y Fr( )g Fo(t)j Fu(and)f Fo(t)e Fr(!)2075 1970 y Fn(=)2075 2030 y Fq(R)2171 2003 y Fr(\001)2257 1970 y Fq(\003)2232 2030 y(R)2296 2003 y Fr( )h Fo(s)h Fu(for)h(ev)m(ery)h(critical)h(pair)e Fr(h)p Fo(s;)15 b(t)p Fr(i)34 b Fu(of)212 2116 y Fr(R)p Fu(.)53 b(By)34 b(en)m(umerating)h(and)f(insp)s(ecting)g(all)h(con)m(v)m(ersions)h(b)s (et)m(w)m(een)e Fo(s)g Fu(and)g Fo(t)g Fu(w)m(e)h(easily)g(obtain)g(a) 212 2228 y(semi-decision)d(pro)s(cedure)e(for)g(the)h(problem)f (whether)h(b)s(oth)f Fo(s)25 b Fr(!)2561 2195 y Fn(=)2561 2255 y Fq(R)2651 2228 y Fr(\001)2732 2195 y Fq(\003)2707 2255 y(R)2771 2228 y Fr( )h Fo(t)31 b Fu(and)f Fo(t)c Fr(!)3279 2195 y Fn(=)3279 2255 y Fq(R)3369 2228 y Fr(\001)3449 2195 y Fq(\003)3424 2255 y(R)3489 2228 y Fr( )f Fo(s)p Fu(.)212 2341 y(Since)32 b Fr(R)g Fu(has)g(\014nitely)g(man)m(y)g (critical)i(pairs)e(w)m(e)g(obtain)h(a)f(semi-decision)h(pro)s(cedure)e (for)h(strong)212 2454 y(con\015uence)e(b)m(y)h(applying)f(the)h (previous)e(pro)s(cedure)h(to)h(all)g(critical)h(pairs)e(in)g (parallel.)327 b Fl(\003)353 2664 y Fu(Linearit)m(y)42 b(is)f(essen)m(tial)i(in)d(the)h(ab)s(o)m(v)m(e)h(pro)s(of.)72 b(W)-8 b(e)42 b(conjecture)g(that)f(for)g(arbitrary)f(TRSs)212 2777 y(strong)31 b(con\015uence)f(is)g(not)h(semi-decidable.)212 3063 y Ft(4)135 b(CR)45 b Fd(\))g Ft(W)l(CR)212 3266 y Fu(Let)31 b Fr(R)452 3280 y Fn(4)491 3266 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)831 3280 y Fn(1)871 3266 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(f)p Fh(B)25 b Fr(!)g Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))p Fo(;)g Fh(B)27 b Fr(!)e Fh(C)p Fr(g)p Fu(.)212 3476 y Ff(Lemma)35 b(4.1)46 b Fs(The)33 b(TRS)g Fr(R)1240 3490 y Fn(4)1279 3476 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(on\015uent)34 b(for)f(every)g(PCP)f(instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 3609 y(Pr)-5 b(o)g(of)53 b Fu(One)30 b(easily)i(c)m(hec)m(ks)g(that)f(all)g(critical)h(pairs)e (of)h Fr(R)2229 3623 y Fn(4)2268 3609 y Fu(\()p Fo(P)13 b Fu(\))31 b(are)g(joinable.)676 b Fl(\003)212 3819 y Ff(Lemma)35 b(4.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 3833 y Fn(4)1279 3819 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e(and)h(only)h(if)e Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 3951 y(Pr)-5 b(o)g(of)52 b Fu(In)27 b(a)i(shortest)g Fr(R)1079 3965 y Fn(4)1118 3951 y Fu(\()p Fo(P)13 b Fu(\)-reduction)30 b(sequence)f(from)f Fh(A)g Fu(to)h Fh(B)f Fu(the)h(rewrite)f(rules)g Fh(B)d Fr(!)g Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))212 4064 y(and)37 b Fh(B)g Fr(!)g Fh(C)g Fu(are)h(not)g(used.)61 b(Hence)38 b Fh(A)f Fr(!)1775 4031 y Fq(\003)1775 4096 y(R)1835 4105 y Fk(4)1870 4096 y Fn(\()p Fm(P)10 b Fn(\))2020 4064 y Fh(B)37 b Fu(if)h(and)f(only)g(if)h Fh(A)f Fr(!)2879 4031 y Fq(\003)2879 4096 y(R)2939 4105 y Fk(1)2973 4096 y Fn(\()p Fm(P)10 b Fn(\))3124 4064 y Fh(B)p Fu(.)62 b(According)212 4189 y(to)34 b(Lemma)g(2.1)h(w)m(e)f(ha)m(v)m(e)h(to)f (sho)m(w)f(that)h Fr(R)1773 4203 y Fn(4)1813 4189 y Fu(\()p Fo(P)13 b Fu(\))34 b(is)g(con\015uen)m(t)f(if)h(and)f(only)h(if)f Fh(A)d Fr(!)3215 4156 y Fq(\003)3215 4220 y(R)3275 4229 y Fk(4)3310 4220 y Fn(\()p Fm(P)10 b Fn(\))3454 4189 y Fh(B)p Fu(.)50 b(In)212 4313 y Fr(R)289 4327 y Fn(4)329 4313 y Fu(\()p Fo(P)13 b Fu(\))39 b(w)m(e)g(ha)m(v)m(e)g Fh(A)g Fr( )f Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))41 b Fr( )e Fh(B)f Fr(!)h Fh(C)p Fu(.)65 b(If)38 b Fr(R)2045 4327 y Fn(4)2085 4313 y Fu(\()p Fo(P)13 b Fu(\))39 b(is)f(con\015uen)m(t)h(then)f Fh(A)g Fr(!)3169 4280 y Fq(\003)3247 4313 y Fh(C)h Fu(whic)m(h)f(is)212 4426 y(equiv)-5 b(alen)m(t)32 b(to)f Fh(A)25 b Fr(!)932 4393 y Fq(\003)996 4426 y Fh(B)p Fu(.)353 4539 y(Con)m(v)m(ersely)-8 b(,)35 b(if)e Fh(A)28 b Fr(!)1100 4506 y Fq(\003)1169 4539 y Fh(B)k Fu(then)g(w)m(e)i(obtain)f(con\015uence)f(b)m(y)h (considering)g(the)g(TRS)e Fr(R)3392 4506 y Fq(0)3392 4563 y Fn(4)3432 4539 y Fu(\()p Fo(P)13 b Fu(\))30 b(=)212 4652 y Fr(R)289 4666 y Fn(4)329 4652 y Fu(\()p Fo(P)13 b Fu(\))27 b Fr([)f(f)p Fh(A)42 b Fr(!)g Fh(C)p Fo(;)15 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s Fu(\))43 b Fr(!)e Fh(C)p Fo(;)15 b Fh(g)q Fu(\()p Fo(x;)g(y)s Fu(\))44 b Fr(!)e Fh(C)p Fr(g)p Fu(.)70 b(One)40 b(easily)i(sho)m(ws)d(that)i Fr(R)3128 4619 y Fq(0)3128 4676 y Fn(4)3168 4652 y Fu(\()p Fo(P)13 b Fu(\))41 b(is)f(linear)212 4765 y(and)33 b(strongly)i(closed) f(and)f(th)m(us)h(\(strongly\))h(con\015uen)m(t)f(b)m(y)g(Theorem)f (2.2.)52 b(Since)34 b(the)g(relations)212 4878 y Fr(!)303 4845 y Fq(\003)303 4909 y(R)363 4918 y Fk(4)398 4909 y Fn(\()p Fm(P)10 b Fn(\))542 4878 y Fu(and)29 b Fr(!)809 4845 y Fq(\003)809 4910 y(R)869 4887 y Fe(0)869 4931 y Fk(4)904 4910 y Fn(\()p Fm(P)10 b Fn(\))1048 4878 y Fu(coincide,)32 b Fr(R)1499 4892 y Fn(4)1538 4878 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)g(con\015uen)m(t.)1415 b Fl(\003)212 5100 y Ff(Corollary)35 b(4.3)46 b Fs(Con\015uenc)-5 b(e)40 b(is)g(not)g(c)-5 b(o-semi-de)g(cidable)41 b(for)f(line)-5 b(ar)41 b(lo)-5 b(c)g(al)5 b(ly)41 b(c)-5 b(on\015uent)41 b(TRSs.)212 5213 y Fl(\003)1920 5462 y Fu(8)p eop %%Page: 9 9 9 8 bop 353 337 a Fu(T)-8 b(o)28 b(sho)m(w)g(the)g (non-semi-decidabilit)m(y)h(of)f(con\015uence)f(for)h(the)f(class)i(of) f(linear)g(lo)s(cally)h(con\015u-)212 450 y(en)m(t)i(TRSs)e(w)m(e)i (need)f(a)h(di\013eren)m(t)g(construction.)41 b(Let)31 b Fr(R)2192 464 y Fn(5)2232 450 y Fu(\()p Fo(P)13 b Fu(\))31 b(consist)g(of)f(the)h(rules)481 645 y(\(1\))256 b Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))34 b Fr(!)c Fh(A)760 b Fu(\(4\))31 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\)\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))481 758 y(\(2\))237 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))33 b Fr(!)d Fh(f)6 b Fu(\()p Fo(\013)1545 772 y Fm(i)1574 758 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)1787 772 y Fm(i)1816 758 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))339 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(b)p Fu(\()p Fo(y)s Fu(\)\))34 b Fr(!)c Fh(A)181 b Fu(if)31 b Fo(a)25 b Fr(6)p Fu(=)g Fo(b)481 871 y Fu(\(2\))31 b Fh(f)6 b Fu(\()p Fo(\013)754 885 y Fm(i)783 871 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)996 885 y Fm(i)1026 871 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))32 b Fr(!)e Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))662 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\))34 b Fr(!)c Fh(A)481 984 y Fu(\(3\))345 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g Fh(c)p Fu(\))32 b Fr(!)e Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))837 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b(a)p Fu(\()p Fo(x)p Fu(\)\))34 b Fr(!)c Fh(A)212 1174 y Fu(with)h Fo(b)g Fu(ranging)h(o)m(v)m(er)g (all)g(elemen)m(ts)h(of)e(the)h(alphab)s(et)f(\000)g(of)g Fo(P)13 b Fu(.)44 b(The)30 b(n)m(um)m(b)s(ers)g(in)h(fron)m(t)g(of)h (some)212 1287 y(of)f(the)f(rewrite)h(rules)f(are)g(used)g(for)g (reference)h(in)f(the)h(pro)s(of)e(of)i(Lemma)f(4.5)i(b)s(elo)m(w.)212 1494 y Ff(Lemma)j(4.4)46 b Fs(The)33 b(TRS)g Fr(R)1240 1508 y Fn(5)1279 1494 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(on\015uent)34 b(for)f(every)g(PCP)f(instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 1626 y(Pr)-5 b(o)g(of)53 b Fu(One)30 b(easily)i(c)m(hec)m(ks)g(that)f (all)g(critical)h(pairs)e(of)h Fr(R)2229 1640 y Fn(5)2268 1626 y Fu(\()p Fo(P)13 b Fu(\))31 b(are)g(joinable.)676 b Fl(\003)212 1833 y Ff(Lemma)35 b(4.5)46 b Fs(The)33 b(TRS)g Fr(R)1240 1847 y Fn(5)1279 1833 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e(and)h(only)h(if)e Fo(P)46 b Fs(admits)34 b(no)f(solution.)212 1965 y(Pr)-5 b(o)g(of)62 b Fu(If)39 b Fo(P)53 b Fu(admits)40 b(a)g(solution)g Fo(\015)46 b Fu(=)40 b Fo(\013)1703 1979 y Fm(i)1727 1988 y Fk(1)1781 1965 y Fr(\001)15 b(\001)g(\001)h Fo(\013)1960 1979 y Fm(i)1984 1987 y Fj(m)2088 1965 y Fu(=)40 b Fo(\014)2250 1979 y Fm(i)2274 1988 y Fk(1)2329 1965 y Fr(\001)15 b(\001)g(\001)h Fo(\014)2501 1979 y Fm(i)2525 1987 y Fj(m)2628 1965 y Fu(for)39 b(some)h Fo(m)h Fl(>)f Fu(1)g(and)f(1)j Fl(6)212 2078 y Fo(i)243 2092 y Fn(1)283 2078 y Fo(;)15 b(:)g(:)g(:)h(;)f(i)515 2092 y Fm(m)608 2078 y Fl(6)25 b Fo(n)30 b Fu(then)g(w)m(e)h(obtain)g (the)f(div)m(erging)h(reductions)f(in)h Fr(R)2587 2092 y Fn(5)2626 2078 y Fu(\()p Fo(P)13 b Fu(\))31 b(consisting)g(of)439 2276 y Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\))30 b Fr(!)1108 2239 y Fn(+)1192 2276 y Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)1452 2290 y Fm(m)1536 2276 y Fr(\001)g(\001)g(\001)i Fo(i)1689 2290 y Fn(1)1728 2276 y Fu(\()p Fh(c)p Fu(\)\))27 b Fr(!)e Fh(A)212 2474 y Fu(and)439 2673 y Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\))30 b Fr(!)25 b Fh(g)q Fu(\()p Fo(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\)\))30 b Fr(!)1734 2635 y Fn(+)1818 2673 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))p Fo(:)212 2871 y Fu(Since)30 b Fh(A)g Fu(and)g Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))33 b(are)e(di\013eren)m(t)g (normal)f(forms,)g Fr(R)2158 2885 y Fn(5)2198 2871 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(not)h(con\015uen)m(t.)353 2984 y(Con)m(v)m(ersely)-8 b(,)42 b(assume)c(that)h Fr(R)1445 2998 y Fn(5)1484 2984 y Fu(\()p Fo(P)13 b Fu(\))39 b(is)f(not)g (con\015uen)m(t;)43 b(w)m(e)38 b(ha)m(v)m(e)i(to)e(pro)m(v)m(e)h(that)g Fo(P)51 b Fu(admits)212 3097 y(a)43 b(solution.)76 b(Consider)41 b(the)i(man)m(y-sorted)g(t)m(yp)s(e)f(declarations)h Fh(A)10 b Fu(:)35 b(1,)45 b Fh(c)10 b Fu(:)35 b(2,)46 b Fo(i)10 b Fu(:)35 b(2)45 b Fr(!)g Fu(2)d(for)g(all)212 3209 y Fo(i)36 b Fr(2)g(f)p Fu(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)p Fr(g)p Fu(,)39 b Fo(a)10 b Fu(:)33 b(2)j Fr(!)g Fu(2)h(for)g(all)g Fo(a)f Fr(2)g Fu(\000,)i Fh(f)16 b Fu(:)33 b(2)25 b Fr(\002)f Fu(2)h Fr(\002)f Fu(2)36 b Fr(!)g Fu(1,)j(and)d Fh(g)11 b Fu(:)34 b(2)25 b Fr(\002)f Fu(2)36 b Fr(!)g Fu(1,)j(whic)m(h)d(is)212 3322 y(compatible)j(with)e Fr(R)975 3336 y Fn(5)1014 3322 y Fu(\()p Fo(P)13 b Fu(\))38 b(b)m(y)f(assigning)h(t)m(yp)s(e)g(2) f(to)h(ev)m(ery)g(v)-5 b(ariable.)63 b(F)-8 b(rom)37 b(Theorem)g(2.4)i(w)m(e)212 3435 y(conclude)33 b(that)g Fr(R)864 3449 y Fn(5)904 3435 y Fu(\()p Fo(P)13 b Fu(\))33 b(admits)f(div)m(erging)i(w)m(ell-sorted)g(reductions)e Fo(u)2758 3402 y Fq(\003)2797 3435 y Fr( )d Fo(t)g Fr(!)3070 3402 y Fq(\003)3138 3435 y Fo(v)36 b Fu(for)c(whic)m(h)g Fo(u)212 3548 y Fu(and)i Fo(v)39 b Fu(do)c(not)g(ha)m(v)m(e)h(a)g (common)f(reduct.)54 b(Since)35 b(w)m(ell-sorted)i(terms)d(of)i(t)m(yp) s(e)f(2)g(are)h(in)e(normal)212 3661 y(form,)26 b Fo(t)f Fu(m)m(ust)g(ha)m(v)m(e)i(t)m(yp)s(e)e(1.)40 b(Hence)26 b(the)f(ro)s(ot)h(sym)m(b)s(ol)f(of)g Fo(t)g Fu(is)h Fh(f)6 b Fu(,)26 b Fh(g)q Fu(,)h(or)f Fh(A)p Fu(,)g(and)e Fo(t)h Fu(do)s(es)g(not)h(con)m(tain)212 3774 y(an)m(y)g(of)g(these)g (sym)m(b)s(ols)g(b)s(elo)m(w)g(the)g(ro)s(ot.)39 b(This)25 b(implies)h(that)g(all)h(rewrite)f(steps)g(in)f(an)m(y)h(reduction)212 3887 y(starting)32 b(from)e Fo(t)g Fu(tak)m(e)i(place)g(at)f(the)g(ro)s (ot)g(p)s(osition.)42 b(Since)30 b Fh(A)h Fu(is)f(in)h(normal)f(form)g (and)g(the)h(TRS)212 4000 y(consisting)38 b(of)g(all)g Fh(g)h Fu(rules)e(is)g(orthogonal)i(and)d(hence)i(con\015uen)m(t,)h(w)m (e)f(conclude)g(that)f(the)h(ro)s(ot)212 4113 y(sym)m(b)s(ol)30 b(of)h Fo(t)f Fu(is)g Fh(f)6 b Fu(.)41 b(W)-8 b(e)32 b(ha)m(v)m(e)439 4311 y Fo(u)517 4273 y Fq(\003)556 4311 y Fr( )25 b Fo(u)724 4273 y Fq(0)30 b(\003)812 4311 y Fr( )25 b Fo(t)h Fr(!)1078 4273 y Fq(\003)1142 4311 y Fo(v)1189 4273 y Fq(0)672 4347 y Fg(|)p 713 4347 189 11 v 189 w({z)p 984 4347 V 189 w(})1238 4311 y Fr(!)1329 4273 y Fq(\003)1394 4311 y Fo(v)212 4535 y Fu(suc)m(h)40 b(that)h(in)g(the)f(underbraced)f(part)h(only)h(rules)f(of)h(t)m(yp)s (e)f(\(2\))i(are)f(used)f(and)f Fo(u)3196 4502 y Fq(0)3262 4535 y Fr(!)3353 4502 y Fq(\003)3434 4535 y Fo(u)i Fu(and)212 4648 y Fo(v)259 4615 y Fq(0)308 4648 y Fr(!)399 4615 y Fq(\003)464 4648 y Fo(v)s Fu(,)28 b(if)f(non-empt)m(y)-8 b(,)29 b(start)e(with)g(a)h(rule)f(of)g(t)m(yp)s(e)g(\(1\))i(or)e(rule) g(\(3\).)40 b(Because)29 b(rules)e(of)g(t)m(yp)s(e)g(\(2\))212 4761 y(are)33 b(rev)m(ersible)g(w)m(e)g(easily)h(obtain)f Fo(u)1499 4728 y Fq(0)1551 4761 y Fr(!)1642 4728 y Fq(\003)1711 4761 y Fo(v)1758 4728 y Fq(0)1814 4761 y Fu(b)m(y)f(using)g(only)h (rules)f(of)h(t)m(yp)s(e)g(\(2\).)48 b(It)33 b(follo)m(ws)h(that)212 4874 y Fo(u)264 4841 y Fq(0)315 4874 y Fu(=)27 b Fh(f)6 b Fu(\()p Fo(u)534 4888 y Fn(1)574 4874 y Fo(;)15 b(u)666 4888 y Fn(2)706 4874 y Fo(;)g(u)798 4888 y Fn(3)838 4874 y Fu(\),)32 b Fo(v)977 4841 y Fq(0)1028 4874 y Fu(=)c Fh(f)6 b Fu(\()p Fo(v)1240 4888 y Fn(1)1280 4874 y Fo(;)15 b(v)1364 4888 y Fn(2)1403 4874 y Fo(;)g(v)1487 4888 y Fn(3)1527 4874 y Fu(\),)33 b(and)e(either)h Fo(v)2101 4888 y Fn(1)2168 4874 y Fu(=)27 b Fo(\013)2324 4888 y Fm(i)2348 4896 y Fj(m)2426 4874 y Fr(\001)15 b(\001)g(\001)i Fo(\013)2606 4888 y Fm(i)2630 4897 y Fk(1)2669 4874 y Fu(\()p Fo(u)2756 4888 y Fn(1)2795 4874 y Fu(\),)33 b Fo(v)2932 4888 y Fn(2)2999 4874 y Fu(=)27 b Fo(\014)3148 4888 y Fm(i)3172 4896 y Fj(m)3250 4874 y Fr(\001)15 