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186 % |
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233 % |
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269 TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 |
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270 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll |
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271 exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 |
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272 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ |
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273 pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get |
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274 div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type |
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275 /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end |
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276 definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup |
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277 sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll |
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278 mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ |
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279 exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} |
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280 forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def |
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281 end |
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282 |
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283 %%EndProcSet |
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284 %%BeginFont: CMTI9 |
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285 %!PS-AdobeFont-1.1: CMTI9 1.0 |
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286 %%CreationDate: 1991 Aug 18 21:08:07 |
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287 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
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288 11 dict begin |
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289 /FontInfo 7 dict dup begin |
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290 /version (1.0) readonly def |
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291 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
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292 /FullName (CMTI9) readonly def |
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293 /FamilyName (Computer Modern) readonly def |
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294 /Weight (Medium) readonly def |
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295 /ItalicAngle -14.04 def |
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296 /isFixedPitch false def |
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297 end readonly def |
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298 /FontName /CMTI9 def |
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299 /PaintType 0 def |
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300 /FontType 1 def |
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301 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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302 /Encoding 256 array |
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303 0 1 255 {1 index exch /.notdef put} for |
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304 dup 0 /.notdef put |
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305 readonly def |
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306 /FontBBox{-35 -250 1148 750}readonly def |
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307 /UniqueID 5000827 def |
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308 currentdict end |
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309 currentfile eexec |
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557 189E16319A4C |
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558 0000000000000000000000000000000000000000000000000000000000000000 |
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559 0000000000000000000000000000000000000000000000000000000000000000 |
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560 0000000000000000000000000000000000000000000000000000000000000000 |
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561 0000000000000000000000000000000000000000000000000000000000000000 |
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562 0000000000000000000000000000000000000000000000000000000000000000 |
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563 0000000000000000000000000000000000000000000000000000000000000000 |
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564 0000000000000000000000000000000000000000000000000000000000000000 |
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565 0000000000000000000000000000000000000000000000000000000000000000 |
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566 cleartomark |
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567 %%EndFont |
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568 %%BeginFont: CMMI6 |
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569 %!PS-AdobeFont-1.1: CMMI6 1.100 |
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570 %%CreationDate: 1996 Jul 23 07:53:52 |
|
571 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
572 11 dict begin |
|
573 /FontInfo 7 dict dup begin |
|
574 /version (1.100) readonly def |
|
575 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
576 /FullName (CMMI6) readonly def |
|
577 /FamilyName (Computer Modern) readonly def |
|
578 /Weight (Medium) readonly def |
|
579 /ItalicAngle -14.04 def |
|
580 /isFixedPitch false def |
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581 end readonly def |
|
582 /FontName /CMMI6 def |
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583 /PaintType 0 def |
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584 /FontType 1 def |
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585 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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586 /Encoding 256 array |
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587 0 1 255 {1 index exch /.notdef put} for |
|
588 dup 0 /.notdef put |
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589 readonly def |
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590 /FontBBox{11 -250 1241 750}readonly def |
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591 /UniqueID 5087381 def |
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592 currentdict end |
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593 currentfile eexec |