b(\001)g(\001)i Fo(\014)3423 4888 y Fm(i)3447 4897 y Fk(1)3486 4874 y Fu(\()p Fo(u)3573 4888 y Fn(2)3613 4874 y Fu(\),)212 4987 y(and)23 b Fo(u)434 5001 y Fn(3)499 4987 y Fu(=)i Fo(i)626 5001 y Fn(1)681 4987 y Fr(\001)15 b(\001)g(\001)h Fo(i)833 5001 y Fm(m)900 4987 y Fu(\()p Fo(v)979 5001 y Fn(3)1018 4987 y Fu(\))24 b(or)g Fo(u)1234 5001 y Fn(1)1298 4987 y Fu(=)h Fo(\013)1452 5001 y Fm(i)1476 5009 y Fj(m)1554 4987 y Fr(\001)15 b(\001)g(\001)i Fo(\013)1734 5001 y Fm(i)1758 5010 y Fk(1)1796 4987 y Fu(\()p Fo(v)1875 5001 y Fn(1)1915 4987 y Fu(\),)26 b Fo(u)2053 5001 y Fn(2)2117 4987 y Fu(=)f Fo(\014)2264 5001 y Fm(i)2288 5009 y Fj(m)2367 4987 y Fr(\001)15 b(\001)g(\001)h Fo(\014)2539 5001 y Fm(i)2563 5010 y Fk(1)2602 4987 y Fu(\()p Fo(v)2681 5001 y Fn(2)2721 4987 y Fu(\),)25 b(and)e Fo(v)3020 5001 y Fn(3)3085 4987 y Fu(=)i Fo(i)3212 5001 y Fn(1)3267 4987 y Fr(\001)15 b(\001)g(\001)h Fo(i)3419 5001 y Fm(m)3486 4987 y Fu(\()p Fo(u)3573 5001 y Fn(3)3613 4987 y Fu(\),)212 5100 y(for)43 b(some)g Fo(m)k Fl(>)f Fu(0)d(and)f(indices)i Fo(i)1468 5114 y Fn(1)1507 5100 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)1740 5114 y Fm(m)1854 5100 y Fr(2)46 b(f)p Fu(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)p Fr(g)p Fu(.)79 b(Without)44 b(loss)f(of)h (generalit)m(y)h(w)m(e)212 5213 y(consider)30 b(the)h(former.)40 b(No)m(w,)31 b(if)f Fo(u)1416 5180 y Fq(0)1465 5213 y Fr(!)1556 5180 y Fq(\003)1620 5213 y Fo(u)h Fu(or)f Fo(v)1861 5180 y Fq(0)1910 5213 y Fr(!)2001 5180 y Fq(\003)2065 5213 y Fo(v)k Fu(is)c(empt)m(y)h(then)f Fo(u)g Fu(and)f Fo(v)34 b Fu(ha)m(v)m(e)d(a)g(common)1920 5462 y(9)p eop %%Page: 10 10 10 9 bop 212 337 a Fu(reduct,)31 b(con)m(tradicting)h(the)f (assumption.)41 b(Hence)31 b(b)s(oth)f Fo(u)2285 304 y Fq(0)2334 337 y Fr(!)2425 304 y Fq(\003)2490 337 y Fo(u)g Fu(and)g Fo(v)2796 304 y Fq(0)2846 337 y Fr(!)2937 304 y Fq(\003)3002 337 y Fo(v)j Fu(are)e(non-empt)m(y)-8 b(.)212 450 y(If)43 b(b)s(oth)g(sequences)h(start)g(with)f(an)g (application)i(of)f(a)g(rule)f(of)h(t)m(yp)s(e)f(\(1\))i(then)e Fo(u)k Fu(=)g Fo(v)k Fu(=)46 b Fh(A)p Fu(,)212 562 y(con)m(tradicting) 36 b(the)e(assumption.)52 b(If)33 b(b)s(oth)h(sequences)g(start)h(with) f(rule)f(\(3\))j(then)d Fo(u)3217 576 y Fn(3)3288 562 y Fu(=)f Fo(v)3435 576 y Fn(3)3505 562 y Fu(=)g Fh(c)p Fu(,)212 675 y(whic)m(h)j(is)g(only)h(p)s(ossible)f(if)g Fo(m)e Fu(=)h(0)h(and)g(th)m(us)g Fo(u)1950 642 y Fq(0)2007 675 y Fu(=)e Fo(v)2158 642 y Fq(0)2181 675 y Fu(.)56 b(Consequen)m(tly)-8 b(,)37 b(one)f(of)g(the)f(sequences)212 788 y Fo(u)264 755 y Fq(0)322 788 y Fr(!)g Fo(u)h Fu(and)f Fo(v)765 755 y Fq(0)823 788 y Fr(!)914 755 y Fq(\003)988 788 y Fo(v)40 b Fu(starts)c(with)g(a)g(rule)g(of)g(t)m(yp)s(e)g(\(1\))h (and)f(the)g(other)g(starts)h(with)e(rule)h(\(3\).)212 901 y(First)31 b(supp)s(ose)e(that)i Fo(u)1025 868 y Fq(0)1073 901 y Fr(!)25 b Fo(u)31 b Fu(starts)g(with)f(a)g(rule)g(of)h (t)m(yp)s(e)g(\(1\).)41 b(Then)30 b(w)m(e)h(ma)m(y)g(write)439 1105 y Fo(u)26 b Fu(=)f Fh(A)f Fr( )i Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)p Fu(\()p Fo(u)1162 1068 y Fq(0)1162 1128 y Fn(3)1204 1105 y Fu(\)\))26 b(=)f Fo(u)1448 1068 y Fq(0)1496 1105 y Fr(!)1587 1068 y Fq(\003)1652 1105 y Fo(v)1699 1068 y Fq(0)1748 1105 y Fu(=)g Fh(f)6 b Fu(\()p Fo(v)1957 1119 y Fn(1)1997 1105 y Fo(;)15 b(v)2081 1119 y Fn(2)2121 1105 y Fo(;)g Fh(c)p Fu(\))26 b Fr(!)f Fh(g)q Fu(\()p Fo(v)2503 1119 y Fn(1)2544 1105 y Fo(;)15 b(v)2628 1119 y Fn(2)2668 1105 y Fu(\))25 b Fr(!)2819 1068 y Fq(\003)2884 1105 y Fo(v)212 1310 y Fu(for)i(some)h Fo(i)e Fr(2)f Fu(\000)i(and)g(th)m(us)g Fo(u)1223 1324 y Fn(1)1288 1310 y Fu(=)e Fo(u)1436 1324 y Fn(2)1501 1310 y Fu(=)g Fo(v)1641 1324 y Fn(3)1705 1310 y Fu(=)g Fh(c)j Fu(and)f Fo(u)2095 1324 y Fn(3)2160 1310 y Fu(=)e Fo(i)p Fu(\()p Fo(u)2374 1277 y Fq(0)2374 1334 y Fn(3)2414 1310 y Fu(\).)40 b(Hence)28 b Fo(v)2825 1324 y Fn(1)2890 1310 y Fu(=)d Fo(\013)3044 1324 y Fm(i)3068 1332 y Fj(m)3146 1310 y Fr(\001)15 b(\001)g(\001)h Fo(\013)3325 1324 y Fm(i)3349 1333 y Fk(1)3388 1310 y Fu(\()p Fh(c)p Fu(\))29 b(and)212 1423 y Fo(v)256 1437 y Fn(2)321 1423 y Fu(=)c Fo(\014)468 1437 y Fm(i)492 1445 y Fj(m)570 1423 y Fr(\001)15 b(\001)g(\001)h Fo(\014)742 1437 y Fm(i)766 1446 y Fk(1)806 1423 y Fu(\()p Fh(c)p Fu(\).)40 b(Because)28 b Fo(u)e Fu(and)g Fo(v)j Fu(are)e(assumed)f(to)h(ha)m(v)m(e)g(no)g(common)f (reduct,)h Fo(v)j Fu(and)c(th)m(us)212 1535 y(also)g Fh(g)q Fu(\()p Fo(v)514 1549 y Fn(1)555 1535 y Fo(;)15 b(v)639 1549 y Fn(2)679 1535 y Fu(\))26 b(do)f(not)h(rewrite)f(to)i Fh(A)e Fu(and)f(this)i(implies)f(that)h(only)g(t)m(yp)s(e)g(\(4\))g (rules)f(are)h(applicable.)212 1648 y(It)35 b(follo)m(ws)h(that)f Fo(v)863 1662 y Fn(1)935 1648 y Fu(=)d Fo(v)1082 1662 y Fn(2)1156 1648 y Fu(and)i(th)m(us)h Fo(\013)1600 1662 y Fm(i)1624 1670 y Fj(m)1702 1648 y Fr(\001)15 b(\001)g(\001)h Fo(\013)1881 1662 y Fm(i)1905 1671 y Fk(1)1976 1648 y Fu(=)32 b Fo(\014)2130 1662 y Fm(i)2154 1670 y Fj(m)2233 1648 y Fr(\001)15 b(\001)g(\001)h Fo(\014)2405 1662 y Fm(i)2429 1671 y Fk(1)2468 1648 y Fu(.)54 b(F)-8 b(rom)35 b Fo(i)p Fu(\()p Fo(u)2908 1615 y Fq(0)2908 1673 y Fn(3)2948 1648 y Fu(\))e(=)f Fo(i)3150 1662 y Fn(1)3205 1648 y Fr(\001)15 b(\001)g(\001)h Fo(i)3357 1662 y Fm(m)3424 1648 y Fu(\()p Fh(c)p Fu(\))36 b(w)m(e)212 1761 y(infer)c(that)g Fo(m)c Fl(>)g Fu(1.)46 b(So)32 b Fo(P)45 b Fu(has)32 b(a)h(solution.)46 b(Next)33 b(supp)s(ose)d(that)j Fo(u)2620 1728 y Fq(0)2672 1761 y Fr(!)28 b Fo(u)k Fu(starts)g(with)g(a)g(rule)g (of)212 1874 y(t)m(yp)s(e)f(\(3\).)41 b(Then)30 b(w)m(e)h(ma)m(y)g (write)439 2078 y Fo(u)517 2041 y Fq(\003)556 2078 y Fr( )25 b Fh(g)q Fu(\()p Fo(u)805 2092 y Fn(1)846 2078 y Fo(;)15 b(u)938 2092 y Fn(2)978 2078 y Fu(\))25 b Fr( )g Fh(f)6 b Fu(\()p Fo(u)1275 2092 y Fn(1)1315 2078 y Fo(;)15 b(u)1407 2092 y Fn(2)1447 2078 y Fo(;)g Fh(c)p Fu(\))27 b(=)e Fo(u)1737 2041 y Fq(0)1785 2078 y Fr(!)1876 2041 y Fq(\003)1941 2078 y Fo(v)1988 2041 y Fq(0)2037 2078 y Fu(=)g Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(i)p Fu(\()p Fo(v)2475 2041 y Fq(0)2472 2101 y Fn(3)2514 2078 y Fu(\)\))26 b Fr(!)f Fh(A)g Fu(=)g Fo(v)212 2283 y Fu(for)30 b(some)g Fo(i)25 b Fr(2)g Fu(\000)30 b(and)f(th)m(us)g Fo(u)1234 2297 y Fn(3)1299 2283 y Fu(=)c Fo(v)1439 2297 y Fn(1)1504 2283 y Fu(=)g Fo(v)1644 2297 y Fn(2)1708 2283 y Fu(=)g Fh(c)30 b Fu(and)g Fo(v)2095 2297 y Fn(3)2159 2283 y Fu(=)25 b Fo(i)p Fu(\()p Fo(v)2368 2250 y Fq(0)2365 2307 y Fn(3)2405 2283 y Fu(\).)41 b(This)29 b(is)h(only)g(p)s(ossible)f (if)h Fo(m)25 b Fu(=)g(0)212 2396 y(but)30 b(then)g Fo(v)630 2410 y Fn(3)695 2396 y Fu(=)25 b Fo(u)843 2410 y Fn(3)912 2396 y Fu(con)m(tradicts)32 b Fo(v)1424 2410 y Fn(3)1489 2396 y Fu(=)25 b Fo(i)p Fu(\()p Fo(v)1698 2363 y Fq(0)1695 2420 y Fn(3)1735 2396 y Fu(\))31 b(and)f Fo(u)2030 2410 y Fn(3)2094 2396 y Fu(=)25 b Fh(c)p Fu(.)1348 b Fl(\003)212 2608 y Ff(Corollary)35 b(4.6)46 b Fs(Con\015uenc)-5 b(e)34 b(is)e(not)h(semi-de)-5 b(cidable)34 b(for)f(line)-5 b(ar)34 b(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(on\015uent)34 b(TRSs.)102 b Fl(\003)212 2895 y Ft(5)135 b(CR)45 b Fd(\))g Ft(GCR)212 3097 y Fu(Let)31 b Fr(R)452 3111 y Fn(6)491 3097 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)831 3064 y Fn(1)831 3122 y(6)871 3097 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(R)1191 3064 y Fn(2)1191 3122 y(6)1260 3097 y Fu(with)30 b Fr(R)1544 3064 y Fn(1)1544 3122 y(6)1584 3097 y Fu(\()p Fo(P)13 b Fu(\))31 b(consisting)g(of)g(the)f(rules)635 3301 y Fh(A)p Fu(\()p Fo(z)t Fu(\))h Fr(!)f Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(z)t Fu(\))481 3413 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fo(y)s Fu(\))p Fo(;)15 b(z)t Fu(\))186 b Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))270 b Fr(!)30 b Fh(A)p Fu(\()p Fo(z)t Fu(\))481 3526 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))428 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))481 3639 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(A)p Fu(\()p Fo(z)t Fu(\))593 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(z)t Fu(\))290 b Fr(!)30 b Fh(B)p Fu(\()p Fo(z)t Fu(\))212 3849 y(and)g Fr(R)466 3816 y Fn(2)466 3873 y(6)536 3849 y Fu(of)g(the)h(rules)494 4052 y Fh(f)6 b Fu(\()p Fh(D)p Fo(;)15 b Fh(D)p Fo(;)g Fh(D)p Fu(\))30 b Fr(!)h Fh(D)338 b(c)30 b Fr(!)h Fh(D)481 4164 y(g)q Fu(\()p Fh(D)p Fo(;)15 b Fh(D)p Fo(;)g Fh(D)p Fu(\))31 b Fr(!)g Fh(D)181 b(A)p Fu(\()p Fh(D)p Fu(\))30 b Fr(!)h Fh(D)692 4277 y Fo(a)p Fu(\()p Fh(D)p Fu(\))f Fr(!)h Fh(D)181 b(B)p Fu(\()p Fh(D)p Fu(\))30 b Fr(!)h Fh(D)212 4487 y Fu(Note)41 b(that)g(the)f (only)h(di\013erence)f(b)s(et)m(w)m(een)h Fr(R)1878 4454 y Fn(1)1878 4511 y(6)1917 4487 y Fu(\()p Fo(P)13 b Fu(\))41 b(and)e Fr(R)2362 4501 y Fn(1)2402 4487 y Fu(\()p Fo(P)13 b Fu(\))41 b(is)f(the)g(addition)g(of)h(an)f(extra)212 4600 y(argumen)m(t)31 b(whic)m(h)f(is)g(simply)g(propagated.)212 4812 y Ff(Lemma)35 b(5.1)46 b Fs(The)33 b(TRS)g Fr(R)1240 4826 y Fn(6)1279 4812 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(gr)-5 b(ound)34 b(c)-5 b(on\015uent)33 b(for)h(every)e(PCP)g (instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 4945 y(Pr)-5 b(o)g(of)55 b Fu(In)31 b Fr(R)669 4912 y Fn(2)669 4969 y(6)736 4945 y Fr(\022)c(R)911 4959 y Fn(6)950 4945 y Fu(\()p Fo(P)13 b Fu(\))33 b(ev)m(ery)f(ground)f(term)h(rewrites)g(to)g Fh(D)p Fu(.)45 b(\(As)32 b(usual)f(w)m(e)h(assume)g(that)g(the)212 5057 y(signature)26 b(of)f(a)h(TRS)f(consists)h(of)f(all)i(function)e (sym)m(b)s(ols)g(o)s(ccurring)g(in)g(its)h(rewrite)f(rules.\))39 b(Hence)212 5170 y Fr(R)289 5184 y Fn(6)329 5170 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(ground)f(con\015uen)m(t.)2317 b Fl(\003)1897 5462 y Fu(10)p eop %%Page: 11 11 11 10 bop 212 337 a Ff(Lemma)35 b(5.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 351 y Fn(6)1279 337 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e(and)h(only)h(if)e Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 469 y(Pr)-5 b(o)g(of)56 b Fu(It)34 b(is)g(not)g(di\016cult)f(to)i(see)f(that)g(the) g(statemen)m(ts)h Fh(A)p Fu(\()p Fo(z)t Fu(\))c Fr(!)2561 436 y Fq(\003)2561 500 y(R)2621 509 y Fk(6)2656 500 y Fn(\()p Fm(P)10 b Fn(\))2800 469 y Fh(B)p Fu(\()p Fo(z)t Fu(\))34 b(and)f Fh(A)e Fr(!)3374 436 y Fq(\003)3374 500 y(R)3434 509 y Fk(1)3468 500 y Fn(\()p Fm(P)10 b Fn(\))3613 469 y Fh(B)212 594 y Fu(are)39 b(equiv)-5 b(alen)m(t.)66 b(According)39 b(to)g(Lemma)f(2.1)i(w)m(e)e(ha)m(v)m(e)i (to)f(sho)m(w)f(that)h Fr(R)2902 608 y Fn(6)2942 594 y Fu(\()p Fo(P)13 b Fu(\))39 b(is)f(con\015uen)m(t)g(if)212 706 y(and)30 b(only)g(if)h Fh(A)p Fu(\()p Fo(z)t Fu(\))26 b Fr(!)966 673 y Fq(\003)966 738 y(R)1026 747 y Fk(6)1060 738 y Fn(\()p Fm(P)10 b Fn(\))1199 706 y Fh(B)p Fu(\()p Fo(z)t Fu(\).)42 b(In)29 b Fr(R)1633 720 y Fn(6)1673 706 y Fu(\()p Fo(P)13 b Fu(\))31 b(w)m(e)g(ha)m(v)m(e)h Fh(A)p Fu(\()p Fo(z)t Fu(\))25 b Fr( )h Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(z)t Fu(\))28 b Fr(!)e Fh(B)p Fu(\()p Fo(z)t Fu(\))31 b(with)f Fh(B)p Fu(\()p Fo(z)t Fu(\))h(in)212 831 y(normal)f(form.)41 b(So)30 b(if)g Fr(R)1057 845 y Fn(6)1097 831 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(con\015uen)m(t)h(then)f(necessarily)h Fh(A)p Fu(\()p Fo(z)t Fu(\))26 b Fr(!)2706 798 y Fq(\003)2770 831 y Fh(B)p Fu(\()p Fo(z)t Fu(\).)353 944 y(Con)m(v)m(ersely)-8 b(,)30 b(if)e Fh(A)p Fu(\()p Fo(z)t Fu(\))e Fr(!)1204 911 y Fq(\003)1268 944 y Fh(B)p Fu(\()p Fo(z)t Fu(\))j(then)e(w)m(e)h (obtain)g(con\015uence)g(b)m(y)g(considering)g(the)g(linear)g(and)212 1057 y(strongly)41 b(closed)f(TRS)g Fr(R)1152 1024 y Fq(0)1152 1081 y Fn(6)1191 1057 y Fu(\()p Fo(P)13 b Fu(\))42 b(=)f Fr(R)1563 1071 y Fn(6)1603 1057 y Fu(\()p Fo(P)13 b Fu(\))27 b Fr([)f(f)p Fh(A)p Fu(\()p Fo(z)t Fu(\))43 b Fr(!)e Fh(B)p Fu(\()p Fo(z)t Fu(\))p Fo(;)15 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))44 b Fr(!)e Fh(B)p Fu(\()p Fo(z)t Fu(\))p Fo(;)15 b Fh(g)q Fu(\()p Fo(x;)g(y)s(;)g(z)t Fu(\))44 b Fr(!)212 1170 y Fh(B)p Fu(\()p Fo(z)t Fu(\))p Fr(g)p Fu(.)3144 b Fl(\003)212 1374 y Ff(Corollary)35 b(5.3)46 b Fs(Con\015uenc)-5 b(e)38 b(is)f(not)h(c)-5 b(o-semi-de)g(cidable)38 b(for)g(line)-5 b(ar)38 b(gr)-5 b(ound-c)g(on\015uent)39 b(TRSs.)212 1487 y Fl(\003)353 1692 y Fu(In)34 b(order)h(to)g(sho)m(w)g(that)g(con\015uence)g(is)g (not)g(semi-decidable)h(for)f(ground)e(con\015uen)m(t)i(TRSs,)212 1805 y(w)m(e)c(consider)f(the)h(TRS)e Fr(R)1147 1819 y Fn(7)1187 1805 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)1527 1772 y Fn(1)1527 1829 y(7)1566 1805 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(R)1886 1772 y Fn(2)1886 1829 y(7)1956 1805 y Fu(with)30 b Fr(R)2240 1772 y Fn(1)2240 1829 y(7)2279 1805 y Fu(\()p Fo(P)13 b Fu(\))31 b(consisting)g(of)g(the)f(rules)481 2000 y(\(1\))h Fh(f)6 b Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))36 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))577 b Fh(A)p Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))481 2112 y(\(2\))254 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))33 b Fr(!)d Fh(f)6 b Fu(\()p Fo(\013)1562 2126 y Fm(i)1591 2112 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)1804 2126 y Fm(i)1833 