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625 0F5B7B569A9C434BC51F73867502C7B793F31A662317782D8FC352DD9B24E9CC |
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630 8EC0669F7C94A445220392867E |
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632 0000000000000000000000000000000000000000000000000000000000000000 |
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633 0000000000000000000000000000000000000000000000000000000000000000 |
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634 0000000000000000000000000000000000000000000000000000000000000000 |
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635 0000000000000000000000000000000000000000000000000000000000000000 |
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636 0000000000000000000000000000000000000000000000000000000000000000 |
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637 0000000000000000000000000000000000000000000000000000000000000000 |
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638 0000000000000000000000000000000000000000000000000000000000000000 |
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639 cleartomark |
|
640 %%EndFont |
|
641 %%BeginFont: CMSY9 |
|
642 %!PS-AdobeFont-1.1: CMSY9 1.0 |
|
643 %%CreationDate: 1991 Aug 15 07:22:27 |
|
644 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
645 11 dict begin |
|
646 /FontInfo 7 dict dup begin |
|
647 /version (1.0) readonly def |
|
648 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
649 /FullName (CMSY9) readonly def |
|
650 /FamilyName (Computer Modern) readonly def |
|
651 /Weight (Medium) readonly def |
|
652 /ItalicAngle -14.035 def |
|
653 /isFixedPitch false def |
|
654 end readonly def |
|
655 /FontName /CMSY9 def |
|
656 /PaintType 0 def |
|
657 /FontType 1 def |
|
658 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
659 /Encoding 256 array |
|
660 0 1 255 {1 index exch /.notdef put} for |
|
661 dup 0 /.notdef put |
|
662 readonly def |
|
663 /FontBBox{-30 -958 1146 777}readonly def |
|
664 /UniqueID 5000819 def |
|
665 currentdict end |
|
666 currentfile eexec |
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671 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A |
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673 65BCA87979C36D14CB552E9A985E48BE4E88ECA16DF418749AF04FDD2B0E1380 |
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674 D281BB2476BB45FF30946B247DFD7F57305FA87E50CA338121C71CDFDF927A9C |
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675 77FF14CB4A1D6D80356FB1171ED38C37702350497B44E42CE31DB2F493807DAA |
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678 1A06C514D91C4C937D4E642503392B1CD1B4BC9CF0AF85959A1BCE4E139BB4EA |
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680 DD2E7241D43631E78179299E32A2C347247F1440F6C3A57ECAEB0082297DF57A |
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681 CFA50AB335FEF49FEC0503D21E97CE8A715937513596FCB0907198203954087E |
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685 A49E4A3B57247E4F81F6827326138068CF90322859FFB6EDEB57229054625827 |
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717 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
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718 11 dict begin |
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796 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
797 11 dict begin |
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950 0000000000000000000000000000000000000000000000000000000000000000 |
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951 cleartomark |
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952 %%EndFont |
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953 %%BeginFont: CMTT10 |
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954 %!PS-AdobeFont-1.1: CMTT10 1.00B |
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955 %%CreationDate: 1992 Apr 26 10:42:42 |
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956 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
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957 11 dict begin |
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958 /FontInfo 7 dict dup begin |
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959 /version (1.00B) readonly def |
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960 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
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961 /FullName (CMTT10) readonly def |
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962 /FamilyName (Computer Modern) readonly def |
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963 /Weight (Medium) readonly def |
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964 /ItalicAngle 0 def |
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965 /isFixedPitch true def |
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966 end readonly def |
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967 /FontName /CMTT10 def |
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968 /PaintType 0 def |
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969 /FontType 1 def |
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970 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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971 /Encoding 256 array |
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972 0 1 255 {1 index exch /.notdef put} for |
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973 dup 0 /.notdef put |
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974 readonly def |
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975 /FontBBox{-4 -235 731 800}readonly def |
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976 /UniqueID 5000832 def |
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977 currentdict end |
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1222 0000000000000000000000000000000000000000000000000000000000000000 |
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1223 0000000000000000000000000000000000000000000000000000000000000000 |
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1224 0000000000000000000000000000000000000000000000000000000000000000 |
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1225 0000000000000000000000000000000000000000000000000000000000000000 |
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1226 0000000000000000000000000000000000000000000000000000000000000000 |
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1227 0000000000000000000000000000000000000000000000000000000000000000 |
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1228 0000000000000000000000000000000000000000000000000000000000000000 |
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1229 cleartomark |
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1230 %%EndFont |
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1231 %%BeginFont: CMSY7 |
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1232 %!PS-AdobeFont-1.1: CMSY7 1.0 |
|
1233 %%CreationDate: 1991 Aug 15 07:21:52 |
|
1234 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1235 11 dict begin |
|
1236 /FontInfo 7 dict dup begin |
|
1237 /version (1.0) readonly def |
|
1238 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1239 /FullName (CMSY7) readonly def |
|
1240 /FamilyName (Computer Modern) readonly def |
|
1241 /Weight (Medium) readonly def |
|
1242 /ItalicAngle -14.035 def |
|
1243 /isFixedPitch false def |
|
1244 end readonly def |
|
1245 /FontName /CMSY7 def |
|
1246 /PaintType 0 def |
|
1247 /FontType 1 def |
|
1248 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1249 /Encoding 256 array |
|
1250 0 1 255 {1 index exch /.notdef put} for |
|
1251 dup 0 /.notdef put |
|
1252 readonly def |
|
1253 /FontBBox{-15 -951 1252 782}readonly def |
|
1254 /UniqueID 5000817 def |
|
1255 currentdict end |
|
1256 currentfile eexec |
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1285 0000000000000000000000000000000000000000000000000000000000000000 |
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1286 0000000000000000000000000000000000000000000000000000000000000000 |
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1287 0000000000000000000000000000000000000000000000000000000000000000 |
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1288 0000000000000000000000000000000000000000000000000000000000000000 |
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1289 0000000000000000000000000000000000000000000000000000000000000000 |
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1290 0000000000000000000000000000000000000000000000000000000000000000 |
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1291 cleartomark |
|
1292 %%EndFont |
|
1293 %%BeginFont: CMSY10 |
|
1294 %!PS-AdobeFont-1.1: CMSY10 1.0 |
|
1295 %%CreationDate: 1991 Aug 15 07:20:57 |
|
1296 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1297 11 dict begin |
|
1298 /FontInfo 7 dict dup begin |
|
1299 /version (1.0) readonly def |
|
1300 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1301 /FullName (CMSY10) readonly def |
|
1302 /FamilyName (Computer Modern) readonly def |
|
1303 /Weight (Medium) readonly def |
|
1304 /ItalicAngle -14.035 def |
|
1305 /isFixedPitch false def |
|
1306 end readonly def |
|
1307 /FontName /CMSY10 def |
|
1308 /PaintType 0 def |
|
1309 /FontType 1 def |
|
1310 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1311 /Encoding 256 array |
|
1312 0 1 255 {1 index exch /.notdef put} for |
|
1313 dup 0 /.notdef put |
|
1314 readonly def |
|
1315 /FontBBox{-29 -960 1116 775}readonly def |
|
1316 /UniqueID 5000820 def |
|
1317 currentdict end |
|
1318 currentfile eexec |
|
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1410 0000000000000000000000000000000000000000000000000000000000000000 |
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1411 0000000000000000000000000000000000000000000000000000000000000000 |
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1412 0000000000000000000000000000000000000000000000000000000000000000 |
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1413 0000000000000000000000000000000000000000000000000000000000000000 |
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1414 0000000000000000000000000000000000000000000000000000000000000000 |
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1415 0000000000000000000000000000000000000000000000000000000000000000 |
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1416 cleartomark |
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1417 %%EndFont |
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1418 %%BeginFont: CMMI10 |
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1419 %!PS-AdobeFont-1.1: CMMI10 1.100 |
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1420 %%CreationDate: 1996 Jul 23 07:53:57 |
|
1421 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1422 11 dict begin |
|
1423 /FontInfo 7 dict dup begin |
|
1424 /version (1.100) readonly def |
|
1425 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1426 /FullName (CMMI10) readonly def |
|
1427 /FamilyName (Computer Modern) readonly def |
|
1428 /Weight (Medium) readonly def |
|
1429 /ItalicAngle -14.04 def |
|
1430 /isFixedPitch false def |
|
1431 end readonly def |
|
1432 /FontName /CMMI10 def |
|
1433 /PaintType 0 def |
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1434 /FontType 1 def |
|
1435 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1436 /Encoding 256 array |
|
1437 0 1 255 {1 index exch /.notdef put} for |
|
1438 dup 0 /.notdef put |
|
1439 readonly def |
|
1440 /FontBBox{-32 -250 1048 750}readonly def |
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1441 /UniqueID 5087385 def |
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1442 currentdict end |
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1443 currentfile eexec |
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1623 0000000000000000000000000000000000000000000000000000000000000000 |
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1624 0000000000000000000000000000000000000000000000000000000000000000 |
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1625 0000000000000000000000000000000000000000000000000000000000000000 |
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1626 0000000000000000000000000000000000000000000000000000000000000000 |
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1627 0000000000000000000000000000000000000000000000000000000000000000 |
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1628 0000000000000000000000000000000000000000000000000000000000000000 |
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1629 0000000000000000000000000000000000000000000000000000000000000000 |
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1630 cleartomark |
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1631 %%EndFont |
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1632 %%BeginFont: CMBX12 |
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1633 %!PS-AdobeFont-1.1: CMBX12 1.0 |
|
1634 %%CreationDate: 1991 Aug 20 16:34:54 |
|
1635 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1636 11 dict begin |
|
1637 /FontInfo 7 dict dup begin |
|
1638 /version (1.0) readonly def |
|
1639 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1640 /FullName (CMBX12) readonly def |
|
1641 /FamilyName (Computer Modern) readonly def |
|
1642 /Weight (Bold) readonly def |
|
1643 /ItalicAngle 0 def |
|
1644 /isFixedPitch false def |
|
1645 end readonly def |
|
1646 /FontName /CMBX12 def |
|
1647 /PaintType 0 def |
|
1648 /FontType 1 def |
|
1649 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1650 /Encoding 256 array |