2112 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))297 b Fh(A)p Fu(\()p Fo(i)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(y)s(;)g(z)t Fu(\))49 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))481 2225 y(\(2\))48 b Fh(f)6 b Fu(\()p Fo(\013)771 2239 y Fm(i)800 2225 y Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)1013 2239 y Fm(i)1043 2225 y Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))32 b Fr(!)e Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))503 b Fh(A)p Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(y)s(;)g(z)t Fu(\))40 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))481 2338 y(\(3\))244 b Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g Fh(c)p Fu(\()p Fo(z)t Fu(\)\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))592 b Fh(A)p Fu(\()p Fo(x;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))32 b Fr(!)e Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))481 2451 y(\(4\))105 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))35 b Fr(!)30 b Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))592 b Fh(A)p Fu(\()p Fo(x;)15 b(i)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))49 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))710 2564 y Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)g(b)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))35 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))62 b(if)30 b Fo(a)25 b Fr(6)p Fu(=)g Fo(b)224 b Fh(A)p Fu(\()p Fo(x;)15 b Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))40 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))709 2677 y Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)g Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))35 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))577 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(a)p Fu(\()p Fo(z)t Fu(\)\))32 b Fr(!)e Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))709 2790 y Fh(g)q Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)g(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))35 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))577 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(i)p Fu(\()p Fo(z)t Fu(\)\))49 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))2369 2903 y Fh(A)p Fu(\()p Fo(x;)g(y)s(;)g Fh(c)p Fu(\()p Fo(z)t Fu(\)\))40 b Fr(!)30 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))212 3104 y(and)30 b Fr(R)466 3071 y Fn(2)466 3128 y(7)536 3104 y Fu(of)g(the)h(rules)489 3299 y Fh(c)p Fu(\()p Fh(D)p Fu(\))56 b Fr(!)f Fh(D)200 b(f)6 b Fu(\()p Fh(D)p Fo(;)15 b(y)s(;)g(z)t Fu(\))57 b Fr(!)e Fh(D)186 b(g)q Fu(\()p Fh(D)p Fo(;)15 b(y)s(;)g(z)t Fu(\))58 b Fr(!)d Fh(D)187 b(A)p Fu(\()p Fh(D)p Fo(;)15 b(y)s(;)g(z)t Fu(\))57 b Fr(!)e Fh(D)481 3412 y Fo(a)p Fu(\()p Fh(D)p Fu(\))h Fr(!)f Fh(D)196 b(f)6 b Fu(\()p Fo(x;)15 b Fh(D)p Fo(;)g(z)t Fu(\))57 b Fr(!)e Fh(D)182 b(g)q Fu(\()p Fo(x;)15 b Fh(D)p Fo(;)g(z)t Fu(\))58 b Fr(!)d Fh(D)183 b(A)p Fu(\()p Fo(x;)15 b Fh(D)p Fo(;)g(z)t Fu(\))57 b Fr(!)e Fh(D)498 3525 y Fo(i)p Fu(\()p Fh(D)p Fu(\))h Fr(!)f Fh(D)182 b(g)q Fu(\()p Fo(x;)15 b(y)s(;)g Fh(D)p Fu(\))57 b Fr(!)e Fh(D)193 b(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g Fh(D)p Fu(\))57 b Fr(!)e Fh(D)182 b(A)p Fu(\()p Fo(x;)15 b(y)s(;)g Fh(D)p Fu(\))56 b Fr(!)f Fh(D)212 3727 y Ff(Lemma)35 b(5.4)46 b Fs(The)33 b(TRS)g Fr(R)1240 3741 y Fn(7)1279 3727 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(gr)-5 b(ound)34 b(c)-5 b(on\015uent)33 b(for)h(every)e(PCP)g(instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 3859 y(Pr)-5 b(o)g(of)53 b Fu(An)30 b(easy)h(induction)f(pro)s(of)g(rev)m(eals)i(that)f(ev)m (ery)g(ground)e(term)i(rewrites)f(to)h Fh(D)p Fu(.)329 b Fl(\003)212 4064 y Ff(Lemma)35 b(5.5)46 b Fs(The)33 b(TRS)g Fr(R)1240 4031 y Fn(1)1240 4088 y(7)1279 4064 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e Fo(P)45 b Fs(admits)34 b(no)g(solution.)212 4196 y(Pr)-5 b(o)g(of)49 b Fu(The)24 b(pro)s(of)h(is)g(similar)h(to)g(the)f (\\if)7 b(")26 b(direction)g(in)f(the)h(pro)s(of)e(of)i(Lemma)f(4.5.)40 b(Supp)s(ose)24 b(that)212 4309 y Fr(R)289 4276 y Fn(1)289 4334 y(7)329 4309 y Fu(\()p Fo(P)13 b Fu(\))22 b(is)h(not)f(con\015uen) m(t.)38 b(Consider)22 b(the)g(man)m(y-sorted)h(t)m(yp)s(e)f (declarations)i Fh(f)6 b Fo(;)15 b Fh(g)q Fo(;)g Fh(A)10 b Fu(:)32 b(2)t Fr(\002)t Fu(2)t Fr(\002)t Fu(2)25 b Fr(!)h Fu(1,)212 4422 y Fh(c)10 b Fu(:)31 b(2)26 b Fr(!)f Fu(2,)k Fo(i)10 b Fu(:)31 b(2)26 b Fr(!)f Fu(2)k(for)f(all)h Fo(i)c Fr(2)g(f)p Fu(1)p Fo(;)15 b(:)g(:)g(:)j(;)d(n)p Fr(g)p Fu(,)29 b(and)f Fo(a)10 b Fu(:)30 b(2)c Fr(!)f Fu(2)k(for)f(all)h Fo(a)c Fr(2)g Fu(\000,)k(whic)m(h)f(is)g(compatible) 212 4535 y(with)i Fr(R)496 4502 y Fn(1)496 4559 y(7)535 4535 y Fu(\()p Fo(P)13 b Fu(\))31 b(b)m(y)f(assigning)h(t)m(yp)s(e)g(2) f(to)h(ev)m(ery)g(v)-5 b(ariable.)41 b(F)-8 b(rom)31 b(Theorem)f(2.4)h(w)m(e)g(conclude)g(that)212 4648 y Fr(R)289 4615 y Fn(1)289 4672 y(7)329 4648 y Fu(\()p Fo(P)13 b Fu(\))34 b(admits)g(div)m(erging)h(w)m(ell-sorted)g (reductions)e Fo(u)2192 4615 y Fq(\003)2231 4648 y Fr( )e Fo(t)g Fr(!)2508 4615 y Fq(\003)2578 4648 y Fo(v)37 b Fu(for)d(whic)m(h)f Fo(u)h Fu(and)f Fo(v)k Fu(do)d(not)212 4761 y(ha)m(v)m(e)e(a)e(common)h(reduct.)353 4874 y(Since)38 b(w)m(ell-sorted)h(terms)f(of)g(t)m(yp)s(e)g(2)g(are)g(in)g(normal)f (form,)j Fo(t)d Fu(m)m(ust)h(ha)m(v)m(e)h(t)m(yp)s(e)f(1.)63 b(Hence)212 4987 y(the)34 b(ro)s(ot)g(sym)m(b)s(ol)f(of)h Fo(t)f Fu(is)h Fh(f)6 b Fu(,)34 b Fh(g)q Fu(,)i(or)d Fh(A)p Fu(,)h(and)f Fo(t)h Fu(do)s(es)f(not)h(ha)m(v)m(e)g(an)m(y)g(of) g(these)g(sym)m(b)s(ols)f(b)s(elo)m(w)h(the)212 5100 y(ro)s(ot.)49 b(This)32 b(implies)i(that)f(all)h(rewrite)g(steps)f(in)f (an)m(y)i(reduction)f(starting)g(from)g Fo(t)g Fu(tak)m(e)i(place)f(at) 212 5213 y(the)40 b(ro)s(ot)g(p)s(osition.)70 b(Because)41 b(the)f(TRS)f(consisting)h(of)g(all)h Fh(g)h Fu(and)d Fh(A)g Fu(rules)h(is)g(con\015uen)m(t,)i(the)1897 5462 y(11)p eop %%Page: 12 12 12 11 bop 212 337 a Fu(ro)s(ot)33 b(sym)m(b)s(ol)f(of)g Fo(t)h Fu(m)m(ust)f(b)s(e)g Fh(f)38 b Fu(and)32 b(hence)g(w)m(e)h(ma)m (y)g(write)g Fo(t)28 b Fu(=)g Fh(f)6 b Fu(\()p Fo(t)2577 351 y Fn(1)2617 337 y Fo(;)15 b(t)2690 351 y Fn(2)2730 337 y Fo(;)g(t)2803 351 y Fn(3)2842 337 y Fu(\).)47 b(Note)34 b(that)f(ev)m(ery)g Fo(t)3645 351 y Fm(i)212 450 y Fu(con)m(tains)f(a)g (single)f(v)-5 b(ariable)32 b Fo(x)1293 464 y Fm(i)1352 450 y Fu(and)f(ev)m(ery)g(reduct)g(of)g Fo(t)g Fu(con)m(tains)h Fo(x)2631 464 y Fm(i)2690 450 y Fu(in)f(its)h Fo(i)p Fu(-th)f(argumen)m(t.)43 b(W)-8 b(e)212 562 y(ha)m(v)m(e)27 b Fo(u)494 529 y Fq(\003)533 562 y Fr( )f Fo(u)702 529 y Fq(0)j(\003)790 562 y Fr( )c Fo(t)g Fr(!)1055 529 y Fq(\003)1119 562 y Fo(v)1166 529 y Fq(0)1215 562 y Fr(!)1306 529 y Fq(\003)1371 562 y Fo(v)k Fu(suc)m(h)c(that)h(in)f Fo(u)1989 529 y Fq(0)k(\003)2077 562 y Fr( )c Fo(t)g Fr(!)2342 529 y Fq(\003)2406 562 y Fo(v)2453 529 y Fq(0)2502 562 y Fu(only)h(rules)f(of)g(t)m(yp)s(e)h(\(2\))g(are)g(used)212 675 y(and)f Fo(u)436 642 y Fq(0)485 675 y Fr(!)576 642 y Fq(\003)640 675 y Fo(u)h Fu(and)f Fo(v)937 642 y Fq(0)986 675 y Fr(!)1077 642 y Fq(\003)1141 675 y Fo(v)s Fu(,)j(if)d(non-empt)m (y)-8 b(,)28 b(start)e(with)f(a)h(rule)f(of)h(t)m(yp)s(e)g(\(1\))h(or)e (rule)h(\(3\).)40 b(Because)212 788 y(rules)e(of)h(t)m(yp)s(e)f(\(2\))i (are)f(rev)m(ersible)g(w)m(e)g(easily)g(obtain)g Fo(u)2233 755 y Fq(0)2295 788 y Fr(!)2386 755 y Fq(\003)2465 788 y Fo(v)2512 755 y Fq(0)2574 788 y Fu(b)m(y)f(using)g(only)g(rules)g(of) h(t)m(yp)s(e)212 901 y(\(2\).)51 b(It)33 b(follo)m(ws)i(that)f Fo(u)1058 868 y Fq(0)1111 901 y Fu(=)c Fh(f)6 b Fu(\()p Fo(u)1333 915 y Fn(1)1373 901 y Fo(;)15 b(u)1465 915 y Fn(2)1505 901 y Fo(;)g(u)1597 915 y Fn(3)1637 901 y Fu(\),)34 b Fo(v)1778 868 y Fq(0)1832 901 y Fu(=)c Fh(f)6 b Fu(\()p Fo(v)2046 915 y Fn(1)2086 901 y Fo(;)15 b(v)2170 915 y Fn(2)2210 901 y Fo(;)g(v)2294 915 y Fn(3)2334 901 y Fu(\),)35 b(and)d(either)i Fo(v)2913 915 y Fn(1)2983 901 y Fu(=)c Fo(\013)3142 915 y Fm(i)3166 923 y Fj(m)3244 901 y Fr(\001)15 b(\001)g(\001)h Fo(\013)3423 915 y Fm(i)3447 924 y Fk(1)3486 901 y Fu(\()p Fo(u)3573 915 y Fn(1)3613 901 y Fu(\),)212 1014 y Fo(v)256 1028 y Fn(2)326 1014 y Fu(=)29 b Fo(\014)477 1028 y Fm(i)501 1036 y Fj(m)580 1014 y Fr(\001)15 b(\001)g(\001)h Fo(\014)752 1028 y Fm(i)776 1037 y Fk(1)815 1014 y Fu(\()p Fo(u)902 1028 y Fn(2)942 1014 y Fu(\),)34 b(and)f Fo(u)1268 1028 y Fn(3)1338 1014 y Fu(=)c Fo(i)1469 1028 y Fn(1)1524 1014 y Fr(\001)15 b(\001)g(\001)h Fo(i)1676 1028 y Fm(m)1743 1014 y Fu(\()p Fo(v)1822 1028 y Fn(3)1862 1014 y Fu(\))34 b(or)f Fo(u)2097 1028 y Fn(1)2166 1014 y Fu(=)d Fo(\013)2325 1028 y Fm(i)2349 1036 y Fj(m)2427 1014 y Fr(\001)15 b(\001)g(\001)i Fo(\013)2607 1028 y Fm(i)2631 1037 y Fk(1)2669 1014 y Fu(\()p Fo(v)2748 1028 y Fn(1)2788 1014 y Fu(\),)35 b Fo(u)2935 1028 y Fn(2)3004 1014 y Fu(=)30 b Fo(\014)3156 1028 y Fm(i)3180 1036 y Fj(m)3258 1014 y Fr(\001)15 b(\001)g(\001)i Fo(\014)3431 1028 y Fm(i)3455 1037 y Fk(1)3494 1014 y Fu(\()p Fo(v)3573 1028 y Fn(2)3613 1014 y Fu(\),)212 1127 y(and)38 b Fo(v)441 1141 y Fn(3)519 1127 y Fu(=)g Fo(i)659 1141 y Fn(1)714 1127 y Fr(\001)15 b(\001)g(\001)h Fo(i)866 1141 y Fm(m)933 1127 y Fu(\()p Fo(u)1020 1141 y Fn(3)1060 1127 y Fu(\),)41 b(for)d(some)h Fo(m)f Fl(>)h Fu(0)f(and)g(indices)g Fo(i)2377 1141 y Fn(1)2417 1127 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)2650 1141 y Fm(m)2756 1127 y Fr(2)38 b(f)p Fu(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(n)p Fr(g)p Fu(.)65 b(Without)212 1240 y(loss)38 b(of)f(generalit)m(y)i(w)m (e)f(consider)f(the)g(former.)60 b(No)m(w,)40 b(if)d Fo(u)2322 1207 y Fq(0)2382 1240 y Fr(!)2473 1207 y Fq(\003)2549 1240 y Fo(u)g Fu(or)g Fo(v)2803 1207 y Fq(0)2863 1240 y Fr(!)2954 1207 y Fq(\003)3030 1240 y Fo(v)j Fu(is)d(empt)m(y)h(then) 212 1353 y Fo(t)j Fu(is)g(con\015uen)m(t,)j(con)m(tradicting)f(the)e (assumption.)73 b(Hence)42 b(b)s(oth)e Fo(u)2656 1320 y Fq(0)2722 1353 y Fr(!)2813 1320 y Fq(\003)2896 1353 y Fo(u)h Fu(and)f Fo(v)3223 1320 y Fq(0)3290 1353 y Fr(!)3381 1320 y Fq(\003)3464 1353 y Fo(v)k Fu(are)212 1466 y(non-empt)m(y)-8 b(.)40 b(If)26 b(b)s(oth)g(sequences)g(start)h(with)f(an)g(application) i(of)f(a)f(rule)g(of)h(t)m(yp)s(e)f(\(1\))i(then,)f(due)f(to)212 1579 y(the)33 b(shap)s(e)f(of)h(the)f Fh(A)h Fu(rules,)g Fo(u)1285 1546 y Fq(0)1341 1579 y Fu(and)f Fo(v)1567 1546 y Fq(0)1623 1579 y Fu(reduce)g(to)i(the)f(common)g(reduct)f Fh(A)p Fu(\()p Fo(x)2985 1593 y Fn(1)3024 1579 y Fo(;)15 b(x)3116 1593 y Fn(2)3156 1579 y Fo(;)g(x)3248 1593 y Fn(3)3288 1579 y Fu(\).)48 b(If)32 b(b)s(oth)212 1692 y(sequences)39 b(start)f(with)g(rule)g(\(3\))h(then)f Fo(u)1684 1706 y Fn(3)1762 1692 y Fu(=)g Fh(c)p Fu(\()p Fo(u)1998 1659 y Fq(0)1998 1716 y Fn(3)2038 1692 y Fu(\))h(and)e Fo(v)2340 1706 y Fn(3)2418 1692 y Fu(=)h Fh(c)p Fu(\()p Fo(v)2649 1659 y Fq(0)2646 1716 y Fn(3)2687 1692 y Fu(\),)i(whic)m(h)e (is)g(only)h(p)s(ossible)212 1804 y(if)k Fo(m)i Fu(=)g(0)e(and)f(th)m (us)g Fo(u)1090 1771 y Fq(0)1159 1804 y Fu(=)j Fo(v)1322 1771 y Fq(0)1346 1804 y Fu(.)77 b(W)-8 b(e)44 b(already)f(observ)m(ed)g (that)g(the)g(TRS)e(consisting)i(of)g(all)h Fh(g)212 1917 y Fu(and)c Fh(A)g Fu(rules)g(is)h(con\015uen)m(t)f(and)g(hence)h Fo(u)1733 1884 y Fq(0)1796 1917 y Fu(and)f Fo(v)2030 1884 y Fq(0)2094 1917 y Fu(ha)m(v)m(e)i(a)f(common)g(reduct.)70 b(Consequen)m(tly)-8 b(,)212 2030 y(one)37 b(of)g(the)g(sequences)g Fo(u)1129 1997 y Fq(0)1188 2030 y Fr(!)f Fo(u)1367 1997 y Fq(00)1445 2030 y Fr(!)1536 1997 y Fq(\003)1611 2030 y Fo(u)h Fu(and)f Fo(v)1930 1997 y Fq(0)1990 2030 y Fr(!)f Fo(v)2163 1997 y Fq(00)2242 2030 y Fr(!)2333 1997 y Fq(\003)2408 2030 y Fo(v)40 b Fu(starts)d(with)g(a)g(rule)f(of)h(t)m(yp)s(e)g(\(1\)) 212 2143 y(and)j(the)h(other)g(starts)g(with)g(rule)f(\(3\).)73 b(First)41 b(supp)s(ose)e(that)j Fo(u)2545 2110 y Fq(0)2610 2143 y Fr(!)h Fo(u)2796 2110 y Fq(00)2881 2143 y Fr(!)2972 2110 y Fq(\003)3054 2143 y Fo(u)d Fu(starts)h(with)g(a)212 2256 y(rule)32 b(of)h(t)m(yp)s(e)f(\(1\).)48 b(This)32 b(implies)h(that)g Fo(u)1668 2223 y Fq(00)1739 2256 y Fu(=)28 b Fh(A)p Fu(\()p Fo(u)1986 2223 y Fq(0)1986 2280 y Fn(1)2026 2256 y Fo(;)15 b(u)2118 2223 y Fq(0)2118 2280 y Fn(2)2158 2256 y Fo(;)g(u)2250 2223 y Fq(0)2250 2280 y Fn(3)2289 2256 y Fu(\))33 b(with)f Fo(u)2618 2270 y Fn(1)2686 2256 y Fu(=)d Fh(c)p Fu(\()p Fo(u)2913 2223 y Fq(0)2913 2280 y Fn(1)2953 2256 y Fu(\),)34 b Fo(u)3099 2270 y Fn(2)3167 2256 y Fu(=)28 b Fh(c)p Fu(\()p Fo(u)3393 2223 y Fq(0)3393 2280 y Fn(2)3433 2256 y Fu(\),)34 b(and)212 2369 y Fo(u)264 2383 y Fn(3)329 2369 y Fu(=)25 b Fo(i)p Fu(\()p Fo(u)543 2336 y Fq(0)543 2393 y Fn(3)583 2369 y Fu(\))d(for)f(some)h Fo(i)k Fr(2)e(f)p Fu(1)p Fo(;)15 b(:)g(:)g(:)j(;)d(n)p Fr(g)p Fu(.)38 b(Moreo)m(v)m(er,)26 b Fo(v)2052 2336 y Fq(00)2120 2369 y Fu(=)f Fh(g)q Fu(\()p Fo(v)2341 2383 y Fn(1)2381 2369 y Fo(;)15 b(v)2465 2383 y Fn(2)2505 2369 y Fo(;)g(v)2592 2336 y Fq(0)2589 2393 y Fn(3)2629 2369 y Fu(\))22 b(with)f Fo(v)2928 2383 y Fn(3)2993 2369 y Fu(=)k Fh(c)p Fu(\()p Fo(v)3211 2336 y Fq(0)3208 2393 y Fn(3)3248 2369 y Fu(\).)38 b(W)-8 b(e)23 b(ha)m(v)m(e)212 2482 y Fo(u)264 2449 y Fq(00)332 2482 y Fr(!)423 2449 y Fq(\003)487 2482 y Fh(A)p Fu(\()p Fo(x)635 2496 y Fn(1)675 2482 y Fo(;)15 b(x)767 2496 y Fn(2)807 2482 y Fo(;)g(x)899 2496 y Fn(3)938 2482 y Fu(\).)40 b(Because)29 b Fo(t)f Fu(is)f(non-con\015uen)m(t,)i Fo(v)2171 2449 y Fq(00)2241 2482 y Fu(cannot)f(reduce)f(to)i Fh(A)p Fu(\()p Fo(x)3078 2496 y Fn(1)3117 2482 y Fo(;)15 b(x)3209 2496 y Fn(2)3249 2482 y Fo(;)g(x)3341 2496 y Fn(3)3381 2482 y Fu(\))28 b(whic)m(h)212 2595 y(implies)35 b(that)h(only)f(rules)f(of)h(t)m(yp)s(e)g(\(4\))h(can)g(b)s(e)e (applied)h(to)g Fo(v)2406 2562 y Fq(00)2449 2595 y Fu(.)54 b(It)35 b(follo)m(ws)h(that)g Fo(v)3180 2609 y Fn(1)3252 2595 y Fu(=)d Fo(\015)5 b Fu(\()p Fh(c)p Fu(\()p Fo(v)3565 2562 y Fq(0)3562 2619 y Fn(1)3603 2595 y Fu(\)\))212 2708 y(and)39 b Fo(v)442 2722 y Fn(2)523 2708 y Fu(=)h Fo(\015)5 b Fu(\()p Fh(c)p Fu(\()p Fo(v)843 2675 y Fq(0)840 2732 y Fn(2)881 2708 y Fu(\)\))40 b(for)g(some)g(string)g Fo(\015)46 b Fr(2)40 b Fu(\000)1896 2675 y Fq(\003)1935 2708 y Fu(.)69 b(Hence)41 b Fo(\015)46 b Fu(=)40 b Fo(\013)2571 2722 y Fm(i)2595 2730 y Fj(m)2674 2708 y Fr(\001)15 b(\001)g(\001)h Fo(\013)2853 2722 y Fm(i)2877 2731 y Fk(1)2957 2708 y Fu(=)40 b Fo(\014)3119 2722 y Fm(i)3143 2730 y Fj(m)3221 2708 y Fr(\001)15 b(\001)g(\001)i Fo(\014)3394 2722 y Fm(i)3418 2731 y Fk(1)3497 2708 y Fu(with)212 2821 y Fo(m)38 b Fl(>)f Fu(1.)64 b(So)38 b Fo(P)51 b Fu(has)37 b(a)i(solution.)63 b(In)38 b(the)g(remaining)g(case)h(w)m(e)f(ha)m(v)m (e)h Fo(u)2797 2788 y Fq(00)2878 2821 y Fu(=)e Fh(g)q Fu(\()p Fo(u)3119 2835 y Fn(1)3160 2821 y Fo(;)15 b(u)3252 2835 y Fn(2)3291 2821 y Fo(;)g(u)3383 2788 y Fq(0)3383 2845 y Fn(3)3423 2821 y Fu(\))39 b(with)212 2934 y Fo(u)264 2948 y Fn(3)334 2934 y Fu(=)31 b Fh(c)p Fu(\()p Fo(u)563 2901 y Fq(0)563 2958 y Fn(3)603 2934 y Fu(\))j(and)f Fo(v)899 2901 y Fq(00)972 2934 y Fu(=)d Fh(A)p Fu(\()p Fo(v)1216 2901 y Fq(0)1213 2958 y Fn(1)1253 2934 y Fo(;)15 b(v)1340 2901 y Fq(0)1337 2958 y Fn(2)1377 2934 y Fo(;)g(v)1464 2901 y Fq(0)1461 2958 y Fn(3)1501 2934 y Fu(\))33 b(with)h Fo(v)1824 2948 y Fn(1)1894 2934 y Fu(=)c Fh(c)p Fu(\()p Fo(v)2117 2901 y Fq(0)2114 2958 y Fn(1)2154 2934 y Fu(\),)35 b Fo(v)2293 2948 y Fn(2)2364 2934 y Fu(=)30 b Fh(c)p Fu(\()p Fo(v)2587 2901 y Fq(0)2584 2958 y Fn(2)2624 2934 y Fu(\),)35 b(and)e Fo(v)2943 2948 y Fn(3)3013 2934 y Fu(=)d Fo(i)p 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Fr(\001)15 b(\001)g(\001)h Fo(\013)2656 3493 y Fm(i)2680 3501 y Fj(m)2785 3479 y Fu(=)42 b Fo(\014)2949 3493 y Fm(i)2973 3502 y Fk(1)3027 3479 y Fr(\001)15 b(\001)g(\001)h Fo(\014)3199 3493 y Fm(i)3223 3501 y Fj(m)3327 3479 y Fu(for)40 b(some)212 3592 y Fo(m)25 b Fl(>)g Fu(1)i(and)f(1)g Fl(6)f Fo(i)856 3606 y Fn(1)895 3592 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)1128 3606 y Fm(m)1221 3592 y Fl(6)25 b Fo(n)p Fu(.)39 b(Then)25 b(w)m(e)i(ha)m(v)m(e)h(the)f(follo)m(wing)h(div)m(erging)f(reductions)f (starting)212 3705 y(from)k Fo(t)25 b Fu(=)g Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\)\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\()p Fo(y)s Fu(\)\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(z)t Fu(\)\))q(:)439 3884 y Fo(t)25 b Fr(!)588 3847 y Fn(+)673 3884 y Fh(f)6 b Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(i)1173 3898 y Fm(m)1258 3884 y Fr(\001)g(\001)g(\001)h Fo(i)1410 3898 y Fn(1)1450 3884 y Fu(\()p Fh(c)p Fu(\()p Fo(z)t Fu(\)\)\))28 b Fr(!)d Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(i)2162 3898 y Fm(m)p Fq(\000)p Fn(1)2335 3884 y Fr(\001)g(\001)g(\001)h Fo(i)2487 3898 y Fn(1)2527 3884 y Fu(\()p Fh(c)p Fu(\()p Fo(z)t Fu(\)\)\))27 b Fr(!)2906 3847 y Fn(+)2990 3884 y Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))212 4064 y(and)439 4243 y Fo(t)25 b Fr(!)h Fh(g)q Fu(\()p Fo(\015)5 b Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\)\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\()p Fo(y)s Fu(\)\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(z)t Fu(\))q(\))31 b Fr(!)1653 4205 y Fn(+)1737 4243 y Fh(g)q Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g Fh(c)p Fu(\()p Fo(z)t Fu(\)\))p Fo(:)212 4422 y Fu(Since)24 b Fh(A)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))25 b(and)e Fh(g)q Fu(\()p Fh(c)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b Fh(c)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g Fh(c)p Fu(\()p Fo(z)t Fu(\)\))29 b(are)24 b(di\013eren)m(t)g(normal)g(forms,)g Fr(R)2842 4436 y Fn(7)2882 4422 y Fu(\()p Fo(P)13 b Fu(\))24 b(is)g(not)g (con\015uen)m(t.)353 4535 y(Con)m(v)m(ersely)-8 b(,)33 b(supp)s(ose)d(that)h Fr(R)1449 4549 y Fn(7)1489 4535 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)g(not)h(con\015uen)m(t.)42 b(If)31 b(w)m(e)g(can)g(sho)m(w)g(that)h Fr(R)3238 4502 y Fn(1)3238 4559 y(7)3277 4535 y Fu(\()p Fo(P)13 b Fu(\))32 b(is)f(not)212 4648 y(con\015uen)m(t)h(then)f(the)h(result)f(follo)m (ws)i(from)e(the)h(preceding)f(lemma.)45 b(F)-8 b(rom)32 b(an)g(insp)s(ection)f(of)h(the)212 4761 y(rewrite)h(rules)f(of)g Fr(R)925 4775 y Fn(7)965 4761 y Fu(\()p Fo(P)13 b Fu(\))33 b(w)m(e)f(easily)i(infer)e(that)h(the)f(set)h(of)g(terms)f(that)h(con)m (tain)h(an)e(o)s(ccurrence)212 4874 y(of)37 b Fh(D)f Fu(is)h(closed)h(under)d(con)m(v)m(ersion.)61 b(Moreo)m(v)m(er,)41 b(it)c(is)g(easy)g(to)h(pro)m(v)m(e)f(b)m(y)g(induction)f(on)h Fo(t)g Fu(that)212 4987 y(ev)m(ery)f(suc)m(h)g(term)f(rewrites)h(to)g Fh(D)p Fu(.)56 b(Hence,)38 b(if)d Fo(t)h Fu(is)f(a)h(non-con\015uen)m (t)g(term)f(then)g Fo(t)h Fu(con)m(tains)g(no)212 5100 y(o)s(ccurrences)26 b(of)g Fh(D)f Fu(and)g(rules)g(of)h Fr(R)1447 5067 y Fn(2)1447 5124 y(7)1512 5100 y Fu(are)g(nev)m(er)g (applied)f(in)g(an)m(y)h(rewrite)g(sequence)g(starting)h(from)212 5213 y Fo(t)p Fu(.)41 b(It)30 b(follo)m(ws)i(that)f Fr(R)982 5180 y Fn(1)982 5237 y(7)1021 5213 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(not)h(con\015uen)m(t.)1771 b Fl(\003)1897 5462 y Fu(12)p eop %%Page: 13 13 13 12 bop 212 337 a Ff(Corollary)35 b(5.7)46 b Fs(Con\015uenc)-5 b(e)34 b(is)e(not)h(semi-de)-5 b(cidable)34 b(for)f(line)-5 b(ar)34 b(gr)-5 b(ound-c)g(on\015uent)34 b(TRSs.)78 b Fl(\003)212 621 y Ft(6)135 b(CR)45 b Fd(\))g Ft(NF)212 824 y Fu(Let)31 b Fr(R)452 838 y Fn(8)491 824 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)831 838 y Fn(1)871 824 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(f)p Fh(B)25 b Fr(!)g Fh(B)p Fr(g)p Fu(.)212 1022 y Ff(Lemma)35 b(6.1)46 b Fs(The)29 b(TRS)h Fr(R)1233 1036 y Fn(8)1272 1022 y Fu(\()p Fo(P)13 b Fu(\))30 b Fs(has)g(the)g(normal)g(form)h(pr)-5 b(op)g(erty)31 b(for)f(every)f(PCP)g(instanc)-5 b(e)29 b Fo(P)13 b Fs(.)212 1154 y(Pr)-5 b(o)g(of)52 b Fu(The)29 b(set)h(of)f(normal)g(forms)g(of)g Fr(R)1639 1168 y Fn(8)1679 1154 y Fu(\()p Fo(P)13 b Fu(\))30 b(coincides)g(with)f(the)g(set)h(of)f (terms)g(that)h(rewrite)g(to)212 1267 y(a)i(normal)f(form.)43 b(In)31 b(other)h(w)m(ords,)f(reducible)g(terms)g(in)h Fr(R)2321 1281 y Fn(8)2360 1267 y Fu(\()p Fo(P)13 b Fu(\))32 b(ha)m(v)m(e)h(no)e(normal)g(form.)43 b(Hence)212 1380 y(the)31 b(normal)f(form)g(prop)s(ert)m(y)g(is)g(trivially)i (satis\014ed.)1560 b Fl(\003)212 1579 y Ff(Lemma)35 b(6.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 1593 y Fn(8)1279 1579 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e(and)h(only)h(if)e Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 1711 y(Pr)-5 b(o)g(of)55 b Fu(Since)32 b(the)h(relations)g Fr(!)1337 1678 y Fq(\003)1337 1742 y(R)1397 1751 y Fk(8)1432 1742 y Fn(\()p Fm(P)10 b Fn(\))1577 1711 y Fu(and)32 b Fr(!)1847 1678 y Fq(\003)1847 1742 y(R)1907 1751 y Fk(1)1942 1742 y Fn(\()p Fm(P)10 b Fn(\))2088 1711 y Fu(coincide,)34 b Fr(R)2541 1725 y Fn(8)2580 1711 y Fu(\()p Fo(P)13 b Fu(\))33 b(is)f(con\015uen)m(t)g(if)h(and)e(only)212 1836 y(if)f Fr(R)372 1850 y Fn(1)412 1836 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(con\015uen)m(t.)41 b(Hence)31 b(the)g(result)f(follo) m(ws)i(from)e(Lemma)g(2.3.)843 b Fl(\003)212 2034 y Ff(Corollary)35 b(6.3)46 b Fs(Con\015uenc)-5 b(e)44 b(is)f(not)g(c)-5 b(o-semi-de)g(cidable)44 b(for)f(line)-5 b(ar)44 b(TRSs)f(with)h(the)f (normal)212 2147 y(form)34 b(pr)-5 b(op)g(erty.)2823 b Fl(\003)353 2345 y Fu(Let)31 b Fr(R)593 2359 y Fn(9)633 2345 y Fu(\()p Fo(P)13 b Fu(\))31 b(b)s(e)e(the)i(union)e(of)i Fr(R)1518 2359 y Fn(5)1557 2345 y Fu(\()p Fo(P)13 b Fu(\))31 b(and)f(the)h(rules)481 2532 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))184 b Fo(a)p Fu(\()p Fo(x)p Fu(\))31 b Fr(!)f Fo(a)p Fu(\()p Fo(x)p Fu(\))182 b Fh(A)30 b Fr(!)g Fh(A)555 2645 y(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))32 b Fr(!)e Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))274 b Fo(i)p Fu(\()p Fo(x)p Fu(\))32 b Fr(!)e Fo(i)p Fu(\()p Fo(x)p Fu(\))220 b Fh(c)30 b Fr(!)g Fh(c)212 2841 y Ff(Lemma)35 b(6.4)46 b Fs(The)29 b(TRS)h Fr(R)1233 2855 y Fn(9)1272 2841 y Fu(\()p Fo(P)13 b Fu(\))30 b Fs(has)g(the)g(normal)g(form)h(pr) -5 b(op)g(erty)31 b(for)f(every)f(PCP)g(instanc)-5 b(e)29 b Fo(P)13 b Fs(.)212 2973 y(Pr)-5 b(o)g(of)57 b Fu(Only)34 b(v)-5 b(ariables)35 b(are)g(in)f(normal)h(form.)52 b(Since)35 b Fr(R)2249 2987 y Fn(9)2288 2973 y Fu(\()p Fo(P)13 b Fu(\))36 b(is)e(non-collapsing,)j(a)e(v)-5 b(ariable)35 b(is)212 3086 y(con)m(v)m(ertible)d(only)f(with)f(itself.)42 b(Hence)31 b(the)f(normal)h(form)f(prop)s(ert)m(y)f(is)i(trivially)g (satis\014ed.)166 b Fl(\003)212 3284 y Ff(Lemma)35 b(6.5)46 b Fs(The)33 b(TRS)g Fr(R)1240 3298 y Fn(9)1279 3284 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(is)g(c)-5 b(on\015uent)34 b(if)e(and)h(only)h(if)e Fo(P)46 b Fs(admits)34 b(no)f(solution.)212 3417 y(Pr)-5 b(o)g(of)53 b Fu(Since)31 b Fr(!)804 3384 y Fq(\003)804 3448 y(R)864 3457 y Fk(9)899 3448 y Fn(\()p Fm(P)10 b Fn(\))1037 3417 y Fu(=)25 b Fr(!)1224 3384 y Fq(\003)1224 3448 y(R)1284 3457 y Fk(5)1319 3448 y Fn(\()p Fm(P)10 b Fn(\))1433 3417 y Fu(,)30 b(the)h(result)f(follo)m (ws)i(from)d(Lemma)i(4.5.)727 b Fl(\003)212 3615 y Ff(Corollary)35 b(6.6)46 b Fs(Con\015uenc)-5 b(e)32 b(is)g(not)g(semi-de)-5 b(cidable)32 b(for)g(line)-5 b(ar)33 b(TRSs)f(with)g(the)g(normal)h (form)212 3728 y(pr)-5 b(op)g(erty.)3043 b Fl(\003)212 4012 y Ft(7)135 b(NF)44 b Fd(\))h Ft(UN)212 4215 y Ff(Lemma)35 b(7.1)46 b Fs(The)33 b(TRS)g Fr(R)1240 4229 y Fn(1)1279 4215 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(has)h(unique)e(normal)i(forms)g (for)f(every)g(PCP)f(instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 4348 y(Pr)-5 b(o)g(of)65 b Fu(Consider)42 b(the)h(con\015uen)m(t)h(TRS) e Fr(R)1756 4315 y Fq(0)1756 4372 y Fn(1)1795 4348 y Fu(\()p Fo(P)13 b Fu(\))44 b(de\014ned)d(in)i(the)g(pro)s(of)g(of)g (Lemma)g(2.3.)80 b(The)212 4460 y(relations)32 b Fr($)673 4427 y Fq(\003)673 4492 y(R)733 4501 y Fk(1)768 4492 y Fn(\()p Fm(P)10 b Fn(\))912 4460 y Fu(and)31 b Fr($)1181 4427 y Fq(\003)1181 4493 y(R)1241 4470 y Fe(0)1241 4514 y Fk(1)1276 4493 y Fn(\()p Fm(P)10 b Fn(\))1420 4460 y Fu(clearly)32 b(coincide.)44 b(Also)32 b(the)f(normal)g(forms)g(of)g (the)g(t)m(w)m(o)h(TRSs)212 4590 y(are)f(the)f(same.)42 b(It)30 b(follo)m(ws)i(that)f Fr(R)1455 4604 y Fn(1)1494 4590 y Fu(\()p Fo(P)13 b Fu(\))31 b(has)f(unique)g(normal)g(forms.)925 b Fl(\003)353 4788 y Fu(W)-8 b(e)23 b(already)f(observ)m(ed)g(\(Lemma)h (2.3\))g(that)f(the)g(TRS)f Fr(R)2322 4802 y Fn(1)2361 4788 y Fu(\()p Fo(P)13 b Fu(\))23 b(has)e(the)h(normal)g(form)f(prop)s (ert)m(y)212 4901 y(if)30 b(and)g(only)h(if)f Fo(P)43 b Fu(has)30 b(a)h(solution.)212 5100 y Ff(Corollary)k(7.2)46 b Fs(The)28 b(normal)i(form)e(pr)-5 b(op)g(erty)31 b(is)d(not)g(c)-5 b(o-semi-de)g(cidable)29 b(for)f(line)-5 b(ar)29 b(TRSs)f(with)212 5213 y(unique)k(normal)i(forms.)2535 b Fl(\003)1897 5462 y Fu(13)p eop %%Page: 14 14 14 13 bop 353 337 a Fu(Let)31 b Fr(R)593 351 y Fn(10)668 337 y Fu(\()p Fo(P)13 b Fu(\))31 b(b)s(e)f(the)g(union)g(of)g Fr(R)1553 351 y Fn(5)1593 337 y Fu(\()p Fo(P)13 b Fu(\))31 b(and)e(the)i(t)m(w)m(o)h(rules)481 509 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))32 b Fr(!)e Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))555 622 y Fh(g)q Fu(\()p Fo(x;)g(y)s Fu(\))32 b Fr(!)e Fh(g)q Fu(\()p Fo(x;)15 b(y)s Fu(\))212 803 y Ff(Lemma)35 b(7.3)46 b Fs(The)33 b(TRS)g Fr(R)1240 817 y Fn(10)1314 803 y Fu(\()p Fo(P)13 b Fu(\))34 b Fs(has)f(unique)f(normal)j(forms)e(for)g(every)g(PCP)f (instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 935 y(Pr)-5 b(o)g(of)56 b Fu(Assume)33 b Fo(n)873 949 y Fn(1)942 935 y Fr($)1033 902 y Fq(\003)1033 967 y(R)1093 976 y Fk(10)1158 967 y Fn(\()p Fm(P)10 b Fn(\))1302 935 y Fo(n)1357 949 y Fn(2)1429 935 y Fu(for)33 b(t)m(w)m(o)i(normal)e(forms)g Fo(n)2366 949 y Fn(1)2405 935 y Fu(,)h Fo(n)2519 949 y Fn(2)2558 935 y Fu(;)h(w)m(e)f(ha)m(v)m(e)g(to)g(pro)m(v)m(e)g Fo(n)3384 949 y Fn(1)3453 935 y Fu(=)c Fo(n)3609 949 y Fn(2)3648 935 y Fu(.)212 1060 y(F)-8 b(or)30 b(a)f(term)g Fo(t)p Fu(,)g(let)h Fo(\036)p Fu(\()p Fo(t)p Fu(\))f(denote)g(the)g (result)g(of)g(replacing)h(all)f(maximal)h(subterms)d(in)i Fo(t)f Fu(with)h(ro)s(ot)212 1173 y(sym)m(b)s(ol)24 b Fh(f)31 b Fu(or)24 b Fh(g)i Fu(b)m(y)e Fh(A)p Fu(.)38 b(An)24 b(insp)s(ection)h(of)f(the)h(rewrite)f(rules)g(of)h Fr(R)2503 1187 y Fn(10)2577 1173 