|
1651 0 1 255 {1 index exch /.notdef put} for |
|
1652 dup 0 /.notdef put |
|
1653 readonly def |
|
1654 /FontBBox{-53 -251 1139 750}readonly def |
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1655 /UniqueID 5000769 def |
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1656 currentdict end |
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1657 currentfile eexec |
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1863 0000000000000000000000000000000000000000000000000000000000000000 |
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1864 0000000000000000000000000000000000000000000000000000000000000000 |
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1865 0000000000000000000000000000000000000000000000000000000000000000 |
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1866 0000000000000000000000000000000000000000000000000000000000000000 |
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1867 0000000000000000000000000000000000000000000000000000000000000000 |
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1868 0000000000000000000000000000000000000000000000000000000000000000 |
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1869 0000000000000000000000000000000000000000000000000000000000000000 |
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1870 cleartomark |
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1871 %%EndFont |
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1872 %%BeginFont: CMMI9 |
|
1873 %!PS-AdobeFont-1.1: CMMI9 1.100 |
|
1874 %%CreationDate: 1996 Jul 23 07:53:55 |
|
1875 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
1876 11 dict begin |
|
1877 /FontInfo 7 dict dup begin |
|
1878 /version (1.100) readonly def |
|
1879 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
1880 /FullName (CMMI9) readonly def |
|
1881 /FamilyName (Computer Modern) readonly def |
|
1882 /Weight (Medium) readonly def |
|
1883 /ItalicAngle -14.04 def |
|
1884 /isFixedPitch false def |
|
1885 end readonly def |
|
1886 /FontName /CMMI9 def |
|
1887 /PaintType 0 def |
|
1888 /FontType 1 def |
|
1889 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
1890 /Encoding 256 array |
|
1891 0 1 255 {1 index exch /.notdef put} for |
|
1892 dup 0 /.notdef put |
|
1893 readonly def |
|
1894 /FontBBox{-29 -250 1075 750}readonly def |
|
1895 /UniqueID 5087384 def |
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1896 currentdict end |
|
1897 currentfile eexec |
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1911 730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F |
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1919 759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3 |
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1920 5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2 |
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1929 DE737E02D2390DF19D2897E974708DFBDA3C802A1E9407CBA645D5A2AFE34142 |
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1931 8B976706C23A57E6F7ECD38C8B1BDE1E6C180F1AD1413F7AE5739674318A0843 |
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1932 DF6FDC96B95E9A6DEE658A1137AA3847C674839F630653DFDE99652DC0ED1C68 |
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1933 1EF74A588654D4ED9339488D7BF24C38866D1591A6C1E19F675D221B55384C8D |
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1934 61C58385B406A786D4C5CCDF34DCCEF62F49928760E5036237C4BE105B5AF62F |
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1935 14C2C6129B04BA48420DDC47A7C7CD24A49E3A1C596DA213F151AA25CCAB9E4F |
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1936 C6116FA2965428375E54F7EE2BEE31132741E69ED2EC6C5E91037931C6ABE035 |
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1938 A8352C36FC2C1B0C95A36C7F36409B730C5E455B4C9B31CCFB1C89F7D34410EC |
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1939 61C462176E8A126BE83D0D8FE0214B559D03D8F2EAFD1B9A9EFEC0E51C39318F |
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1940 B986612C4D1FCBB5828D6CD3C06662C268813EE181B6A0B3BA674474379979B8 |
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1941 846FE4D02953FBEC269885B1BCB9B596751BCF0479233264DEB0D6D101A9848C |
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1942 D3D81DCD773053F92AB20A86E320953AF16CE56C59BC80E06408EAB2CC3D6FEF |
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1943 579CD48E05FB6842F2E77B6E2CB0921307747D350757CC56ADFAE41F096B56B4 |
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1944 48DA9EDD036B17A6CD3EA367135774025A66F3EF8F95C7F24F760D20A8F137AA |
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1945 4C841E0B3DC090B372D7F97C2C5C55C354209B7FBB23EE682B3F9D6150978B1F |
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1946 D7BD91C4BA734D860DDEFDD63FC82481B8A44CCFDA0A4979150528CCBC6477AA |
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1947 818E01F8E123743F8A48F8BBFA80560A58A147B9F772AACF16D6724453D4C02F |
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1948 700DE56AE001F7313AFEEDC030C1CFD457C505C2B50EAC8385067A67B4DF08B0 |
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1949 6F3991B9F979B0F58D299C74124990DA1CD0029C996ED4067A6B728670FD48E1 |
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1950 56FA01AC81D418F3AE636938EA1C65B9D9102CEFA74CD16C85FECC964FD0244B |
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1951 7C041B3B260F0144363D74A8E6FBEE772751AA7CB1D6E646F3E0722E8E64288D |
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1952 9AF535786D64A075035066C7085B5281BBEBA2BFA7D71A5512F98195BD5E15EE |
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1953 E31A616136C7A7411EF9C63AFEBFDBFE7962AF66EA87C6D3FD7E4A8960350CBE |
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1954 46E32368F95FFEB188B9984832A6D830B2BE2FBDB372DD5E3C4C6B91617587A8 |
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2006 cleartomark |
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2007 %%EndFont |
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2008 %%BeginFont: CMBX9 |
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2009 %!PS-AdobeFont-1.1: CMBX9 1.0 |
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2010 %%CreationDate: 1991 Aug 20 16:36:25 |
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2011 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
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2012 11 dict begin |
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2013 /FontInfo 7 dict dup begin |
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2014 /version (1.0) readonly def |
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2015 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
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2016 /FullName (CMBX9) readonly def |