y Fu(\()p Fo(P)13 b Fu(\))25 b(rev)m(eals)h(that)f Fo(\036)p Fu(\()p Fo(s)p Fu(\))h(=)f Fo(\036)p Fu(\()p Fo(t)p Fu(\))212 1285 y(if)34 b Fo(s)d Fr(!)464 1304 y Fq(R)524 1313 y Fk(10)589 1304 y Fn(\()p Fm(P)10 b Fn(\))734 1285 y Fo(t)p Fu(.)52 b(It)34 b(follo)m(ws)i(that)e Fo(\036)p Fu(\()p Fo(n)1593 1299 y Fn(1)1633 1285 y Fu(\))e(=)f Fo(\036)p Fu(\()p Fo(n)1946 1299 y Fn(2)1985 1285 y Fu(\).)53 b(Since)34 b Fo(n)2394 1299 y Fn(1)2433 1285 y Fu(,)h Fo(n)2548 1299 y Fn(2)2621 1285 y Fu(are)f(normal)g(forms,)h(they)f(do)212 1398 y(not)d(con)m(tain)g(the)g(sym)m(b)s(ols)f Fh(f)36 b Fu(and)30 b Fh(g)q Fu(,)h(and)f(th)m(us)g Fo(n)1973 1412 y Fn(1)2037 1398 y Fu(=)25 b Fo(\036)p Fu(\()p Fo(n)2277 1412 y Fn(1)2317 1398 y Fu(\))h(=)e Fo(\036)p Fu(\()p Fo(n)2617 1412 y Fn(2)2657 1398 y Fu(\))i(=)f Fo(n)2869 1412 y Fn(2)2908 1398 y Fu(.)670 b Fl(\003)212 1582 y Ff(Lemma)35 b(7.4)46 b Fs(The)38 b(TRS)g Fr(R)1250 1596 y Fn(10)1325 1582 y Fu(\()p Fo(P)13 b Fu(\))38 b Fs(has)h(the)f(normal) i(form)e(pr)-5 b(op)g(erty)41 b(if)c(and)i(only)g(if)e Fo(P)51 b Fs(admits)212 1695 y(no)33 b(solution.)212 1827 y(Pr)-5 b(o)g(of)55 b Fu(Assume)32 b(that)g Fo(P)45 b Fu(admits)33 b(a)f(solution.)47 b(Then)31 b(w)m(e)i(ha)m(v)m(e)g(a)f (con)m(v)m(ersion)i Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))31 b Fr($)3346 1794 y Fq(\003)3346 1859 y(R)3406 1868 y Fk(10)3471 1859 y Fn(\()p Fm(P)10 b Fn(\))3613 1827 y Fh(A)212 1952 y Fu(as)29 b(in)g(the)g(\014rst)f(part)h(of)g(the) g(pro)s(of)f(of)h(Lemma)h(4.5.)41 b(Since)29 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fu(\))31 b(do)s(es)e(not)g(rewrite)g (to)h Fh(A)e Fu(and)h Fh(A)212 2065 y Fu(is)i(a)f(normal)h(form,)f(w)m (e)h(conclude)f(that)h Fr(R)1709 2079 y Fn(10)1784 2065 y Fu(\()p Fo(P)13 b Fu(\))31 b(do)s(es)f(not)h(ha)m(v)m(e)g(the)g (normal)f(form)g(prop)s(ert)m(y)-8 b(.)353 2178 y(Con)m(v)m(ersely)g(,) 28 b(assume)c(that)i Fo(P)38 b Fu(admits)25 b(no)f(solution.)40 b(Then)24 b(according)i(to)f(Lemma)g(4.5)h Fr(R)3492 2192 y Fn(5)3532 2178 y Fu(\()p Fo(P)13 b Fu(\))212 2290 y(is)32 b(con\015uen)m(t.)46 b(Since)33 b Fr(!)1068 2257 y Fq(\003)1068 2322 y(R)1128 2331 y Fk(10)1193 2322 y Fn(\()p Fm(P)10 b Fn(\))1335 2290 y Fu(=)27 b Fr(!)1524 2257 y Fq(\003)1524 2322 y(R)1584 2331 y Fk(5)1619 2322 y Fn(\()p Fm(P)10 b Fn(\))1765 2290 y Fu(the)32 b(TRS)f Fr(R)2215 2304 y Fn(10)2290 2290 y Fu(\()p Fo(P)13 b Fu(\))33 b(is)f(con\015uen)m(t)g(and)g(hence)g(has)g(the)212 2403 y(normal)e(form)g(prop)s(ert)m(y)-8 b(.)2509 b Fl(\003)212 2587 y Ff(Corollary)35 b(7.5)46 b Fs(The)40 b(normal)h(form)f(pr)-5 b(op)g(erty)43 b(is)c(not)h(semi-de)-5 b(cidable)40 b(for)g(line)-5 b(ar)40 b(TRSs)g(with)212 2700 y(unique)32 b(normal)i(forms.)2535 b Fl(\003)212 2982 y Ft(8)135 b(UN)45 b Fd(\))g Ft(CON)212 3184 y Fu(Let)31 b Fr(R)452 3198 y Fn(11)527 3184 y Fu(\()p Fo(P)13 b Fu(\))31 b(b)s(e)e(the)i(union)f(of)g Fr(R)1412 3198 y Fn(2)1452 3184 y Fu(\()p Fo(P)13 b Fu(\))31 b(and)e(the)i(rules) 481 3357 y Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t(;)g(w)r Fu(\))33 b Fr(!)d Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t(;)g(w)r Fu(\))184 b Fo(a)p Fu(\()p Fo(x)p Fu(\))31 b Fr(!)g Fo(a)p Fu(\()p Fo(x)p Fu(\))182 b Fh(c)31 b Fr(!)f Fh(c)576 3470 y(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))33 b Fr(!)d Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))296 b Fo(i)p Fu(\()p Fo(x)p Fu(\))31 b Fr(!)g Fo(i)p Fu(\()p Fo(x)p Fu(\))212 3646 y Ff(Lemma)k(8.1)46 b Fh(A)25 b Fr($)939 3613 y Fq(\003)939 3678 y(R)999 3687 y Fk(11)1064 3678 y Fn(\()p Fm(P)10 b Fn(\))1203 3646 y Fh(B)32 b Fs(if)g(and)i(only)f(if)f Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 3779 y(Pr)-5 b(o)g(of)53 b Fu(Immediate)31 b(consequence)h(of)e(Lemma)h (2.5)g(as)g Fr($)2218 3746 y Fq(\003)2218 3810 y(R)2278 3819 y Fk(11)2343 3810 y Fn(\()p Fm(P)10 b Fn(\))2482 3779 y Fu(=)25 b Fr($)2669 3746 y Fq(\003)2669 3810 y(R)2729 3819 y Fk(2)2763 3810 y Fn(\()p Fm(P)10 b Fn(\))2877 3779 y Fu(.)701 b Fl(\003)212 3974 y Ff(Lemma)35 b(8.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 3988 y Fn(11)1314 3974 y Fu(\()p Fo(P)13 b Fu(\))34 b Fs(is)e(c)-5 b(onsistent)34 b(for)f(every)g(PCP)f(instanc)-5 b(e)33 b Fo(P)13 b Fs(.)212 4106 y(Pr)-5 b(o)g(of)53 b Fu(T)-8 b(rivial,)32 b(as)e Fr(R)982 4120 y Fn(11)1057 4106 y Fu(\()p Fo(P)13 b Fu(\))31 b(lac)m(ks)h(collapsing)g(rules.)1518 b Fl(\003)212 4290 y Ff(Lemma)35 b(8.3)46 b Fs(The)31 b(TRS)f Fr(R)1235 4304 y Fn(11)1310 4290 y Fu(\()p Fo(P)13 b Fu(\))31 b Fs(has)h(unique)d(normal)j(forms)g(if)e(and)h(only)h(if)e Fo(P)43 b Fs(do)-5 b(es)32 b(not)f(have)212 4403 y(a)i(solution.)212 4535 y(Pr)-5 b(o)g(of)67 b Fu(According)45 b(to)g(Lemma)g(8.1)h(w)m(e)f (ha)m(v)m(e)h(to)f(sho)m(w)g(that)g Fr(R)2585 4549 y Fn(11)2660 4535 y Fu(\()p Fo(P)13 b Fu(\))45 b(admits)g(t)m(w)m(o)h (di\013eren)m(t)212 4648 y(con)m(v)m(ertible)c(normal)e(forms)f(if)h (and)f(only)h(if)g Fh(A)g Fu(and)f Fh(B)g Fu(are)i(con)m(v)m(ertible.) 71 b(Since)40 b Fh(A)f Fu(and)g Fh(B)h Fu(are)212 4761 y(normal)30 b(forms,)g(the)h(\\if)7 b(")31 b(direction)g(is)g(trivial.) 353 4874 y(Con)m(v)m(ersely)-8 b(,)36 b(supp)s(ose)c(that)i Fr(R)1457 4888 y Fn(11)1532 4874 y Fu(\()p Fo(P)13 b Fu(\))34 b(admits)f(t)m(w)m(o)i(di\013eren)m(t)f(con)m(v)m(ertible)i (normal)d(forms)g Fo(t)3609 4888 y Fn(1)3648 4874 y Fu(,)212 4987 y Fo(t)245 5001 y Fn(2)284 4987 y Fu(.)47 b(The)32 b(only)g(normal)g(forms)g(of)g Fr(R)1491 5001 y Fn(11)1566 4987 y Fu(\()p Fo(P)13 b Fu(\))33 b(are)g Fh(A)p Fu(,)f Fh(B)p Fu(,)h(and)f(v)-5 b(ariables.)47 b(Because)34 b Fr(R)3155 5001 y Fn(11)3229 4987 y Fu(\()p Fo(P)13 b Fu(\))33 b(is)g(non-)212 5100 y(collapsing,)40 b(v)-5 b(ariables)37 b(are)h(con)m(v)m(ertible)g(only)f(to)h(themselv)m(es.)61 b(Hence)37 b Fo(t)2818 5114 y Fn(1)2893 5100 y Fu(=)f Fh(A)g Fu(and)g Fo(t)3313 5114 y Fn(2)3388 5100 y Fu(=)g Fh(B)g Fu(or)212 5213 y(vice-v)m(ersa.)2981 b Fl(\003)1897 5462 y Fu(14)p eop %%Page: 15 15 15 14 bop 212 337 a Ff(Corollary)35 b(8.4)46 b Fs(The)38 b(pr)-5 b(op)g(erty)40 b(of)e(having)g(unique)f(normal)i(forms)g(is)e (not)h(semi-de)-5 b(cidable)39 b(for)212 450 y(line)-5 b(ar)34 b(c)-5 b(onsistent)34 b(TRSs.)2464 b Fl(\003)212 662 y Ff(Theorem)35 b(8.5)46 b Fs(The)g(pr)-5 b(op)g(erty)48 b(of)d(having)h(unique)f(normal)i(forms)f(is)f(c)-5 b(o-semi-de)g (cidable)47 b(for)212 775 y(TRSs.)212 907 y(Pr)-5 b(o)g(of)52 b Fu(By)29 b(en)m(umerating)g(all)h(con)m(v)m(ersions)g(w)m(e)f(can)g (easily)h(\014nd)e(out)h(if)f(there)h(exists)h(a)f(con)m(v)m(ersion)212 1020 y(b)s(et)m(w)m(een)g(di\013eren)m(t)g(normal)f(forms.)39 b(Hence)29 b(the)g(prop)s(ert)m(y)f(of)g(not)g(ha)m(ving)h(unique)f (normal)g(forms)212 1133 y(is)j(semi-decidable.)2699 b Fl(\003)212 1420 y Ft(9)135 b(CON)45 b Fd(\))f Ft(CON)1229 1376 y Fv(!)212 1623 y Fu(Let)31 b Fr(R)452 1637 y Fn(12)527 1623 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)867 1637 y Fn(2)906 1623 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(R)1226 1590 y Fn(1)1226 1647 y(12)1331 1623 y Fu(with)30 b Fr(R)1615 1590 y Fn(1)1615 1647 y(12)1720 1623 y Fu(consisting)h(of)g(the)f(t)m (w)m(o)i(rules)481 1826 y Fh(e)p Fu(\()p Fh(A)p Fo(;)15 b(x)p Fu(\))31 b Fr(!)f Fh(C)481 1939 y(e)p Fu(\()p Fh(B)p Fo(;)15 b(x)p Fu(\))31 b Fr(!)f Fo(x)212 2144 y Ff(Lemma)35 b(9.1)46 b Fh(A)25 b Fr($)939 2111 y Fq(\003)939 2176 y(R)999 2185 y Fk(12)1064 2176 y Fn(\()p Fm(P)10 b Fn(\))1203 2144 y Fh(B)32 b Fs(if)g(and)i(only)f(if)f Fo(P)46 b Fs(admits)34 b(a)f(solution.)212 2296 y(Pr)-5 b(o)g(of)49 b Fu(W)-8 b(e)27 b(sho)m(w)f(that)g(in)f(a)h(shortest)g(con)m(v)m (ersion)h(b)s(et)m(w)m(een)g Fh(A)e Fu(and)g Fh(B)g Fu(rules)h(of)f Fr(R)3066 2263 y Fn(1)3066 2320 y(12)3167 2296 y Fu(are)h(not)g(used.) 212 2409 y(The)39 b(desired)g(result)h(then)f(follo)m(ws)i(from)e (Lemma)h(2.5.)70 b(Supp)s(ose)38 b(to)i(the)g(con)m(trary)g(that)h(in)e (a)212 2522 y(shortest)28 b(con)m(v)m(ersion)h(b)s(et)m(w)m(een)f Fh(A)g Fu(and)f Fh(B)g Fu(rules)g(of)h Fr(R)2086 2489 y Fn(1)2086 2546 y(12)2189 2522 y Fu(are)g(used.)39 b(It)28 b(is)f(not)h(di\016cult)g(to)g(see)h(that)212 2635 y(this)f(is)f(only)h (p)s(ossible)f(if)h(the)f(con)m(v)m(ersion)i(con)m(tains)g(an)f (outermost)g Fh(e)g Fu(sym)m(b)s(ol)f(that)h(is)g(in)m(tro)s(duced)212 2748 y(and)i(eliminated)h(b)m(y)g(a)f(rule)g(of)h Fr(R)1398 2715 y Fn(1)1398 2772 y(12)1473 2748 y Fu(.)40 b(In)30 b(other)h(w)m(ords,)f(there)g(exists)h(a)g(fragmen)m(t)439 2952 y Fo(C)7 b Fu([)p Fo(t)p Fu(])25 b Fr( )h Fo(C)7 b Fu([)p Fh(e)p Fu(\()p Fo(t)941 2966 y Fn(1)980 2952 y Fo(;)15 b(t)1053 2966 y Fn(2)1093 2952 y Fu(\)])26 b Fr($)1270 2914 y Fq(\003)1334 2952 y Fo(C)1406 2914 y Fq(0)1429 2952 y Fu([)p Fh(e)p Fu(\()p Fo(t)1562 2914 y Fq(0)1562 2974 y Fn(1)1603 2952 y Fo(;)15 b(t)1676 2914 y Fq(0)1676 2974 y Fn(2)1715 2952 y Fu(\)])736 3010 y Fg(|)p 777 3010 439 11 v 439 w({z)p 1298 3010 V 439 w(})1801 2952 y Fr(!)25 b Fo(C)1989 2914 y Fq(0)2012 2952 y Fu([)p Fo(t)2070 2914 y Fq(0)2093 2952 y Fu(])1439 b(\(1\))212 3205 y(suc)m(h)30 b(that)h(ev)m(ery)g(step)f(in)g(the)g (underbraced)e(part)i(is)h(of)f(the)g(form)g Fo(C)2656 3219 y Fn(1)2695 3205 y Fu([)p Fh(e)p Fu(\()p Fo(s)2838 3219 y Fn(1)2878 3205 y Fo(;)15 b(s)2961 3219 y Fn(2)3001 3205 y Fu(\)])26 b Fr($)f Fo(C)3268 3219 y Fn(2)3307 3205 y Fu([)p Fh(e)p Fu(\()p Fo(s)3450 3172 y Fq(0)3450 3229 y Fn(1)3490 3205 y Fo(;)15 b(s)3573 3172 y Fq(0)3573 3229 y Fn(2)3613 3205 y Fu(\)])212 3318 y(with)28 b(no)g(o)s (ccurrences)g(of)g Fh(e)g Fu(ab)s(o)m(v)m(e)i(the)e(displa)m(y)m(ed)g (ones)h(and)e(either)i(\(i\))g(a)f(rewrite)g(rule)g(of)g Fr(R)3492 3332 y Fn(2)3532 3318 y Fu(\()p Fo(P)13 b Fu(\))212 3431 y(is)25 b(applied)g(ab)s(o)m(v)m(e)h(the)f(displa)m(y)m(ed)h(o)s (ccurrences)f(of)g Fh(e)p Fu(,)h(\(ii\))g(a)g(rewrite)f(rule)g(is)g (applied)f(to)i(a)f(subterm)212 3543 y(of)34 b Fo(C)384 3557 y Fn(1)424 3543 y Fu([)p Fh(e)p Fu(\()p Fo(s)567 3557 y Fn(1)607 3543 y Fo(;)15 b(s)690 3557 y Fn(2)729 3543 y Fu(\)])35 b(\()p Fo(C)924 3557 y Fn(2)963 3543 y Fu([)p Fh(e)p Fu(\()p Fo(s)1106 3510 y Fq(0)1106 3568 y Fn(1)1146 3543 y Fo(;)15 b(s)1229 3510 y Fq(0)1229 3568 y Fn(2)1269 3543 y Fu(\)]\))35 b(disjoin)m(t)f(from)f Fh(e)p Fu(\()p Fo(s)2062 3557 y Fn(1)2102 3543 y Fo(;)15 b(s)2185 3557 y Fn(2)2225 3543 y Fu(\))34 b(\()p Fh(e)p Fu(\()p Fo(s)2447 3510 y Fq(0)2447 3568 y Fn(1)2487 3543 y Fo(;)15 b(s)2570 3510 y Fq(0)2570 3568 y Fn(2)2609 3543 y Fu(\)\),)36 b(or)e(\(iii\))h(a)g(rewrite)f(rule)f(is)212 3656 y(applied)27 b(to)i(one)e(the)h(argumen)m(ts)g(of)f(the)h(displa)m (y)m(ed)g(o)s(ccurrences)f(of)h Fh(e)p Fu(.)40 b(Because)29 b Fr(R)3154 3670 y Fn(2)3193 3656 y Fu(\()p Fo(P)13 b Fu(\))28 b(is)g(linear)212 3769 y(and)37 b(v)-5 b(ariable)39 b(preserving,)g(w)m(e)f(can)g(shift)f(all)i(\(i\))g(steps)e(in)h(the)f (underbraced)f(part)i(in)f(fron)m(t)h(of)212 3882 y Fo(C)7 b Fu([)p Fo(t)p Fu(].)41 b(W)-8 b(e)31 b(do)f(the)h(same)g(with)f(all)h (\(ii\))h(steps.)40 b(The)30 b(result)g(is)h(a)g(new)e(fragmen)m(t)439 4086 y Fo(C)7 b Fu([)p Fo(t)p Fu(])25 b Fr($)710 4049 y Fq(\003)775 4086 y Fo(C)847 4049 y Fq(0)870 4086 y Fu([)p Fo(t)p Fu(])g Fr( )h Fo(C)1167 4049 y Fq(0)1189 4086 y Fu([)p Fh(e)p Fu(\()p Fo(t)1322 4100 y Fn(1)1363 4086 y Fo(;)15 b(t)1436 4100 y Fn(2)1475 4086 y Fu(\)])26 b Fr($)1652 4049 y Fq(\003)1717 4086 y Fo(C)1789 4049 y Fq(0)1811 4086 y Fu([)p Fh(e)p Fu(\()p Fo(t)1944 4049 y Fq(0)1944 4109 y Fn(1)1985 4086 y Fo(;)15 b(t)2058 4049 y Fq(0)2058 4109 y Fn(2)2097 4086 y Fu(\)])1095 4145 y Fg(|)p 1136 4145 450 11 v 450 w({z)p 1668 4145 V 450 w(})2183 4086 y Fr(!)25 b Fo(C)2371 4049 y Fq(0)2394 4086 y Fu([)p Fo(t)2452 4049 y Fq(0)2476 4086 y Fu(])1056 b(\(2\))212 4338 y(of)33 b(the)h(same)f(length)h(as)f(\(1\))h(suc)m(h)f (that)h(all)g(steps)f(in)g(the)g(underbraced)e(part)i(tak)m(e)i(place)f (b)s(elo)m(w)212 4451 y(the)d(displa)m(y)m(ed)g(o)s(ccurrences)f(of)h Fh(e)p Fu(.)41 b(So)30 b Fo(t)1623 4465 y Fn(1)1687 4451 y Fr($)1778 4418 y Fq(\003)1843 4451 y Fo(t)1876 4418 y Fq(0)1876 4475 y Fn(1)1946 4451 y Fu(and)f Fo(t)2155 4465 y Fn(2)2220 4451 y Fr($)2311 4418 y Fq(\003)2375 4451 y Fo(t)2408 4418 y Fq(0)2408 4475 y Fn(2)2448 4451 y Fu(.)40 b(There)30 b(are)h(four)f(p)s(ossibilities:)323 4638 y(1.)46 b Fo(t)472 4652 y Fn(1)537 4638 y Fu(=)25 b Fo(t)666 4605 y Fq(0)666 4663 y Fn(1)730 4638 y Fu(=)g Fh(B)p Fu(,)30 b Fo(t)c Fu(=)f Fo(t)1130 4652 y Fn(2)1169 4638 y Fu(,)30 b(and)g Fo(t)1434 4605 y Fq(0)1482 4638 y Fu(=)25 b Fo(t)1611 4605 y Fq(0)1611 4663 y Fn(2)1651 4638 y Fu(,)323 4826 y(2.)46 b Fo(t)472 4840 y Fn(1)537 4826 y Fu(=)25 b Fo(t)666 4793 y Fq(0)666 4850 y Fn(1)730 4826 y Fu(=)g Fh(A)30 b Fu(and)g Fo(t)25 b Fu(=)g Fo(t)1281 4793 y Fq(0)1329 4826 y Fu(=)g Fh(C)p Fu(,)323 5014 y(3.)46 b Fo(t)472 5028 y Fn(1)537 5014 y Fu(=)25 b Fh(B)p Fu(,)30 b Fo(t)782 4981 y Fq(0)782 5038 y Fn(1)847 5014 y Fu(=)24 b Fh(A)p Fu(,)31 b Fo(t)25 b Fu(=)g Fo(t)1246 5028 y Fn(2)1285 5014 y Fu(,)31 b(and)e Fo(t)1550 4981 y Fq(0)1599 5014 y Fu(=)c Fh(C)p Fu(,)323 5201 y(4.)46 b Fo(t)472 5215 y Fn(1)537 5201 y Fu(=)25 b Fh(A)p Fu(,)30 b Fo(t)782 5168 y Fq(0)782 5225 y Fn(1)847 5201 y Fu(=)24 b Fh(B)p Fu(,)31 b Fo(t)25 b Fu(=)g Fh(C)p