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2017 /FamilyName (Computer Modern) readonly def |
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2018 /Weight (Bold) readonly def |
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2019 /ItalicAngle 0 def |
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2020 /isFixedPitch false def |
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2021 end readonly def |
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2022 /FontName /CMBX9 def |
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2023 /PaintType 0 def |
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2024 /FontType 1 def |
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2025 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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2026 /Encoding 256 array |
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2027 0 1 255 {1 index exch /.notdef put} for |
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2028 dup 0 /.notdef put |
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2029 readonly def |
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2031 /UniqueID 5000767 def |
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2032 currentdict end |
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2165 0000000000000000000000000000000000000000000000000000000000000000 |
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2166 cleartomark |
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2167 %%EndFont |
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2168 %%BeginFont: CMTT9 |
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2169 %!PS-AdobeFont-1.1: CMTT9 1.0 |
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2170 %%CreationDate: 1991 Aug 20 16:46:24 |
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2171 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
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2172 11 dict begin |
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2173 /FontInfo 7 dict dup begin |
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2174 /version (1.0) readonly def |
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2175 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
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2176 /FullName (CMTT9) readonly def |
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2177 /FamilyName (Computer Modern) readonly def |
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2178 /Weight (Medium) readonly def |
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2179 /ItalicAngle 0 def |
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2180 /isFixedPitch true def |
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2181 end readonly def |
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2182 /FontName /CMTT9 def |
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2183 /PaintType 0 def |
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2184 /FontType 1 def |
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2185 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
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2186 /Encoding 256 array |
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2187 0 1 255 {1 index exch /.notdef put} for |
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2188 dup 0 /.notdef put |
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2189 readonly def |
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2416 0000000000000000000000000000000000000000000000000000000000000000 |
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2417 0000000000000000000000000000000000000000000000000000000000000000 |
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2418 0000000000000000000000000000000000000000000000000000000000000000 |
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2419 0000000000000000000000000000000000000000000000000000000000000000 |
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2420 0000000000000000000000000000000000000000000000000000000000000000 |
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2421 0000000000000000000000000000000000000000000000000000000000000000 |
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2422 0000000000000000000000000000000000000000000000000000000000000000 |
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2423 cleartomark |
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2424 %%EndFont |
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2425 %%BeginFont: CMR9 |
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2426 %!PS-AdobeFont-1.1: CMR9 1.0 |
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2427 %%CreationDate: 1991 Aug 20 16:39:59 |
|
2428 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2429 11 dict begin |
|
2430 /FontInfo 7 dict dup begin |
|
2431 /version (1.0) readonly def |
|
2432 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2433 /FullName (CMR9) readonly def |
|
2434 /FamilyName (Computer Modern) readonly def |
|
2435 /Weight (Medium) readonly def |
|
2436 /ItalicAngle 0 def |
|
2437 /isFixedPitch false def |
|
2438 end readonly def |
|
2439 /FontName /CMR9 def |
|
2440 /PaintType 0 def |
|
2441 /FontType 1 def |
|
2442 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2443 /Encoding 256 array |
|
2444 0 1 255 {1 index exch /.notdef put} for |
|
2445 dup 0 /.notdef put |
|
2446 readonly def |
|
2447 /FontBBox{-39 -250 1036 750}readonly def |
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2448 /UniqueID 5000792 def |
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2449 currentdict end |
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2450 currentfile eexec |
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2761 0000000000000000000000000000000000000000000000000000000000000000 |
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2762 0000000000000000000000000000000000000000000000000000000000000000 |
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2763 0000000000000000000000000000000000000000000000000000000000000000 |
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2764 0000000000000000000000000000000000000000000000000000000000000000 |
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2765 cleartomark |
|
2766 %%EndFont |
|
2767 %%BeginFont: CMR6 |
|
2768 %!PS-AdobeFont-1.1: CMR6 1.0 |
|
2769 %%CreationDate: 1991 Aug 20 16:39:02 |
|
2770 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2771 11 dict begin |
|
2772 /FontInfo 7 dict dup begin |
|
2773 /version (1.0) readonly def |
|
2774 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2775 /FullName (CMR6) readonly def |
|
2776 /FamilyName (Computer Modern) readonly def |
|
2777 /Weight (Medium) readonly def |
|
2778 /ItalicAngle 0 def |
|
2779 /isFixedPitch false def |
|
2780 end readonly def |
|
2781 /FontName /CMR6 def |
|
2782 /PaintType 0 def |
|
2783 /FontType 1 def |
|
2784 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2785 /Encoding 256 array |
|
2786 0 1 255 {1 index exch /.notdef put} for |
|
2787 dup 0 /.notdef put |
|
2788 readonly def |
|
2789 /FontBBox{-20 -250 1193 750}readonly def |
|
2790 /UniqueID 5000789 def |
|
2791 currentdict end |
|
2792 currentfile eexec |
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2835 C8492D74F1F8821811A109842FF74FFE4CBB994B16F4F736A743808F775C13B3 |
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2836 7623DCB63EEC224C8B930F2B1886F2BFDCD3FCE095C2BD94FE612B8B59456FCE |
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2837 |
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2841 0000000000000000000000000000000000000000000000000000000000000000 |
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2842 0000000000000000000000000000000000000000000000000000000000000000 |
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2843 0000000000000000000000000000000000000000000000000000000000000000 |
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2844 0000000000000000000000000000000000000000000000000000000000000000 |
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2845 0000000000000000000000000000000000000000000000000000000000000000 |
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2846 cleartomark |
|
2847 %%EndFont |
|
2848 %%BeginFont: CMR7 |
|
2849 %!PS-AdobeFont-1.1: CMR7 1.0 |
|
2850 %%CreationDate: 1991 Aug 20 16:39:21 |
|
2851 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2852 11 dict begin |
|
2853 /FontInfo 7 dict dup begin |
|
2854 /version (1.0) readonly def |
|
2855 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2856 /FullName (CMR7) readonly def |
|
2857 /FamilyName (Computer Modern) readonly def |
|
2858 /Weight (Medium) readonly def |
|
2859 /ItalicAngle 0 def |
|
2860 /isFixedPitch false def |
|
2861 end readonly def |
|
2862 /FontName /CMR7 def |
|
2863 /PaintType 0 def |
|
2864 /FontType 1 def |
|
2865 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2866 /Encoding 256 array |
|
2867 0 1 255 {1 index exch /.notdef put} for |
|
2868 dup 0 /.notdef put |
|
2869 readonly def |
|
2870 /FontBBox{-27 -250 1122 750}readonly def |
|
2871 /UniqueID 5000790 def |
|
2872 currentdict end |
|
2873 currentfile eexec |
|
2874 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
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2877 D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
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2878 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 |
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2912 F39C3E3A29F9BAA3AEBA89CC3E7B134E79A2DA39D449D98FE4ADE1A0ACDA79C4 |
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2913 1E2C78CDCC5BDD192DAA27F17DC4368EBC54CC009802018E50519F69025DE7E8 |
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2914 FB61694AEA672E59D00F6BA0CA9509FB8DC6544CB92B70BA0AE393A43959C90E |
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2915 8857B0DA53A8D26426CAE0DCF2F3033BAC62502E08447B09CC18B2525AB65EC2 |
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2916 C63074FF725168D115137ECDAB1EEB083711F647F757C9806056811466D59087 |
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2917 B93D6B71594046B3B60A7D5789D6B99FC14764428A251F0A54ED |
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2918 0000000000000000000000000000000000000000000000000000000000000000 |
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2919 0000000000000000000000000000000000000000000000000000000000000000 |
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2920 0000000000000000000000000000000000000000000000000000000000000000 |
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2921 0000000000000000000000000000000000000000000000000000000000000000 |
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2922 0000000000000000000000000000000000000000000000000000000000000000 |
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2923 0000000000000000000000000000000000000000000000000000000000000000 |
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2924 0000000000000000000000000000000000000000000000000000000000000000 |
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2925 0000000000000000000000000000000000000000000000000000000000000000 |
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2926 cleartomark |
|
2927 %%EndFont |
|
2928 %%BeginFont: CMR10 |
|
2929 %!PS-AdobeFont-1.1: CMR10 1.00B |
|
2930 %%CreationDate: 1992 Feb 19 19:54:52 |
|
2931 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
2932 11 dict begin |
|
2933 /FontInfo 7 dict dup begin |
|
2934 /version (1.00B) readonly def |
|
2935 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
2936 /FullName (CMR10) readonly def |
|
2937 /FamilyName (Computer Modern) readonly def |
|
2938 /Weight (Medium) readonly def |
|
2939 /ItalicAngle 0 def |
|
2940 /isFixedPitch false def |
|
2941 end readonly def |
|
2942 /FontName /CMR10 def |
|
2943 /PaintType 0 def |
|
2944 /FontType 1 def |
|
2945 /FontMatrix [0.001 0 0 0.001 0 0] readonly def |
|
2946 /Encoding 256 array |
|
2947 0 1 255 {1 index exch /.notdef put} for |
|
2948 dup 0 /.notdef put |
|
2949 readonly def |
|
2950 /FontBBox{-251 -250 1009 969}readonly def |
|
2951 /UniqueID 5000793 def |
|
2952 currentdict end |
|
2953 currentfile eexec |
|
2954 D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 |
|
2955 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 |
|
2956 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F |
|
2957 D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 |
|
2958 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 |
|
2959 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 |
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2960 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F |
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2961 D1F017CE45884D76EF2CB9BC5821FD25365DDEA6E45F332B5F68A44AD8A530F0 |
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2962 92A36FAC8D27F9087AFEEA2096F839A2BC4B937F24E080EF7C0F9374A18D565C |
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2963 295A05210DB96A23175AC59A9BD0147A310EF49C551A417E0A22703F94FF7B75 |
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2976 B9E88A4024C3326AD2F34F47B54739B48825AD6699F7D117EA4C4AEC4440BF6D |
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2978 97BE2BC7CA32B4534075F6491BE959D1F635463E71679E527F4F456F774B2AF8 |
|
2979 FEF3D8C63B2F8B99FE0F73BA44B3CF15A613471EA3C7A1CD783D3EB41F4ACEE5 |
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3323 0000000000000000000000000000000000000000000000000000000000000000 |
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3324 0000000000000000000000000000000000000000000000000000000000000000 |