Fu(,)30 b(and)g Fo(t)1536 5168 y Fq(0)1584 5201 y Fu(=)25 b Fo(t)1713 5168 y Fq(0)1713 5225 y Fn(2)1753 5201 y Fu(.)1897 5462 y(15)p eop %%Page: 16 16 16 15 bop 212 337 a Fu(In)30 b(the)g(\014rst)g(case)h(w)m(e)g(obtain)g (the)g(shorter)f(fragmen)m(t)439 541 y Fo(C)7 b Fu([)p Fo(t)p Fu(])25 b Fr($)710 503 y Fq(\003)775 541 y Fo(C)847 503 y Fq(0)870 541 y Fu([)p Fo(t)p Fu(])g(=)g Fo(C)1146 503 y Fq(0)1169 541 y Fu([)p Fo(t)1227 555 y Fn(2)1267 541 y Fu(])g Fr($)1408 503 y Fq(\003)1473 541 y Fo(C)1545 503 y Fq(0)1568 541 y Fu([)p Fo(t)1626 503 y Fq(0)1626 563 y Fn(2)1665 541 y Fu(])h(=)e Fo(C)1883 503 y Fq(0)1906 541 y Fu([)p Fo(t)1964 503 y Fq(0)1988 541 y Fu(])212 745 y(con)m(tradicting)40 b(the)e(fact)g(that)h(the)f(giv)m(en)g(con)m (v)m(ersion)h(b)s(et)m(w)m(een)g Fh(A)e Fu(and)g Fh(B)g Fu(is)h(shortest.)63 b(In)37 b(the)212 858 y(second)31 b(case)g(w)m(e)g(obtain)g(the)f(shorter)g(fragmen)m(t)439 1062 y Fo(C)7 b Fu([)p Fo(t)p Fu(])25 b Fr($)710 1025 y Fq(\003)775 1062 y Fo(C)847 1025 y Fq(0)870 1062 y Fu([)p Fo(t)p Fu(])g(=)g Fo(C)1146 1025 y Fq(0)1169 1062 y Fu([)p Fh(C)p Fu(])h(=)f Fo(C)1471 1025 y Fq(0)1494 1062 y Fu([)p Fo(t)1552 1025 y Fq(0)1575 1062 y Fu(])p Fo(:)212 1266 y Fu(In)39 b(the)i(third)e(case)i(w)m(e)f(ha)m(v)m(e)h (the)g(shorter)e(con)m(v)m(ersion)j Fh(B)e Fu(=)h Fo(t)2487 1280 y Fn(1)2568 1266 y Fr($)2659 1233 y Fq(\003)2740 1266 y Fo(t)2773 1233 y Fq(0)2773 1291 y Fn(1)2853 1266 y Fu(=)g Fh(A)f Fu(b)s(et)m(w)m(een)g Fh(A)g Fu(and)212 1379 y Fh(B)p Fu(,)34 b(again)g(con)m(tradicting)h(the)e(fact)h(that)g (the)f(giv)m(en)h(con)m(v)m(ersion)h(b)s(et)m(w)m(een)f Fh(A)e Fu(and)h Fh(B)g Fu(is)g(shortest.)212 1492 y(Finally)-8 b(,)32 b(in)e(the)h(fourth)e(case)j(w)m(e)f(obtain)f(a)h(con)m (tradiction)h(in)e(the)h(same)g(w)m(a)m(y)-8 b(.)627 b Fl(\003)212 1705 y Ff(Lemma)35 b(9.2)46 b Fs(The)33 b(TRS)g Fr(R)1240 1719 y Fn(12)1314 1705 y Fu(\()p Fo(P)13 b Fu(\))34 b Fs(is)e(c)-5 b(onsistent)34 b(with)g(r)-5 b(esp)g(e)g(ct)34 b(to)f(r)-5 b(e)g(duction.)212 1837 y(Pr)g(o)g(of)51 b Fu(If)26 b Fr(R)637 1851 y Fn(12)712 1837 y Fu(\()p Fo(P)13 b Fu(\))28 b(is)f(not)h(consisten)m(t)g(with)f (resp)s(ect)h(to)f(reduction)h(then)f(there)g(m)m(ust)g(b)s(e)g (di\013eren)m(t)212 1950 y(v)-5 b(ariables)31 b Fo(x)f Fu(and)g Fo(y)j Fu(and)d(a)h(term)f Fo(t)g Fu(suc)m(h)g(that)439 2154 y Fo(x)c Fr( )f Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(x)p Fu(\))922 2117 y Fq(\003)962 2154 y Fr( )25 b Fo(t)g Fr(!)1227 2117 y Fq(\003)1291 2154 y Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(y)s Fu(\))27 b Fr(!)e Fo(y)212 2359 y Fu(b)s(ecause)30 b Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(x)p Fu(\))27 b Fr(!)e Fo(x)30 b Fu(is)h(the)f(only)h(collapsing)g(rule)f(in)h Fr(R)2269 2373 y Fn(8)2308 2359 y Fu(\()p Fo(P)13 b Fu(\).)41 b(Only)30 b(terms)g(in)g(the)h(set)439 2563 y Fo(S)495 2577 y Fm(x)564 2563 y Fu(=)25 b Fr(f)p Fh(e)p Fu(\()p Fo(t)813 2577 y Fn(1)854 2563 y Fo(;)15 b(x)p Fu(\))p Fo(;)g Fh(e)p Fu(\()p Fo(t)1129 2577 y Fn(1)1170 2563 y Fo(;)g Fh(e)p Fu(\()p Fo(t)1318 2577 y Fn(2)1358 2563 y Fo(;)g(x)p Fu(\)\))p Fo(;)g Fh(e)p Fu(\()p Fo(t)1668 2577 y Fn(1)1710 2563 y Fo(;)g Fh(e)p Fu(\()p Fo(t)1858 2577 y Fn(2)1898 2563 y Fo(;)g Fh(e)p Fu(\()p Fo(t)2046 2577 y Fn(3)2087 2563 y Fo(;)g(x)p Fu(\)\)\))p Fo(;)g Fr(\001)g(\001)g(\001)28 b(j)d Fo(t)2540 2577 y Fm(i)2593 2563 y Fr(!)2684 2525 y Fq(\003)2749 2563 y Fh(B)p Fr(g)212 2767 y Fu(rewrite)31 b(to)g Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(x)p Fu(\).)42 b(Similarly)-8 b(,)31 b(only)f(terms)h(in)f(the)g(set) 439 2971 y Fo(S)495 2985 y Fm(y)562 2971 y Fu(=)25 b Fr(f)p Fh(e)p Fu(\()p Fo(t)811 2985 y Fn(1)851 2971 y Fo(;)15 b(y)s Fu(\))p Fo(;)g Fh(e)p Fu(\()p Fo(t)1122 2985 y Fn(1)1163 2971 y Fo(;)g Fh(e)p Fu(\()p Fo(t)1311 2985 y Fn(2)1352 2971 y Fo(;)g(y)s Fu(\)\))p Fo(;)g Fh(e)p Fu(\()p Fo(t)1658 2985 y Fn(1)1699 2971 y Fo(;)g Fh(e)p Fu(\()p Fo(t)1847 2985 y Fn(2)1887 2971 y Fo(;)g Fh(e)p Fu(\()p Fo(t)2035 2985 y Fn(3)2076 2971 y Fo(;)g(y)s Fu(\)\)\))p Fo(;)g Fr(\001)g(\001)g(\001)28 b(j)d Fo(t)2525 2985 y Fm(i)2579 2971 y Fr(!)2670 2934 y Fq(\003)2734 2971 y Fh(B)p Fr(g)212 3176 y Fu(rewrite)31 b(to)g Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(y)s Fu(\).)41 b(Ho)m(w)m(ev)m(er,)33 b(as)e Fo(S)1517 3190 y Fm(x)1581 3176 y Fr(\\)19 b Fo(S)1717 3190 y Fm(y)1784 3176 y Fu(=)25 b Fr(;)p Fu(,)31 b(term)f Fo(t)g Fu(do)s(es)g(not)h(exist.)766 b Fl(\003)212 3388 y Ff(Lemma)35 b(9.3)46 b Fs(The)33 b(TRS)g Fr(R)1240 3402 y Fn(12)1314 3388 y Fu(\()p Fo(P)13 b Fu(\))34 b Fs(is)e(c)-5 b(onsistent)34 b(if)e(and)i(only)f(if)g Fo(P)45 b Fs(do)-5 b(es)34 b(not)f(have)g(a)g(solution.)212 3521 y(Pr)-5 b(o)g(of)51 b Fu(According)29 b(to)g(Lemma)f(9.1)h(w)m(e)f (ha)m(v)m(e)h(to)g(sho)m(w)f(that)g Fr(R)2419 3535 y Fn(12)2494 3521 y Fu(\()p Fo(P)13 b Fu(\))29 b(is)f(inconsisten)m(t)h (if)f(and)f(only)212 3634 y(if)j Fh(A)g Fu(and)g Fh(B)g Fu(are)h(con)m(v)m(ertible.)43 b(If)29 b Fh(A)c Fr($)1571 3601 y Fq(\003)1571 3665 y(R)1631 3674 y Fk(12)1696 3665 y Fn(\()p Fm(P)10 b Fn(\))1835 3634 y Fh(B)30 b Fu(then)g Fr(R)2210 3648 y Fn(12)2285 3634 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(inconsisten)m(t)i(as)439 3850 y Fo(x)26 b Fr( )f Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(x)p Fu(\))26 b Fr($)1013 3812 y Fq(\003)1078 3850 y Fh(e)p Fu(\()p Fh(A)p Fo(;)15 b(x)p Fu(\))26 b Fr(!)f Fh(C)g Fr( )h Fh(e)p Fu(\()p Fh(A)p Fo(;)15 b(y)s Fu(\))26 b Fr($)2059 3812 y Fq(\003)2123 3850 y Fh(e)p Fu(\()p Fh(B)p Fo(;)15 b(y)s Fu(\))27 b Fr(!)e Fo(y)212 4054 y Fu(for)31 b(di\013eren)m(t)h(v) -5 b(ariables)32 b Fo(x)f Fu(and)f Fo(y)s Fu(.)43 b(If)31 b Fr(R)1635 4068 y Fn(12)1710 4054 y Fu(\()p Fo(P)13 b Fu(\))32 b(is)f(inconsisten)m(t)h(then)f(all)h(terms)f(are)h(con)m(v) m(ertible.)212 4167 y(In)e(particular,)h Fh(A)25 b Fr($)948 4134 y Fq(\003)948 4198 y(R)1008 4207 y Fk(12)1073 4198 y Fn(\()p Fm(P)10 b Fn(\))1212 4167 y Fh(B)p Fu(.)2305 b Fl(\003)212 4379 y Ff(Corollary)35 b(9.4)46 b Fs(Consistency)33 b(is)g(not)g(semi-de)-5 b(cidable)33 b(for)g(line)-5 b(ar)33 b(terminating)h(TRSs)f(that)g(ar)-5 b(e)212 4492 y(c)g(onsistent)34 b(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)e(r)-5 b(e)g(duction.)1975 b Fl(\003)212 4705 y Ff(Theorem)35 b(9.5)46 b Fs(Consistency)34 b(is)e(c)-5 b(o-semi-de)g(cidable)34 b(for)f(TRSs.)212 4837 y(Pr)-5 b(o)g(of)58 b Fu(By)36 b(en)m(umerating)g(and)e(insp)s(ecting)i(all)g(con)m(v)m(ersions)g(w)m (e)g(easily)h(obtain)e(a)h(semi-decision)212 4950 y(pro)s(cedure)25 b(for)i(the)f(problem)g(whether)g(there)h(exists)g(a)g(con)m(v)m (ersion)g(b)s(et)m(w)m(een)g(di\013eren)m(t)g(v)-5 b(ariables.)212 5063 y(Hence)31 b(inconsistency)g(is)g(semi-decidable.)1875 b Fl(\003)1897 5462 y Fu(16)p eop %%Page: 17 17 17 16 bop 212 337 a Ft(10)135 b(UN)45 b Fd(\))g Ft(UN)1107 293 y Fv(!)212 540 y Fu(Let)31 b Fr(R)452 554 y Fn(13)527 540 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)867 554 y Fn(11)941 540 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(R)1261 507 y Fn(1)1261 564 y(12)1356 540 y Fr([)g(f)p Fh(e)p Fu(\()p Fo(x;)15 b(y)s Fu(\))27 b Fr(!)e Fh(e)p Fu(\()p Fo(x;)15 b(y)s Fu(\))p Fo(;)g Fh(C)27 b Fr(!)e Fh(C)p Fr(g)p Fu(.)212 717 y Ff(Lemma)35 b(10.1)46 b Fh(A)25 b Fr($)991 684 y Fq(\003)991 749 y(R)1051 758 y Fk(13)1116 749 y Fn(\()p Fm(P)10 b Fn(\))1255 717 y Fh(B)32 b Fs(if)g(and)i(only)f (if)g Fo(P)45 b Fs(admits)34 b(a)f(solution.)212 850 y(Pr)-5 b(o)g(of)53 b Fu(Immediate)31 b(consequence)h(of)e(Lemma)h(9.1) g(as)g Fr($)2218 817 y Fq(\003)2218 881 y(R)2278 890 y Fk(13)2343 881 y Fn(\()p Fm(P)10 b Fn(\))2482 850 y Fu(=)25 b Fr($)2669 817 y Fq(\003)2669 881 y(R)2729 890 y Fk(12)2794 881 y Fn(\()p Fm(P)10 b Fn(\))2907 850 y Fu(.)671 b Fl(\003)212 1039 y Ff(Lemma)35 b(10.2)46 b Fs(The)33 b(TRS)f Fr(R)1291 1053 y Fn(13)1366 1039 y Fu(\()p Fo(P)13 b Fu(\))33 b Fs(has)g(unique)f(normal)i(forms)f(with)g (r)-5 b(esp)g(e)g(ct)34 b(to)f(r)-5 b(e)g(duction)34 b(for)212 1152 y(every)f(PCP)e(instanc)-5 b(e)34 b Fo(P)13 b Fs(.)212 1285 y(Pr)-5 b(o)g(of)56 b Fu(By)33 b(induction)g(on)g(the)g (structure)f(of)i(terms)e(w)m(e)i(can)f(easily)h(pro)m(v)m(e)g(that)g (ev)m(ery)f(term)h(has)212 1398 y(at)d(most)g(one)g(normal)f(form.)2372 b Fl(\003)212 1576 y Ff(Lemma)35 b(10.3)46 b Fs(The)27 b(TRS)g Fr(R)1280 1590 y Fn(13)1355 1576 y Fu(\()p Fo(P)13 b Fu(\))27 b Fs(has)h(unique)e(normal)i(forms)g(if)e(and)i(only)g(if)e Fo(P)40 b Fs(do)-5 b(es)27 b(not)h(have)212 1689 y(a)33 b(solution.)212 1821 y(Pr)-5 b(o)g(of)50 b Fu(Since)27 b(the)h(normal)f(forms)f(of)i Fr(R)1590 1835 y Fn(13)1664 1821 y Fu(\()p Fo(P)13 b Fu(\))28 b(and)e Fr(R)2083 1835 y Fn(11)2158 1821 y Fu(\()p Fo(P)13 b Fu(\))28 b(coincide,)h(the)e (\\if)7 b(")28 b(direction)g(follo)m(ws)212 1934 y(from)33 b(Lemma)h(8.3.)52 b(Supp)s(ose)31 b(that)k Fr(R)1587 1948 y Fn(13)1662 1934 y Fu(\()p Fo(P)13 b Fu(\))34 b(admits)g(t)m(w)m (o)h(di\013eren)m(t)f(con)m(v)m(ertible)h(normal)f(forms)212 2047 y Fo(t)245 2061 y Fn(1)284 2047 y Fu(,)f Fo(t)375 2061 y Fn(2)415 2047 y Fu(.)46 b(According)33 b(to)g(Lemma)f(10.1)i(it) f(su\016ces)f(to)h(sho)m(w)f(that)h Fh(A)e Fu(and)h Fh(B)g Fu(are)g(con)m(v)m(ertible.)49 b(The)212 2160 y(only)29 b(normal)f(forms)g(of)h Fr(R)1144 2174 y Fn(13)1218 2160 y Fu(\()p Fo(P)13 b Fu(\))30 b(are)e Fh(A)p Fu(,)h Fh(B)p Fu(,)g(and)f(v)-5 b(ariables.)40 b(If)28 b Fo(t)2476 2174 y Fn(1)2544 2160 y Fu(and)g Fo(t)2752 2174 y Fn(2)2820 2160 y Fu(are)h(di\013eren)m(t)g(v)-5 b(ariables)212 2273 y(then)27 b(w)m(e)i(obtain)f(a)g(con)m(v)m(ersion)h(b)s(et)m(w)m (een)f Fh(A)f Fu(and)g Fh(B)g Fu(b)m(y)h(substituting)f Fh(A)h Fu(for)f(all)i(o)s(ccurrences)e(of)h Fo(t)3634 2287 y Fn(1)212 2386 y Fu(and)g Fh(B)h Fu(for)f(all)i(o)s(ccurrences)f (of)g Fh(A)f Fu(in)h(the)g(con)m(v)m(ersion)h Fo(t)2151 2400 y Fn(1)2215 2386 y Fr($)2306 2353 y Fq(\003)2371 2386 y Fo(t)2404 2400 y Fn(2)2443 2386 y Fu(.)40 b(If)29 b(one)g(of)g(the)g(normal)g(forms)f Fo(t)3609 2400 y Fn(1)3648 2386 y Fu(,)212 2498 y Fo(t)245 2512 y Fn(2)314 2498 y Fu(is)i(a)g(v)-5 b(ariable)30 b(and)g(the)f(other)h(is)g Fh(A)f Fu(\()p Fh(B)p Fu(\))h(then)g(w)m(e)g(obtain)g(a)g(con)m(v)m (ersion)h(b)s(et)m(w)m(een)g Fh(A)e Fu(and)g Fh(B)g Fu(b)m(y)212 2611 y(substituting)h Fh(B)g Fu(\()p Fh(A)p Fu(\))h(for)f(all)h(o)s (ccurrences)f(of)h(the)g(v)-5 b(ariable)31 b(in)f(the)g(con)m(v)m (ersion)i Fo(t)3064 2625 y Fn(1)3129 2611 y Fr($)3220 2578 y Fq(\003)3284 2611 y Fo(t)3317 2625 y Fn(2)3356 2611 y Fu(.)222 b Fl(\003)212 2789 y Ff(Corollary)35 b(10.4)46 b Fs(The)33 b(pr)-5 b(op)g(erty)35 b(of)e(having)f(unique)g (normal)i(forms)f(is)g(not)g(semi-de)-5 b(cidable)33 b(for)212 2902 y(line)-5 b(ar)34 b(TRSs)f(that)h(have)f(unique)f (normal)i(forms)g(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)e(r)-5 b(e)g(duction.)632 b Fl(\003)353 3080 y Fu(In)40 b(Section)h(8)g(w)m(e) g(already)g(observ)m(ed)g(that)g(the)g(the)f(prop)s(ert)m(y)g(of)h(ha)m (ving)g(unique)e(normal)212 3193 y(forms)30 b(is)g(co-semi-decidable.) 212 3474 y Ft(11)135 b(UN)689 3431 y Fv(!)822 3474 y Fd(\))44 b Ft(CON)1289 3431 y Fv(!)212 3677 y Fu(Let)31 b Fr(R)452 3691 y Fn(14)527 3677 y Fu(\()p Fo(P)13 b Fu(\))26 b(=)f Fr(R)867 3691 y Fn(1)906 3677 y Fu(\()p Fo(P)13 b Fu(\))21 b Fr([)f(f)p Fh(A)25 b Fr(!)g Fh(C)p Fr(g)p Fu(.)212 3855 y Ff(Lemma)35 b(11.1)46 b Fs(The)37 b(TRS)f Fr(R)1299 3869 y Fn(14)1374 3855 y Fu(\()p Fo(P)13 b Fu(\))37 b Fs(is)f(c)-5 b(onsistent)37 b(with)g(r)-5 b(esp)g(e)g(ct)38 b(to)f(r)-5 b(e)g(duction)37 b(for)g(every)f(PCP)212 3968 y(instanc)-5 b(e)34 b Fo(P)13 b Fs(.)212 4100 y(Pr)-5 b(o)g(of)53 b Fu(T)-8 b(rivial,)32 b(as)e Fr(R)982 4114 y Fn(14)1057 4100 y Fu(\()p Fo(P)13 b Fu(\))31 b(lac)m(ks)h(collapsing) g(rules.)1518 b Fl(\003)212 4278 y Ff(Lemma)35 b(11.2)46 b Fs(The)38 b(TRS)g Fr(R)1302 4292 y Fn(14)1376 4278 y Fu(\()p Fo(P)13 b Fu(\))38 b Fs(has)h(unique)d(normal)k(forms)e(with) h(r)-5 b(esp)g(e)g(ct)39 b(to)f(r)-5 b(e)g(duction)39 b(if)212 4391 y(and)34 b(only)f(if)f Fo(P)46 b Fs(do)-5 b(es)34 b(not)f(have)g(a)g(solution.)212 4524 y(Pr)-5 b(o)g(of)57 b Fu(If)34 b Fo(P)47 b Fu(admits)35 b(a)g(solution)f(then)h Fh(A)c Fr(!)1804 4491 y Fq(\003)1804 4555 y(R)1864 4564 y Fk(1)1899 4555 y Fn(\()p Fm(P)10 b Fn(\))2044 4524 y Fh(B)34 b Fu(and)g(th)m(us)g Fh(A)e Fr(!)2708 4491 y Fq(\003)2708 4555 y(R)2768 4564 y Fk(14)2833 4555 y Fn(\()p Fm(P)10 b Fn(\))2978 4524 y Fh(B)34 b Fu(b)m(y)g(Lemma)h(2.1.) 212 4648 y(Since)e Fh(A)c Fr(!)633 4667 y Fq(R)693 4676 y Fk(14)758 4667 y Fn(\()p Fm(P)10 b Fn(\))902 4648 y Fh(C)33 b Fu(and)f(b)s(oth)g Fh(B)h Fu(and)f Fh(C)h Fu(are)h(normal)e (forms,)i Fr(R)2575 4662 y Fn(14)2649 4648 y Fu(\()p Fo(P)13 b Fu(\))34 b(do)s(es)f(not)g(ha)m(v)m(e)h(unique)212 4761 y(normal)c(forms)g(with)g(resp)s(ect)h(to)g(reduction.)353 4874 y(Con)m(v)m(ersely)-8 b(,)25 b(supp)s(ose)c(that)h Fo(P)35 b Fu(do)s(es)22 b(not)g(ha)m(v)m(e)h(a)f(solution.)39 b(Then,)22 b(b)m(y)g(Lemma)g(2.1,)j Fh(A)g Fr(!)3465 4841 y Fq(\003)3465 4905 y(R)3525 4914 y Fk(1)3560 4905 y Fn(\()p Fm(P)10 b Fn(\))212 4987 y Fh(B)33 b Fu(do)s(es)g(not)g (hold.)49 b(This)32 b(immediately)i(implies)g(that)g Fh(A)29 b Fr(!)2347 4954 y Fq(\003)2347 5018 y(R)2407 5027 y Fk(14)2472 5018 y Fn(\()p Fm(P)10 b Fn(\))2616 4987 y Fh(B)32 b Fu(do)s(es)h(not)g(hold.)49 b(No)m(w)34 b(one)212 5100 y(easily)44 b(sho)m(ws)f(b)m(y)g(structural)g(induction) f(that)i(ev)m(ery)g(term)f(has)f(at)i(most)f(one)h(normal)e(form.)212 5213 y(Hence)31 b Fr(R)559 5227 y Fn(14)634 5213 y Fu(\()p Fo(P)13 b Fu(\))31 b(has)f(unique)g(normal)g(forms)g(with)g(resp)s(ect) g(to)h(reduction.)752 b Fl(\003)1897 5462 y Fu(17)p eop %%Page: 18 18 18 17 bop 212 337 a Ff(Corollary)35 b(11.3)46 b Fs(The)33 b(pr)-5 b(op)g(erty)36 b(of)d(having)g(unique)f(normal)i(forms)g(with)f (r)-5 b(esp)g(e)g(ct)35 b(to)e(r)-5 b(e)g(duction)212 450 y(is)33 b(not)g(semi-de)-5 b(cidable)33 b(for)h(line)-5 b(ar)33 b(TRSs)g(that)h(ar)-5 b(e)34 b(c)-5 b(onsistent)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g(r)-5 b(e)g(duction.)123 b Fl(\003)212 648 y Ff(Theorem)35 b(11.4)46 b Fs(The)35 b(pr)-5 b(op)g(erty)38 b(of)d(having)g(unique)f(normal)j(forms)e(with)h (r)-5 b(esp)g(e)g(ct)36 b(to)g(r)-5 b(e)g(duction)212 761 y(is)33 b(c)-5 b(o-semi-de)g(cidable)33 b(for)g(TRSs.)212 893 y(Pr)-5 b(o)g(of)58 b Fu(By)36 b(en)m(umerating)g(and)e(insp)s (ecting)i(all)g(con)m(v)m(ersions)g(w)m(e)g(easily)h(obtain)e(a)h (semi-decision)212 1006 y(pro)s(cedure)f(for)g(the)h(problem)g(whether) f(there)h(exists)g(a)g(con)m(v)m(ersion)h(of)f(the)g(form)g Fo(s)3252 973 y Fq(\003)3291 1006 y Fr( )e(\001)h(!)3567 973 y Fq(\003)3641 1006 y Fo(t)212 1119 y Fu(with)f Fo(s)f Fu(and)h Fo(t)f Fu(di\013eren)m(t)i(normal)f(forms.)51 b(Hence)35 b(the)f(prop)s(ert)m(y)f(of)h(not)h(ha)m(ving)f(unique)f (normal)212 1232 y(forms)d(with)g(resp)s(ect)g(to)h(reduction)g(is)f (semi-decidable.)1415 b Fl(\003)212 1516 y Ft(12)135 b(W)l(CR,)46 b(GCR,)f(CON)1584 1473 y Fv(!)212 1719 y Fu(In)36 b(this)g(section)h(w)m(e)g(in)m(v)m(estigate)i (\(semi-\)decidabilit)m(y)g(issues)d(for)g(the)g(three)h(prop)s(erties) e(in)h(the)212 1832 y(con\015uence)30 b(hierarc)m(h)m(y)h(that)g(do)g (not)f(app)s(ear)g(in)g(an)g Fo(X)38 b Fu(p)s(osition)31 b(of)f(an)g(implication)i Fo(X)h Fr(\))25 b Fo(Y)20 b Fu(.)353 1945 y(W)-8 b(e)32 b(start)f(with)f(lo)s(cal)h(con\015uence.) 41 b(F)-8 b(rom)31 b(Lemma)g(2.3)g(w)m(e)g(obtain)g(the)f(follo)m(wing) i(result.)212 2143 y Ff(Corollary)j(12.1)46 b Fs(L)-5 b(o)g(c)g(al)35 b(c)-5 b(on\015uenc)g(e)33 b(is)g(not)g(c)-5 b(o-semi-de)g(cidable)34 b(for)f(line)-5 b(ar)34 b(TRSs.)397 b Fl(\003)212 2341 y Ff(Theorem)35 b(12.2)46 b Fs(L)-5 b(o)g(c)g(al)35 b(c)-5 b(on\015uenc)g(e)33 b(is)g(semi-de)-5 b(cidable)33 b(for)h(TRSs.)212 2473 y(Pr)-5 b(o)g(of)51 b Fu(Let)29 b Fr(R)f Fu(b)s(e)f(an)h(arbitrary)g(\014nite)g(TRS.)g(By)g (en)m(umerating)h(and)e(insp)s(ecting)h(all)h(con)m(v)m(ersions)212 2586 y(b)s(et)m(w)m(een)34 b(t)m(w)m(o)g(terms)f Fo(s)f Fu(and)h Fo(t)g Fu(w)m(e)g(easily)h(obtain)g(a)f(semi-decision)h(pro)s (cedure)e(for)h(the)g(problem)212 2699 y(whether)g Fo(s)h Fu(and)f Fo(t)h Fu(are)h(joinable.)52 b(Applying)33 b(this)h(pro)s (cedure)f(in)h(parallel)h(to)g(the)f(\014nitely)g(man)m(y)212 2812 y(critical)e(pairs)e(of)h Fr(R)f Fu(yields)h(a)g(semi-decision)g (pro)s(cedure)e(for)h(lo)s(cal)i(con\015uence.)561 b Fl(\003)353 3010 y Fu(Next)29 b(w)m(e)f(consider)g(ground)e (con\015uence.)40 b(In)27 b(the)h(in)m(tro)s(duction)g(w)m(e)g(already) g(men)m(tioned)h(that)212 3123 y(ground)i(con\015uence)h(is)g (undecidable)g(for)g(terminating)h(TRSs.)44 b(Actually)34 b(from)d(the)h(pro)s(of)g(in)f([9)q(,)212 3236 y(Theorem)f(3.3])h(it)g (follo)m(ws)g(that)g(ground)e(con\015uence)h(is)g(not)h (semi-decidable.)41 b(F)-8 b(rom)31 b(Lemma)f(2.3)212 3349 y(w)m(e)h(obtain)g(the)f(follo)m(wing)i(result.)212 3547 y Ff(Corollary)j(12.3)46 b Fs(Gr)-5 b(ound)35 b(c)-5 b(on\015uenc)g(e)33 b(is)g(not)g(c)-5 b(o-semi-de)g(cidable)34 b(for)f(line)-5 b(ar)34 b(TRSs.)301 b Fl(\003)353 3745 y Fu(Finally)-8 b(,)32 b(w)m(e)f(consider)f(consistency)i(with)e(resp)s (ect)g(to)h(reduction.)212 3943 y Ff(Theorem)k(12.4)46 b Fs(Consistency)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g(r)-5 b(e)g(duction)34 b(is)e(c)-5 b(o-semi-de)g(cidable)34 b(for)f(TRSs.)212 4076 y(Pr)-5 b(o)g(of)58 b Fu(By)36 b(en)m(umerating)g(and)e(insp)s(ecting)i(all)g(con)m(v)m(ersions)g(w)m (e)g(easily)h(obtain)e(a)h(semi-decision)212 4189 y(pro)s(cedure)f(for) g(the)h(problem)g(whether)f(there)h(exists)g(a)g(con)m(v)m(ersion)h(of) f(the)g(form)g Fo(s)3252 4156 y Fq(\003)3291 4189 y Fr( )e(\001)h(!) 3567 4156 y Fq(\003)3641 4189 y Fo(t)212 4302 y Fu(with)26 b Fo(s)h Fu(and)f Fo(t)g Fu(di\013eren)m(t)h(v)-5 b(ariables.)40 b(Hence)28 b(the)f(prop)s(ert)m(y)f(of)h(not)g(b)s(eing)f(consisten)m (t)i(with)e(resp)s(ect)212 4414 y(to)31 b(reduction)g(is)f (semi-decidable.)2184 b Fl(\003)353 4613 y Fu(Let)31 b(the)g(TRS)e Fr(R)963 4627 y Fn(15)1038 4613 y Fu(\()p Fo(P)13 b Fu(\))31 b(consist)g(of)f(the)h(rules)481 4799 y Fh(f)6 b Fu(\()p Fo(\013)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(\014)5 b Fu(\()p Fo(y)s Fu(\))p Fo(;)15 b(z)t Fu(\))34 b Fr(!)c Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))292 b Fh(g)q Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t Fu(\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))756 4912 y Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(z)t Fu(\))33 b Fr(!)d Fo(z)836 b Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(z)t Fu(\))34 b Fr(!)c Fo(z)616 5025 y Fh(h)p Fu(\()p Fo(x;)15 b(y)s(;)g(z)t(;)g(w)r Fu(\))33 b Fr(!)d Fh(f)6 b Fu(\()p Fo(x;)15 b(y)s(;)g(z)t Fu(\))184 b Fh(h)p Fu(\()p Fo(a)p Fu(\()p Fo(x)p Fu(\))p Fo(;)15 b(a)p Fu(\()p Fo(y)s Fu(\))p Fo(;)g(z)t(;)g(w)r Fu(\))34 b Fr(!)c Fh(g)q Fu(\()p Fo(x;)15 b(y)s(;)g(w)r Fu(\))212 5213 y(Note)32 b(that)f Fr(R)706 5227 y Fn(15)781 5213 y Fu(\()p Fo(P)13 b Fu(\))31 b(is)f(terminating)h(for)f(ev)m(ery)h(PCP) f(instance)h Fo(P)13 b Fu(.)1897 5462 y(18)p eop %%Page: 19 19 19 18 bop 212 337 a Ff(Lemma)35 b(12.5)46 b Fs(The)40 b(TRS)f Fr(R)1305 351 y Fn(15)1380 337 y Fu(\()p Fo(P)13 b Fu(\))40 b Fs(is)f(c)-5 b(onsistent)41 b(with)f(r)-5 b(esp)g(e)g(ct)41 b(to)e(r)-5 b(e)g(duction)41 b(if)e(and)h(only)g Fo(P)212 450 y Fs(admits)34 b(a)f(solution.)212 582 y(Pr)-5 b(o)g(of)62 b Fu(If)39 b Fo(P)53 b Fu(admits)40 b(a)g(solution)g Fo(\015)46 b Fu(=)40 b Fo(\013)1703 596 y Fm(i)1727 605 y Fk(1)1781 582 y Fr(\001)15 b(\001)g(\001)h Fo(\013)1960 596 y Fm(i)1984 604 y Fj(m)2088 582 y Fu(=)40 b Fo(\014)2250 596 y Fm(i)2274 605 y Fk(1)2329 582 y Fr(\001)15 b(\001)g(\001)h Fo(\014)2501 596 y Fm(i)2525 604 y Fj(m)2628 582 y Fu(for)39 b(some)h Fo(m)h Fl(>)f Fu(1)g(and)f(1)j Fl(6)212 695 y Fo(i)243 709 y Fn(1)283 695 y Fo(;)15 b(:)g(:)g(:)h(;)f(i)515 709 y Fm(m)609 695 y Fl(6)25 b Fo(n)31 b Fu(then)f(w)m(e)h(ha)m(v)m(e)h (the)f(follo)m(wing)h(div)m(erging)g(reductions)f(in)f Fr(R)2902 709 y Fn(15)2977 695 y Fu(\()p Fo(P)13 b Fu(\))31 b(starting)h(from)212 808 y(the)f(term)f Fo(t)25 b Fu(=)g Fh(h)p Fu(\()p Fo(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(x;)g(y)s Fu(\):)439 1004 y Fo(x)26 b Fr( )f Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(x)p Fu(\))976 966 y Fq(\003)1016 1004 y Fr( )25 b Fh(f)6 b Fu(\()p Fo(\015)f Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)5 b Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(x)p Fu(\))29 b Fr( )c Fo(t)g Fr(!)h Fh(g)q Fu(\()p Fo(\015)2145 966 y Fq(0)2169 1004 y Fu(\()p Fh(c)p Fu(\))p Fo(;)15 b(\015)2371 966 y Fq(0)2396 1004 y Fu(\()p Fh(c)p Fu(\))p Fo(;)g(y)s Fu(\))27 b Fr(!)2747 966 y Fq(\003)2812 1004 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(y)s Fu(\))28 b Fr(!)d Fo(y)s(:)212 1200 y Fu(Here)42 b Fo(\015)490 1167 y Fq(0)554 1200 y Fu(is)f(the)g(string)g Fo(\015)46 b Fu(min)m(us)40 b(its)h(\014rst)f(sym)m(b)s(ol.)72 b(Hence)42 b Fr(R)2532 1214 y Fn(15)2607 1200 y Fu(\()p Fo(P)13 b Fu(\))41 b(is)g(not)g(consisten)m(t)i(with)212 1313 y(resp)s(ect)30 b(to)i(reduction.)353 1425 y(Con)m(v)m(ersely)-8 b(,)37 b(assume)e(that)g Fr(R)1433 1439 y Fn(15)1508 1425 y Fu(\()p Fo(P)13 b Fu(\))35 b(is)f(not)h(consisten)m(t)h(with)e (resp)s(ect)h(to)g(reduction.)53 b(Then)212 1538 y Fo(x)289 1505 y Fq(\003)329 1538 y Fr( )25 b Fo(t)h Fr(!)595 1505 y Fq(\003)659 1538 y Fo(y)34 b Fu(for)c(some)h(term)f Fo(t)g Fu(and)g(distinct)h(v)-5 b(ariables)31 b Fo(x)g Fu(and)f Fo(y)s Fu(.)40 b(Without)32 b(loss)f(of)f(generalit)m(y)212 1651 y(w)m(e)23 b(assume)f(that)g(the)h(size)g(of)f Fo(t)g Fu(is)g(minimal.)38 b(This)22 b(immediately)h(implies)g(that)f(the)h (ro)s(ot)f(sym)m(b)s(ol)g(of)212 1764 y Fo(t)h Fu(m)m(ust)g(b)s(e)f Fh(f)6 b Fu(,)25 b Fh(g)q Fu(,)g(or)e Fh(h)p Fu(.)38 b(W)-8 b(e)24 b(sho)m(w)f(that)h(it)f(m)m(ust)g(b)s(e)f Fh(h)p Fu(.)38 b(If)23 b Fo(t)i Fu(=)g Fh(f)6 b Fu(\()p Fo(t)2420 1778 y Fn(1)2460 1764 y Fo(;)15 b(t)2533 1778 y Fn(2)2572 1764 y Fo(;)g(t)2645 1778 y Fn(3)2685 1764 y Fu(\))23 b(then)g(the)g(reduction)g(from)212 1877 y Fo(t)j Fu(to)i Fo(x)e Fu(m)m(ust)h(b)s(e)f(of)g(the)h(form)f Fo(t)f Fr(!)1411 1844 y Fq(\003)1476 1877 y Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(t)1738 1844 y Fq(0)1738 1901 y Fn(3)1779 1877 y Fu(\))26 b Fr(!)f Fo(t)1989 1844 y Fq(0)1989 1901 y Fn(3)2054 1877 y Fr(!)2145 1844 y Fq(\003)2209 1877 y Fo(x)i Fu(with)f Fo(t)2524 1891 y Fn(3)2589 1877 y Fr(!)2680 1844 y Fq(\003)2744 1877 y Fo(t)2777 1844 y Fq(0)2777 1901 y Fn(3)2816 1877 y Fu(.)40 b(Hence)27 b Fo(t)3180 1891 y Fn(3)3245 1877 y Fr(!)3336 1844 y Fq(\003)3400 1877 y Fo(x)p Fu(.)40 b(The)212 1990 y(same)35 b(reasoning)g(yields)g Fo(t)1146 2004 y Fn(3)1218 1990 y Fr(!)1309 1957 y Fq(\003)1381 1990 y Fo(y)s Fu(,)g(con)m(tradicting)i(the)e(minimalit)m(y)h(of)f Fo(t)p Fu(.)53 b(A)35 b(similar)g(argumen)m(t)212 2103 y(rev)m(eals)d(that)f(the)f(ro)s(ot)h(sym)m(b)s(ol)f(of)h Fo(t)f Fu(cannot)h(b)s(e)f Fh(g)q Fu(.)41 b(Hence)31 b Fo(t)25 b Fu(=)g Fh(h)p Fu(\()p Fo(t)2610 2117 y Fn(1)2650 2103 y Fo(;)15 b(t)2723 2117 y Fn(2)2762 2103 y Fo(;)g(t)2835 2117 y Fn(3)2875 2103 y Fo(;)g(t)2948 2117 y Fn(4)2988 2103 y Fu(\).)41 b(W)-8 b(e)31 b(m)m(ust)g(ha)m(v)m(e)439 2299 y Fo(x)517 2261 y Fq(\003)556 2299 y Fr( )25 b Fo(u)749 2261 y Fq(\003)789 2299 y Fr( )g Fh(h)p Fu(\()p Fo(t)1020 2313 y Fn(1)1060 2299 y Fo(;)15 b(t)1133 2313 y Fn(2)1172 2299 y Fo(;)g(t)1245 2313 y Fn(3)1285 2299 y Fo(;)g(t)1358 2313 y Fn(4)1398 2299 y Fu(\))25 b Fr(!)1549 2261 y Fq(\003)1614 2299 y Fo(u)1666 2261 y Fq(0)1714 2299 y Fr(!)1805 2261 y Fq(\003)1870 2299 y Fo(y)1642 b Fu(\(3\))212 2495 y(with)34 b Fo(t)456 2509 y Fn(3)527 2495 y Fr(!)618 2462 y Fq(\003)689 2495 y Fo(u)g Fu(or)g Fo(t)923 2509 y Fn(4)994 2495 y Fr(!)1085 2462 y Fq(\003)1156 2495 y Fo(u)g Fu(and)g Fo(t)1456 2509 y Fn(3)1527 2495 y Fr(!)1618 2462 y Fq(\003)1689 2495 y Fo(u)1741 2462 y Fq(0)1798 2495 y Fu(or)g Fo(t)1946 2509 y Fn(4)2017 2495 y Fr(!)2108 2462 y Fq(\003)2179 2495 y Fo(u)2231 2462 y Fq(0)2255 2495 y Fu(.)52 b(If)33 b Fo(t)2459 2509 y Fn(3)2530 2495 y Fr(!)2621 2462 y Fq(\003)2692 2495 y Fo(u)h Fu(and)g Fo(t)2992 2509 y Fn(3)3063 2495 y Fr(!)3154 2462 y Fq(\003)3225 2495 y Fo(u)3277 2462 y Fq(0)3334 2495 y Fu(then)g Fo(t)g Fu(is)212 2608 y(not)i(minimal)g(as)g Fo(x)941 2575 y Fq(\003)980 2608 y Fr( )f Fo(u)1192 2575 y Fq(\003)1231 2608 y Fr( )f Fo(t)1389 2622 y Fn(3)1463 2608 y Fr(!)1554 2575 y Fq(\003)1627 2608 y Fo(u)1679 2575 y Fq(0)1737 2608 y Fr(!)g Fo(y)s Fu(.)56 b(Lik)m(ewise,)39 b Fo(t)c Fu(is)h(not)g(minimal)g(if)f Fo(t)3201 2622 y Fn(4)3275 2608 y Fr(!)3366 2575 y Fq(\003)3439 2608 y Fo(u)h Fu(and)212 2721 y Fo(t)245 2735 y Fn(4)321 2721 y Fr(!)412 2688 y Fq(\003)487 2721 y Fo(u)539 2688 y Fq(0)563 2721 y Fu(.)60 b(No)m(w)38 b(supp)s(ose)d(that)j Fo(t)1447 2735 y Fn(3)1522 2721 y Fr(!)1613 2688 y Fq(\003)1689 2721 y Fo(u)f Fu(and)f Fo(t)1994 2735 y Fn(4)2070 2721 y Fr(!)2161 2688 y Fq(\003)2236 2721 y Fo(u)2288 2688 y Fq(0)2312 2721 y Fu(.)60 b(\(In)37 b(the)g(remaining)g(case)h Fo(t)3379 2735 y Fn(4)3455 2721 y Fr(!)3546 2688 y Fq(\003)3621 2721 y Fo(u)212 2833 y Fu(and)32 b Fo(t)424 2847 y Fn(3)491 2833 y Fr(!)582 2800 y Fq(\003)650 2833 y Fo(u)702 2800 y Fq(0)758 2833 y Fu(w)m(e)g(obtain)h(a)g(PCP)e(solution)i(in)f (exactly)i(the)f(same)f(w)m(a)m(y)-8 b(.