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3325 0000000000000000000000000000000000000000000000000000000000000000 |
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3326 0000000000000000000000000000000000000000000000000000000000000000 |
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3327 0000000000000000000000000000000000000000000000000000000000000000 |
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3328 0000000000000000000000000000000000000000000000000000000000000000 |
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3329 0000000000000000000000000000000000000000000000000000000000000000 |
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3330 0000000000000000000000000000000000000000000000000000000000000000 |
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3331 cleartomark |
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3332 %%EndFont |
|
3333 %%BeginFont: CMMIB10 |
|
3334 %!PS-AdobeFont-1.1: CMMIB10 1.100 |
|
3335 %%CreationDate: 1996 Jul 23 07:54:00 |
|
3336 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. |
|
3337 11 dict begin |
|
3338 /FontInfo 7 dict dup begin |
|
3339 /version (1.100) readonly def |
|
3340 /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def |
|
3341 /FullName (CMMIB10) readonly def |
|
3342 /FamilyName (Computer Modern) readonly def |
|
3343 /Weight (Bold) readonly def |
|
3344 /ItalicAngle -14.04 def |
|
3345 /isFixedPitch false def |
|
3346 end readonly def |
|
3347 /FontName /CMMIB10 def |
|
3348 /PaintType 0 def |
|
3349 /FontType 1 def |
|
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3404 TeXDict begin 39158280 55380996 1000 600 600 (aprolog.dvi) |
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3457 %%EndProlog |
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3460 TeXDict begin |
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3462 end |
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3464 %%Page: 1 1 |
|
3465 TeXDict begin 1 0 bop 550 448 a Fs(System)45 b(Description:)h |
|
3466 Fr(\013)p Fs(Prolog,)g(a)g(F)-11 b(resh)44 b(Approac)l(h)614 |
|
3467 598 y(to)i(Logic)f(Programming)h(Mo)t(dulo)e Fr(\013)p |
|
3468 Fs(-Equiv)-7 b(alence)1275 886 y Fq(James)27 b(Cheney)1796 |
|
3469 856 y Fp(1)1861 886 y Fq(and)g(Christian)g(Urban)2616 |
|
3470 856 y Fp(2)1621 1029 y Fo(1)1694 1060 y Fn(Cornell)g(Univ)n(ersit)n(y) |
|
3471 1532 1152 y Fm(jcheney@cs.cornell.edu)1516 1211 y Fo(2)1589 |
|
3472 1243 y Fn(Univ)n(ersit)n(y)e(of)i(Cam)n(bridge)1415 1334 |
|
3473 y Fm(Christian.Urban@cl.cam.ac.uk)759 1667 y Fl(Abstract.)43 |
|
3474 b Fk(\013)p Fn(Prolog)31 b(is)f(a)g(protot)n(yp)r(e)f(logic)j |
|
3475 (programming)e(language)h(with)f(a)759 1758 y(built-in)36 |
|
3476 b(notion)h(of)h(binders)f(and)f(uni\014cation)h(mo)r(dulo)g |
|
3477 Fk(\013)p Fn(-equiv)l(alence.)g(It)f(is)759 1850 y(based)d(on)f(a)h |
|
3478 (mild)f(extension)g(of)i(\014rst-order)e(Horn)f(form)n(ulae:)j(instead) |
|
3479 f(of)g(the)759 1941 y(usual)26 b(\014rst-order)g(terms)g(and)f |
|
3480 (\014rst-order)g(uni\014cation,)h Fk(\013)p Fn(Prolog)i(uses)e(nominal) |
|
3481 759 2032 y(terms)k(and)f(nominal)h(uni\014cation)g(in)n(tro)r(duced)f |
|
3482 (in)g([3].)i(In)e(this)h(pap)r(er,)f(w)n(e)h(giv)n(e)759 |
|
3483 2124 y(three)20 b(examples)g(that)g(demonstrate)g(the)g(adv)l(an)n |
|
3484 (tages)g(of)h Fk(\013)p Fn(Prolog)h(and)d(describ)r(e)759 |
|
3485 2215 y(our)26 b(curren)n(t)f(implemen)n(tation.)523 2493 |
|
3486 y Fj(1)112 b(In)m(tro)s(duction)523 2690 y Fq(Logic)23 |
|
3487 b(programming)g(is)h(particularly)f(suited)i(for)f(implemen)n(ting)h |
|
3488 (inference)f(rules)g(de\014n-)523 2790 y(ing)d(relations)f(o)n(v)n(er)g |
|
3489 (terms.)h(Man)n(y)g(in)n(teresting)f(examples,)h(ho)n(w)n(ev)n(er,)f |
|
3490 (in)n(v)n(olv)n(e)f(terms)i(with)523 2889 y(b)r(ound)27 |
|
3491 b(v)-5 b(ariables)26 b(and)g Fi(\013)p Fq(-equiv)-5 b(alence.)26 |
|
3492 b(F)-7 b(or)27 b(example)f(the)h(t)n(yping)f(relation)g(for)g |
|
3493 Fi(\025)p Fq(-terms)523 2989 y(is)i(often)f(form)n(ulated)g(with)h(the) |
|
3494 g(inference)g(rules)813 3142 y Fi(x)9 b Fq(:)g Fi(A)25 |
|
3495 b Fh(2)e Fi(\000)p 813 3165 316 4 v 825 3238 a(\000)30 |
|
3496 b(.)18 b(x)9 b Fq(:)g Fi(A)1294 3145 y(\000)30 b(.)18 |
|
3497 b(M)g Fq(:)9 b Fi(A)23 b Fh(!)g Fi(B)88 b(\000)30 b(.)18 |
|
3498 b(N)g Fq(:)9 b Fi(A)p 1294 3165 935 4 v 1554 3238 a(\000)30 |
|
3499 b(.)18 b(M)9 b(N)17 b Fq(:)9 b Fi(B)2394 3123 y Fh(f)p |
|
3500 Fi(x)23 b Fq(:)g Fi(A)p Fh(g)18 b([)h Fi(\000)30 b(.)18 |
|
3501 b(M)g Fq(:)9 b Fi(B)p 2394 3164 694 4 v 2415 3238 a(\000)31 |
|
3502 b(.)17 b(\025x:M)i Fq(:)9 b Fi(A)24 b Fh(!)f Fi(B)523 |
|
3503 3402 y Fq(with)d(the)h(implicit)f(side-condition)f(that)h(in)g |
|
3504 Fi(\000)32 b Fq(\(a)20 b(set)f(of)h(t)n(yp)r(e)g(assignmen)n(ts)f(for)g |
|
3505 (v)-5 b(ariables\))523 3502 y(no)40 b(v)-5 b(ariable)38 |
|
3506 b(has)i(more)f(than)h(one)f(assignmen)n(t.)g(This)h(side-condition)f |
|
3507 (is)g(violated)g(if)523 3601 y(the)d(third)f(rule)g(is)h(applied)f(b)r |
|
3508 (ottom-up)h(and)f Fi(\000)47 b Fq(in)36 b(the)g(conclusion)e(already)g |
|
3509 (con)n(tains)523 3701 y(an)j(assignmen)n(t)f(for)g Fi(x)p |
|
3510 Fq(.)i(In)f(this)g(case)f(the)h(b)r(ound)h(v)-5 b(ariable)36 |
|
3511 b Fi(x)h Fq(in)g Fi(\025x:M)47 b Fq(needs)37 b(to)g(b)r(e)523 |
|
3512 3801 y(renamed)26 b(to)h(a)f(new)h Fi(x)1245 3771 y Fg(0)1296 |
|
3513 3801 y Fq(so)f(that)h(a)f(t)n(yp)r(e)h(assignmen)n(t)f(for)g |
|
3514 Fi(x)2430 3771 y Fg(0)2481 3801 y Fq(can)g(b)r(e)i(safely)e(added)h(to) |
|
3515 f Fi(\000)12 b Fq(.)523 3900 y(It)32 b(is)g(understo)r(o)r(d)g(that)g |
|
3516 (this)g(renaming)f(do)r(es)h(not)g(a\013ect)g(the)g(`meaning')g(of)g |
|
3517 (the)g(term,)523 4000 y(b)r(ecause)27 b(terms)h(are)e(iden)n(ti\014ed)i |
|
3518 (mo)r(dulo)g Fi(\013)p Fq(-equiv)-5 b(alence.)648 4100 |
|
3519 y(Ho)n(w)n(ev)n(er,)21 b(this)j(in)n(tuitiv)n(e)g(accoun)n(t)e(of)i |
|
3520 Fi(\013)p Fq(-equiv)-5 b(alence)23 b(do)r(es)g(not)g(carry)f(o)n(v)n |
|
3521 (er)g(to)h(a)g(cor-)523 4199 y(rect)g(declarativ)n(e)f(implemen)n |
|
3522 (tation)i(in)f(Prolog.)f(F)-7 b(or)23 b(example,)g(if)h(w)n(e)f(na)-9 |
|
3523 b(\177)-32 b(\020v)n(ely)21 b(implemen)n(t)523 4299 y(the)28 |
|
3524 b Fi(\025)p Fq(-abstraction)e(t)n(yping)i(rule)f(with)h(the)g(clause) |
|
3525 591 4461 y Ff(type)42 b(Gamma)f(\(lam)h(X)i(M\))e(\(arrow)f(A)j(B\))e |
|
3526 (:-)h(type)f(\(pair)g(X)h(A\)::Gamma)d(M)j(B.)523 4625 |
|
3527 y Fq(where)32 b Ff(X)g Fq(is)g(a)g(\(logical\))g(v)-5 |
|
3528 b(ariable,)31 b(then)i(w)n(e)f(get)g(incorrect)g(answ)n(ers)e(for)i |
|
3529 Fi(\025)p Fq(-terms)g(suc)n(h)523 4725 y(as)25 b Fi(\025x:\025x:)p |
|
3530 Fq(\()p Fi(x)h(x)p Fq(\).)h(F)-7 b(or)25 b(this)i(un)n(t)n(yp)r(eable)e |
|
3531 (term,)h(the)h(na)-9 b(\177)-32 b(\020v)n(e)24 b(implemen)n(tation)i |
|
3532 (erroneously)523 4825 y(returns)i(t)n(w)n(o)f(p)r(ossible)h(t)n(yp)r |
|
3533 (es:)g Fi(\034)34 b Fh(!)24 b Fq(\()p Fi(\034)34 b Fh(!)24 |
|
3534 b Fi(\033)s Fq(\))h Fh(!)f Fi(\033)31 b Fq(and)d(\()p |
|
3535 Fi(\034)34 b Fh(!)25 b Fi(\033)s Fq(\))f Fh(!)g Fi(\034)34 |
|
3536 b Fh(!)24 b Fi(\033)s Fq(.)29 b(Although)523 4924 y(this)e(problem)e |
|
3537 (can)h(b)r(e)h(\014xed)f(b)n(y)g(judicious)g(use)g(of)g(cut,)h |
|
3538 (\014rst-order)d(terms)i(are)f(un)n(wieldy)p eop end |
|
3539 %%Page: 2 2 |
|
3540 TeXDict begin 2 1 bop 523 448 a Fq(for)27 b(implemen)n(ting)h |
|
3541 (relations)f(ab)r(out)g(syn)n(tax)g(with)h(binders)g(correctly)-7 |
|
3542 b(.)26 b(The)i(traditional)523 548 y(solution)j(for)f(this)i(problem)f |
|
3543 (is)g(to)g(use)g(a)g(higher-order)d(logic)j(programming)e(language,)523 |
|
3544 648 y(suc)n(h)h(as)f Fi(\025)p Fq(Prolog)f(\(see)h(for)h(example)f |
|
3545 ([1]\),)h(with)g(higher-order)d(terms.)j(Suc)n(h)g(languages)523 |
|
3546 747 y(pro)n(vide)k(elegan)n(t)h(mec)n(hanisms)g(for)g(dealing)g(with)h |
|
3547 (binders,)f(but)h(they)g(also)e(in)n(tro)r(duce)523 847 |
|
3548 y(new)24 b(problems:)f(uni\014cation)h(for)f(higher-order)f(terms)h(is) |
|
3549 h(in)g(general)f(undecidable)h(and)f(if)523 946 y(uni\014ers)j(exist)h |
|
3550 (for)f(a)g(problem)g(they)h(ma)n(y)f(not)h(b)r(e)g(subsumed)f(b)n(y)h |
|
3551 (a)f(single)g(most)g(general)523 1046 y(uni\014er.)648 |
|
3552 1155 y(In)i(this)g(pap)r(er)g(w)n(e)g(shall)g(presen)n(t)f(a)h(logic)f |
|
3553 (programming)g(language)f(called)i Fi(\013)p Fq(Prolog)523 |
|
3554 1255 y(in)g(whic)n(h)f(one)h(can)f(implemen)n(t)h(the)g(rules)f(giv)n |
|
3555 (en)g(ab)r(o)n(v)n(e)f(correctly)g(as)h(follo)n(ws:)532 |
|
3556 1455 y Fm(type)40 b(Gamma)h(\(var)f(X\))g(A)f(:-)h(member)g(\(pair)h(X) |
|
3557 e(A\))h(Gamma.)532 1566 y(type)g(Gamma)h(\(app)f(M)f(N\))h(B)f(:-)h |
|
3558 (type)g(Gamma)h(M)e(\(arrow)i(A)e(B\),)h(type)g(Gamma)h(N)e(A.)532 |
|
3559 1677 y(type)h(Gamma)h(\(lam)f(x.M\))g(\(arrow)h(A)e(B\))h(/)f(x#Gamma)i |
|
3560 (:-)f(type)g(\(pair)g(x)g(A\)::Gamma)h(M)e(B.)532 1812 |
|
3561 y(member)i(A)e(A::Tail.)532 1924 y(member)i(A)e(B::Tail)i(:-)f(member)g |
|
3562 (A)g(Tail.)523 2115 y Fq(Tw)n(o)i(no)n(v)n(el)g(features)h(of)f |
|
3563 Fi(\013)p Fq(Prolog)g(are)g(illustrated)g(b)n(y)h(the)g(third)h |
|
3564 (clause.)e(First,)h Fi(\025)p Fq(-)523 2215 y(abstractions)26 |
|
3565 b(are)h(translated)g(to)h Fe(nominal)i(terms)k Fq(whic)n(h)28 |
|
3566 b(include)g(an)g Fe(abstr)l(action)35 b Fq(con-)523 2315 |
|
3567 y(struct)h Ff(a.t)p Fq(.)f(Th)n(us,)g Ff(\(lam)42 b(x.M\))35 |
|
3568 b Fq(is)h(in)n(tended)g(to)g(unify)g(with)h(an)n(y)e |
|
3569 Fi(\025)p Fq(-abstraction,)g(for)523 2414 y(example)24 |
|
3570 b Ff(\(lam)42 b(x.var)g(x\))p Fq(,)24 b Ff(\(lam)42 b(y.app)f(\(var)h |
|
3571 (x\))h(\(var)f(y\)\))24 b Fq(and)g(so)g(on.)h(Note)g(that)523 |
|
3572 2514 y(binders)38 b(in)g(nominal)g(terms)f(ha)n(v)n(e)g(concrete)g(v)-5 |
|
3573 b(alues)38 b(that)g(are)f(not)h(restricted)g(to)g(the)523 |
|
3574 2613 y(scop)r(e)28 b(of)h(the)f(abstraction.)g(These)g(v)-5 |
|
3575 b(alues)28 b(can)g(b)r(e)h(therefore)e(used)i(in)g(the)f(b)r(o)r(dy)h |
|
3576 (of)g(the)523 2713 y(clause,)36 b(for)g(example)h(to)f(add)h |
|
3577 Ff(\(pair)k(x)j(A\))36 b Fq(to)g(the)h(con)n(text)g Ff(Gamma)p |
|
3578 Fq(.)d(In)j(con)n(trast,)f(in)523 2813 y(higher-order)f(abstract)i(syn) |
|
3579 n(tax)g(binders)h(are)e(anon)n(ymous)h(and)g(cannot)h(escap)r(e)f |
|
3580 (their)523 2912 y(scop)r(e.)30 b(So,)f(in)i Fi(\025)p |
|
3581 Fq(Prolog,)d(a)h(binder)h(from)g(the)g(head)g(cannot)f(b)r(e)i(used)f |
|
3582 (in)g(the)g(b)r(o)r(dy)g(of)g(a)523 3012 y(clause.)648 |
|
3583 3121 y(Second,)25 b Fe(fr)l(eshness)j(c)l(onstr)l(aints)p |
|
3584 Fq(,)d(here)g Ff(x#Gamma)p Fq(,)d(can)j(b)r(e)h(attac)n(hed)e(to)i(the) |
|
3585 f(heads)g(of)523 3221 y(clauses;)h(the)h(freshness)f(constrain)n(t)f |
|
3586 Ff(x#Gamma)f Fq(ensures)i(that)h(the)g(v)-5 b(ariable)26 |
|
3587 b Ff(Gamma)e Fq(is)j(not)523 3321 y(instan)n(tiated)c(with)g(a)g(term)g |
|
3588 (con)n(taining)f Ff(x)g Fq(freely)-7 b(.)23 b(Since)g(the)g(third)h |
|
3589 (clause)e(is)h(in)n(tended)g(to)523 3420 y(implemen)n(t)31 |
|
3590 b(the)g(t)n(yp)r(e)g(inference)f(rule)g(for)g Fi(\025)p |
|
3591 Fq(-abstractions,)g(its)g(op)r(erational)f(b)r(eha)n(viour)523 |
|
3592 3520 y(is)36 b(giv)n(en)g(b)n(y:)g(c)n(ho)r(ose)g(fresh)g(names)g(for)g |
|
3593 Ff(Gamma)p Fq(,)f Ff(x)p Fq(,)h Ff(M)p Fq(,)g Ff(A)g |
|
3594 Fq(and)h Ff(B)f Fq(\(this)h(is)f(standard)g(in)523 3619 |
|
3595 y(Prolog-lik)n(e)21 b(languages\),)i(unify)i(the)f(head)g(of)g(the)h |
|
3596 (clause)f(with)g(the)h(goal)e(form)n(ula,)g(apply)523 |
|
3597 3719 y(the)29 b(resulting)f(uni\014er)h(to)g(the)g(b)r(o)r(dy)g(of)g |
|
3598 (the)g(clause)f(and)h(mak)n(e)f(sure)g(that)h Ff(Gamma)e |
|
3599 Fq(is)i(not)523 3819 y(substituted)23 b(with)g(a)f(term)h(that)g(con)n |
|
3600 (tains)e(free)h(o)r(ccurrences)f(of)i(the)g(fresh)f(name)g(w)n(e)g(ha)n |
|
3601 (v)n(e)523 3918 y(c)n(hosen)27 b(for)g Ff(x)p Fq(.)648 |
|
3602 4028 y(T)-7 b(o)29 b(calculate)g(uni\014ers,)h Fi(\013)p |
|
3603 Fq(Prolog)e(emplo)n(ys)h(the)i(nominal)e(uni\014cation)h(algorithm)f |
|
3604 (of)523 4127 y(Urban)22 b(et)h(al.)f(\(see)g([3])g(for)g(the)h |
|
3605 (details\).)f(Nominal)h(uni\014cation)f(retains)g(the)g(main)h(adv)-5 |
|
3606 b(an-)523 4227 y(tages)19 b(of)h(\014rst-order)e(uni\014cation:)h(all)h |
|
3607 (uni\014cation)g(problems)f(are)g(decidable)g(and)h(solv)-5 |
|
3608 b(able)523 4327 y(problems)27 b(p)r(ossess)g(most)g(general)g |
|
3609 (uni\014ers.)g(Moreo)n(v)n(er,)e(the)k(nominal)e(uni\014cation)h(algo-) |
|
3610 523 4426 y(rithm)k(pro)r(duces)g(uni\014ers)g(that)g(mak)n(e)g(t)n(w)n |
|
3611 (o)f(terms)h(equal)g(mo)r(dulo)g Fi(\013)p Fq(-equiv)-5 |
|
3612 b(alence.)32 b(F)-7 b(or)523 4526 y(example)30 b(if)g(w)n(e)g(unify)h |
|
3613 Ff(\(lam)42 b(x.M\))p Fq(,)28 b(where)i Ff(M)g Fq(is)f(a)h(logical)f(v) |
|
3614 -5 b(ariable,)29 b(with)i(an)n(y)e(term)h(of)523 4625 |
|
3615 y(the)i(form)e Ff(\(lam)42 b(y.t\))p Fq(|)p Ff(t)29 b |
|
3616 Fq(b)r(eing)j(the)f(represen)n(tation)f(of)h(an)f(arbitrary)f |
|
3617 Fi(\025)p Fq(-term|then)523 4725 y Ff(M)f Fq(will)g(b)r(e)g(instan)n |
|
3618 (tiated)g(suc)n(h)g(that)g(all)g(o)r(ccurrences)f(b)r(ound)h(b)n(y)g |
|
3619 Ff(y)g Fq(in)g Ff(t)g Fq(are)f(b)r(ound)h(in)g Ff(M)523 |
|
3620 4825 y Fq(b)n(y)d Ff(x)p Fq(.)g(This)g(is)h(ac)n(hiev)n(ed)e(in)h(the)h |
|
3621 (nominal)f(uni\014cation)g(algorithm)g(b)n(y)g(t)n(w)n(o)f(means:)h |
|
3622 (First,)523 4924 y(b)r(ound)k(names)e(are)h(renamed)f(via)h(a)g(sw)n |
|
3623 (apping)f(op)r(eration.)g(F)-7 b(or)28 b(instance)g(applying)g(the)p |
|
3624 eop end |
|
3625 %%Page: 3 3 |
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3626 TeXDict begin 3 2 bop 523 448 a Fq(sw)n(apping)27 b(\()p |
|
3627 Ff(x)c(