\))48 b(Then)32 b(w)m(e)g(obtain)h(the)212 2946 y(div)m(ergence)439 3142 y Fo(x)517 3105 y Fq(\003)556 3142 y Fr( )25 b Fh(h)p Fu(\()p Fo(t)787 3156 y Fn(1)827 3142 y Fo(;)15 b(t)900 3156 y Fn(2)939 3142 y Fo(;)g(x;)g(y)s Fu(\))27 b Fr(!)1272 3105 y Fq(\003)1336 3142 y Fo(y)2176 b Fu(\(4\))212 3338 y(from)23 b(\(3\))h(b)m(y)g(replacing)g Fo(t)1094 3352 y Fn(3)1156 3338 y Fu(b)m(y)g Fo(x)f Fu(and)g Fo(t)1554 3352 y Fn(4)1616 3338 y Fu(b)m(y)g Fo(y)s Fu(.)39 b(W)-8 b(e)24 b(claim)h(that)e Fo(t)2454 3352 y Fn(1)2517 3338 y Fu(and)g Fo(t)2720 3352 y Fn(2)2782 3338 y Fu(con)m(tain)i(no)e(o)s (ccurrences)212 3451 y(of)37 b Fh(f)6 b Fu(,)38 b Fh(g)q Fu(,)h(and)d Fh(h)p Fu(.)59 b(The)36 b(reason)h(is)g(that)g(an)m(y)g (subterm)e(in)i Fo(t)2311 3465 y Fn(1)2386 3451 y Fu(or)g Fo(t)2537 3465 y Fn(2)2613 3451 y Fu(with)f(one)h(of)g(these)g (function)212 3564 y(sym)m(b)s(ols)g(at)g(its)h(ro)s(ot)f(p)s(osition)g (can)g(b)s(e)f(replaced)i(b)m(y)e(a)i(fresh)e(v)-5 b(ariable)37 b(without)g(a\013ecting)i(the)212 3677 y(p)s(ossibilit)m(y)e(to)g(p)s (erform)f(the)g(ab)s(o)m(v)m(e)i(div)m(erging)g(reductions;)h(just)d (drop)g(all)h(rewrite)g(steps)g(that)212 3790 y(to)s(ok)c(place)h(at)f (or)f(b)s(elo)m(w)h(the)f(replaced)h(subterm.)46 b(It)33 b(follo)m(ws)g(that)g(all)g(steps)g(in)f(\(4\))h(tak)m(e)h(place)212 3903 y(at)d(the)g(ro)s(ot)f(p)s(osition.)41 b(W)-8 b(e)32 b(ma)m(y)f(therefore)g(write)f(\(4\))i(as)439 4099 y Fo(x)26 b Fr( )f Fh(f)6 b Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(x)p Fu(\))976 4061 y Fq(\003)1016 4099 y Fr( )25 b Fh(f)6 b Fu(\()p Fo(t)1234 4113 y Fn(1)1274 4099 y Fo(;)15 b(t)1347 4113 y Fn(2)1386 4099 y Fo(;)g(x)p Fu(\))26 b Fr( )f Fh(h)p Fu(\()p Fo(t)1770 4113 y Fn(1)1810 4099 y Fo(;)15 b(t)1883 4113 y Fn(2)1923 4099 y Fo(;)g(x;)g(y)s Fu(\))26 b Fr(!)f Fh(g)q Fu(\()p Fo(t)2394 4061 y Fq(0)2394 4121 y Fn(1)2434 4099 y Fo(;)15 b(t)2507 4061 y Fq(0)2507 4121 y Fn(2)2547 4099 y Fo(;)g(y)s Fu(\))26 b Fr(!)2787 4061 y Fq(\003)2852 4099 y Fh(g)q Fu(\()p Fh(c)p Fo(;)15 b Fh(c)p Fo(;)g(y)s Fu(\))28 b Fr(!)d Fo(y)212 4294 y Fu(with)44 b Fo(t)466 4308 y Fn(1)554 4294 y Fu(=)k Fo(a)p Fu(\()p Fo(t)789 4261 y Fq(0)789 4319 y Fn(1)829 4294 y Fu(\))c(and)g Fo(t)1132 4308 y Fn(2)1220 4294 y Fu(=)k Fo(a)p Fu(\()p Fo(t)1455 4261 y Fq(0)1455 4319 y Fn(2)1495 4294 y Fu(\))d(for)f(some)g Fo(a)49 b Fr(2)f Fu(\000.)83 b(No)m(w)45 b Fo(\013)2619 4308 y Fm(i)2643 4317 y Fk(1)2697 4294 y Fr(\001)15 b(\001)g(\001)h Fo(\013)2876 4308 y Fm(i)2900 4316 y Fj(m)2963 4294 y Fu(\()p Fh(c)p Fu(\))50 b(=)e Fo(t)3275 4308 y Fn(1)3363 4294 y Fu(=)g Fo(t)3515 4308 y Fn(2)3603 4294 y Fu(=)212 4407 y Fo(\014)263 4421 y Fm(i)287 4430 y Fk(1)341 4407 y Fr(\001)15 b(\001)g(\001)i Fo(\014)514 4421 y Fm(i)538 4429 y Fj(m)601 4407 y Fu(\()p Fh(c)p Fu(\))31 b(for)f(some)h Fo(m)25 b Fl(>)g Fu(1)31 b(and)f(1)25 b Fl(6)g Fo(i)1760 4421 y Fn(1)1800 4407 y Fo(;)15 b(:)g(:)g(:)i(;)e(i)2033 4421 y Fm(m)2125 4407 y Fl(6)25 b Fo(n)p Fu(.)41 b(Hence)31 b Fo(P)43 b Fu(admits)31 b(a)f(solution.)178 b Fl(\003)212 4612 y Ff(Corollary)35 b(12.6)46 b Fs(Consistency)38 b(with)f(r)-5 b(esp)g(e)g(ct)38 b(to)e(r)-5 b(e)g(duction)38 b(is)e(not)h(semi-de)-5 b(cidable)37 b(for)g(line)-5 b(ar)212 4725 y(terminating)34 b(TRSs.)2650 b Fl(\003)212 5010 y Ft(13)135 b(Conclusions)45 b(and)g(F)-11 b(urther)44 b(Researc)l(h)212 5213 y Fu(The)30 b(results)g(pro)m(v)m(ed)h(in)f(this)g(pap)s(er)f(are)i(summarized)f (in)g(the)h(follo)m(wing)g(tables:)1897 5462 y(19)p eop %%Page: 20 20 20 19 bop 553 463 a Fo(X)37 b Fr(\))31 b Fo(Y)397 b Fu(\(1\))124 b(\(2\))85 b(\(3\))129 b(\(4\))p 424 518 1613 4 v 452 636 a(SCR)29 b Fr(\))i Fu(CR)516 b(3)p Fo(:)p Fu(4)1551 603 y Fc(a)1676 636 y Fu(3)p Fo(:)p Fu(3)503 806 y(CR)29 b Fr(\))i Fu(W)m(CR)192 b(4)p Fo(:)p Fu(6)359 b(4)p Fo(:)p Fu(3)503 975 y(CR)29 b Fr(\))i Fu(GCR)211 b(5)p Fo(:)p Fu(7)359 b(5)p Fo(:)p Fu(3)503 1144 y(CR)29 b Fr(\))i Fu(NF)288 b(6)p Fo(:)p Fu(6)359 b(6)p Fo(:)p Fu(3)508 1314 y(NF)30 b Fr(\))h Fu(UN)279 b(7)p Fo(:)p Fu(5)359 b(7)p Fo(:)p Fu(2)499 1483 y(UN)30 b Fr(\))h Fu(CON)210 b(8)p Fo(:)p Fu(4)603 b(8)p Fo(:)p Fu(5)431 1652 y(CON)29 b Fr(\))i Fu(CON)991 1617 y Fq(!)1200 1652 y Fu(9)p Fo(:)p Fu(4)1315 1619 y Fc(b)1920 1652 y Fu(9)p Fo(:)p Fu(5)499 1822 y(UN)f Fr(\))h Fu(UN)923 1786 y Fq(!)1157 1822 y Fu(10)p Fo(:)p Fu(4)603 b(8)p Fo(:)p Fu(5)424 1991 y(UN)560 1955 y Fq(!)665 1991 y Fr(\))31 b Fu(CON)991 1955 y Fq(!)1157 1991 y Fu(11)p Fo(:)p Fu(3)558 b(11)p Fo(:)p Fu(4)2326 463 y Fo(X)263 b Fu(\(1)2744 430 y Fq(0)2768 463 y Fu(\))129 b(\(2\))106 b(\(3)3233 430 y Fq(0)3257 463 y Fu(\))129 b(\(4\))p 2218 518 1319 4 v 2218 636 a(W)m(CR)445 b(12)p Fo(:)p Fu(2)85 b(12)p Fo(:)p Fu(1)2228 806 y(GCR)241 b([9)q(])2769 773 y Fm(b)3131 806 y Fu(12)p Fo(:)p Fu(3)2227 975 y(CON)2432 939 y Fq(!)2607 975 y Fu(12)p Fo(:)p Fu(6)2767 942 y Fm(b)3376 975 y Fu(12)p Fo(:)p Fu(4)p 348 2103 1276 4 v 452 2157 a Fb(a)486 2189 y Fa(only)26 b(for)g(linear)h(TRSs) 455 2251 y Fb(b)486 2283 y Fa(ev)n(en)e(for)i(terminating)f(TRSs)212 2461 y Fu(Here)42 b(\(1\))g(means)f(that)g Fo(X)49 b Fu(is)41 b(not)g(a)g(semi-decidable)h(prop)s(ert)m(y)f(of)g(linear)g (TRSs)f(that)i(satisfy)212 2574 y(prop)s(ert)m(y)37 b Fo(Y)20 b Fu(,)40 b(\(2\))f(means)f(that)g Fo(X)46 b Fu(is)38 b(semi-decidable,)j(\(3\))e(means)f(that)g Fo(X)45 b Fu(is)38 b(not)g(a)h(co-semi-)212 2687 y(decidable)28 b(prop)s(ert)m(y)e(of)h(linear)h(TRSs)e(that)h(satisfy)h(prop)s(ert)m (y)e Fo(Y)20 b Fu(,)28 b(\(4\))g(means)f(that)h Fo(X)34 b Fu(is)28 b(co-semi-)212 2800 y(decidable,)f(and)c(\(1)882 2767 y Fq(0)906 2800 y Fu(\))i(and)f(\(3)1217 2767 y Fq(0)1241 2800 y Fu(\))h(mean)f(the)h(same)f(as)h(\(1\))g(and)f(\(3\))i (without)e(the)h(quali\014cation)g(\\that)212 2913 y(satisfy)31 b(prop)s(ert)m(y)f Fo(Y)20 b Fu(".)353 3026 y(In)35 b(Section)h(3)f(w)m (e)h(already)g(men)m(tioned)g(the)f(op)s(en)f(problem)h(whether)g (strong)g(con\015uence)g(is)212 3139 y(semi-decidable)j(for)f (arbitrary)g(\(\014nite\))h(TRSs.)60 b(Ev)m(en)37 b(for)g(terminating)h (TRSs)e(it)i(is)f(unkno)m(wn)212 3252 y(whether)i(strong)g (con\015uence)g(is)g(\(semi-\)decidable.)69 b(All)39 b(other)h(prop)s(erties)e(in)h(the)g(con\015uence)212 3364 y(hierarc)m(h)m(y)-8 b(,)32 b(except)f(GCR,)f(CON)g(and)g(CON)1788 3331 y Fq(!)1862 3364 y Fu(,)h(are)f(equiv)-5 b(alen)m(t)32 b(and)e(decidable)h(in)f(the)g(presence)212 3477 y(of)e(termination.)41 b(W)-8 b(e)28 b(do)g(not)g(kno)m(w)g(whether)f(Corollary)h(12.3)h(can)f (b)s(e)f(strengthened)h(to)g(\(linear\))212 3590 y(terminating)j(TRSs.) 353 3703 y(In)g(our)g(companion)h(pap)s(er)e([4)q(])i(w)m(e)g(presen)m (t)f(relativ)m(e)j(undecidabilit)m(y)e(results)g(for)f(prop)s(erties) 212 3816 y(related)26 b(to)f(termination.)40 b(Most)25 b(of)g(these)g(results)g(are)g(obtained)g(for)g(TRSs)e(consisting)j(of) f(a)g(single)212 3929 y(rewrite)i(rule.)39 b(F)-8 b(or)27 b(the)f(con\015uence)g(hierarc)m(h)m(y)h(w)m(e)g(do)f(not)h(ha)m(v)m(e) g(results)f(for)g(single)h(rewrite)f(rules.)353 4042 y(Another)31 b(problem)g(is)g(whether)g(the)g(results)g(obtained)h(in)f (this)g(pap)s(er)f(can)i(b)s(e)e(strengthened)212 4155 y(to)36 b(string)f(rewriting)g(systems.)55 b(The)35 b(pap)s(ers)f([13)q (,)h(16)q(,)g(19)q(])g(con)m(tain)i(some)e(kind)g(of)g(relativ)m(e)i (un-)212 4268 y(decidabilit)m(y)31 b(results)f(for)f(string)h (rewriting.)41 b(In)29 b([16)q(])h(it)g(is)g(pro)m(v)m(ed)g(that)g(a)g (n)m(um)m(b)s(er)f(of)h(prop)s(erties)212 4381 y(are)36 b(undecidable)g(for)f(string)h(rewriting)g(systems)g(for)f(whic)m(h)h (the)g(w)m(ord)f(problem)h(is)f(decidable.)212 4494 y(One)25 b(of)h(these)g(prop)s(erties)g(is)f(the)h(negation)h(of)f(CON,)f (meaning)h(that)h(all)f(strings)g(are)g(con)m(v)m(ertible.)212 4606 y(In)31 b([13)q(])g(it)h(is)f(pro)m(v)m(ed)g(that)h(the)g(prop)s (ert)m(y)e(of)h(ha)m(ving)h(\014nite)f(deriv)-5 b(ation)32 b(t)m(yp)s(e)f(is)h(undecidable)e(for)212 4719 y(string)i(rewriting)f (systems)h(for)f(whic)m(h)h(the)g(w)m(ord)f(problem)g(is)h(p)s (olynomially)g(decidable.)45 b(In)31 b([19)q(])212 4832 y(it)41 b(is)g(sho)m(wn)g(that)g(con\015uence)g(is)g(undecidable)f(for) h(string)g(rewriting)f(systems)h(whic)m(h)g(ha)m(v)m(e)h(a)212 4945 y(decidable)31 b(w)m(ord)f(problem.)353 5058 y(Besides)c(ground)e (con\015uence,)j(one)f(can)f(extend)h(the)f(con\015uence)h(hierarc)m(h) m(y)g(b)m(y)f(in)m(tro)s(ducing)g(a)212 5171 y(prop)s(ert)m(y)k (ground-)p Fo(X)36 b Fu(for)29 b(ev)m(ery)i(other)e Fo(X)7 b Fu(,)31 b(except)f(CON)f(and)g(CON)2654 5138 y Fq(!)2728 5171 y Fu(.)40 b(Ground-)p Fo(X)c Fu(is)30 b(obtained)1897 5462 y(20)p eop %%Page: 21 21 21 20 bop 212 337 a Fu(from)29 b Fo(X)36 b Fu(b)m(y)29 b(replacing)h(the)f(quan)m(ti\014cation)i(o)m(v)m(er)f(terms)f(b)m(y)g (quan)m(ti\014cation)i(o)m(v)m(er)f(ground)e(terms.)212 450 y(This)i(giv)m(es)h(rise)g(to)g(man)m(y)f(new)g(relativ)m(e)j (undecidabilit)m(y)e(questions.)212 734 y Ft(Ac)l(kno)l(wledgemen)l(ts) 212 937 y Fu(W)-8 b(e)39 b(are)f(grateful)h(to)f(an)g(anon)m(ymous)f (referee)i(for)e(the)h(suggestion)h(to)g(expand)e(our)g(results)g(b)m (y)212 1049 y(in)m(v)m(estigating)29 b(semi-decidabilit)m(y)-8 b(.)41 b(Aart)26 b(Middeldorp)f(is)h(partially)g(supp)s(orted)e(b)m(y)h (the)h(Gran)m(t-in-)212 1162 y(Aid)j(for)f(Scien)m(ti\014c)h(Researc)m (h)h(C\(2\))f(11680338)j(of)d(the)g(Ministry)f(of)h(Education,)h (Science,)g(Sp)s(orts)212 1275 y(and)g(Culture)f(of)i(Japan.)212 1559 y Ft(References)258 1762 y Fu([1])46 b(T.)26 b(Aoto)i(and)e(Y.)h (T)-8 b(o)m(y)m(ama.)36 b(P)m(ersistency)27 b(of)g(con\015uence.)34 b Fs(Journal)c(of)g(Universal)f(Computer)399 1875 y(Scienc)-5 b(e)p Fu(,)30 b(3\(11\):1134{1147)q(,)36 b(1997.)258 2058 y([2])46 b(F.)36 b(Baader)h(and)e(T.)g(Nipk)m(o)m(w.)58 b Fs(T)-7 b(erm)38 b(R)-5 b(ewriting)38 b(and)h(A)n(l)5 b(l)37 b(That)p Fu(.)57 b(Cam)m(bridge)36 b(Univ)m(ersit)m(y)399 2171 y(Press,)30 b(1998.)258 2353 y([3])46 b(N.)33 b(Dersho)m(witz)h (and)f(J.-P)-8 b(.)33 b(Jouannaud.)47 b(Rewrite)33 b(systems.)49 b(In)32 b(J.)g(v)-5 b(an)33 b(Leeu)m(w)m(en,)i(editor,)399 2466 y Fs(Handb)-5 b(o)g(ok)47 b(of)f(The)-5 b(or)g(etic)g(al)47 b(Computer)g(Scienc)-5 b(e)p Fu(,)48 b(v)m(olume)c(B,)h(pages)g (243{320.)i(Elsevier,)399 2579 y(1990.)258 2762 y([4])f(A.)23 b(Geser,)i(A.)e(Middeldorp,)h(E.)f(Ohlebusc)m(h,)h(and)e(H.)h(Zan)m (tema.)29 b(Relativ)m(e)c(undecidabilit)m(y)f(in)399 2875 y(term)30 b(rewriting.)f(Part)h(1:)41 b(The)29 b(termination)h (hierarc)m(h)m(y)-8 b(.)41 b Fs(Information)34 b(and)f(Computation)p Fu(.)399 2988 y(To)d(app)s(ear.)258 3170 y([5])46 b(A.)32 b(Geser,)i(A.)e(Middeldorp,)g(E.)h(Ohlebusc)m(h,)e(and)h(H.)g(Zan)m (tema.)47 b(Relativ)m(e)35 b(undecidabilit)m(y)399 3283 y(in)f(term)g(rewriting.)53 b(In)33 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings) 39 b(of)d(the)h(10th)h(A)n(nnual)e(Confer)-5 b(enc)g(e)37 b(of)g(the)f(Eur)-5 b(op)g(e)g(an)399 3396 y(Asso)g(ciation)27 b(for)g(Computer)g(Scienc)-5 b(e)25 b(L)-5 b(o)g(gic)p Fu(,)26 b(v)m(olume)e(1258)g(of)g Fs(L)-5 b(e)g(ctur)g(e)26 b(Notes)g(in)g(Computer)399 3509 y(Scienc)-5 b(e)p Fu(,)30 b(pages)h(150{166,)j(Berlin,)d(1997.)h(Springer-V)-8 b(erlag.)258 3692 y([6])46 b(G.)22 b(Huet.)27 b(Con\015uen)m(t)21 b(reductions:)36 b(Abstract)22 b(prop)s(erties)f(and)g(applications)i (to)f(term)g(rewrit-)399 3805 y(ing)30 b(systems.)41 b Fs(Journal)34 b(of)f(the)g(A)n(CM)p Fu(,)c(27:797{821,)35 b(1980.)258 3987 y([7])46 b(G.)37 b(Huet)h(and)e(D.S.)i(Lankford.)59 b(On)36 b(the)h(uniform)f(halting)i(problem)e(for)h(term)g(rewriting) 399 4100 y(systems.)k(Rapp)s(ort)29 b(Lab)s(oria)i(283,)h(INRIA,)e (1978.)258 4283 y([8])46 b(G.)39 b(Huet)h(and)e(D.C.)h(Opp)s(en.)64 b(Equations)39 b(and)g(rewrite)g(rules:)57 b(A)39 b(surv)m(ey)-8 b(.)66 b(In)38 b(R.)h(Bo)s(ok,)399 4396 y(editor,)30 b Fs(F)-7 b(ormal)34 b(L)-5 b(anguage)32 b(The)-5 b(ory:)42 b(Persp)-5 b(e)g(ctives)32 b(and)h(Op)-5 b(en)31 b(Pr)-5 b(oblems)p Fu(,)31 b(pages)f(349{405.)399 4509 y(Academic)i(Press,)e (1980.)258 4691 y([9])46 b(D.)41 b(Kapur,)g(P)-8 b(.)40 b(Narendran,)i(and)e(F.)g(Otto.)70 b(On)40 b(ground-con\015uence)f(of)h (term)g(rewriting)399 4804 y(systems.)h Fs(Information)34 b(and)g(Computation)p Fu(,)f(86\(1\):14{31,)i(1980.)212 4987 y([10])47 b(J.W.)27 b(Klop.)35 b(T)-8 b(erm)26 b(rewriting)h (systems.)35 b(In)26 b(S.)g(Abramski,)h(D.)g(Gabba)m(y)-8 b(,)29 b(and)d(T.)h(Maibaum,)399 5100 y(editors,)41 b Fs(Handb)-5 b(o)g(ok)41 b(of)f(L)-5 b(o)g(gic)40 b(in)g(Computer)h (Scienc)-5 b(e)p Fu(,)40 b(v)m(olume)f(2,)h(pages)f(1{116.)h(Oxford)399 5213 y(Univ)m(ersit)m(y)32 b(Press,)e(1992.)1897 5462 y(21)p eop %%Page: 22 22 22 21 bop 212 337 a Fu([11])47 b(J.W.)27 b(Klop,)g(A.)g(Middeldorp,)g (Y.)g(T)-8 b(o)m(y)m(ama,)29 b(and)d(R.)h(de)f(V)-8 b(rijer.)34 b(Mo)s(dularit)m(y)27 b(of)g(con\015uence:)399 450 y(A)j(simpli\014ed)g (pro)s(of.)40 b Fs(Information)35 b(Pr)-5 b(o)g(c)g(essing)34 b(L)-5 b(etters)p Fu(,)31 b(49:101{109,)k(1994.)212 637 y([12])47 b(Y.)c(Matiy)m(asevic)m(h)j(and)c(G.)i(Senizergues.)77 b(Decision)45 b(problems)d(for)g(semi-Th)m(ue)h(systems)399 750 y(with)31 b(a)h(few)f(rules.)44 b(In)31 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)36 b(of)e(the)g(11th)h(IEEE)e(A)n(nnual)g(Symp) -5 b(osium)36 b(on)e(L)-5 b(o)g(gic)34 b(in)399 863 y(Computer)g (Scienc)-5 b(e)p Fu(,)30 b(pages)h(523{531,)j(1996.)212 1051 y([13])47 b(F.)37 b(Otto)h(and)e(A.)h(Sattler-Klein.)61 b(The)36 b(prop)s(ert)m(y)h(FDT)g(is)g(undecidable)f(for)h(\014nitely)g (pre-)399 1164 y(sen)m(ted)29 b(monoids)f(that)h(ha)m(v)m(e)g(p)s (olynomial-time)h(decidable)f(w)m(ord)f(problems.)37 b Fs(International)399 1276 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b(enc)g(e)46 b(on)f(R)-5 b(ewriting)399 3531 y(T)e(e)i(chniques)36 b(and)g(Applic)-5 b(ations)p Fu(,)37 b(v)m(olume)e(1379)g(of)f Fs(L)-5 b(e)g(ctur)g(e)37 b(Notes)f(in)f(Computer)i(Scienc)-5 b(e)p Fu(,)399 3644 y(pages)31 b(166{180,)j(Berlin,)d(1998.)h (Springer-V)-8 b(erlag.)212 3832 y([20])47 b(H.)34 b(Zan)m(tema.)51 b(T)-8 b(ermination)34 b(of)g(term)f(rewriting:)47 b(In)m(terpretation) 35 b(and)e(t)m(yp)s(e)h(elimination.)399 3945 y Fs(Journal)g(of)e(Symb) -5 b(olic)34 b(Computation)p Fu(,)f(17:23{50,)h(1994.)1897 5462 y(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF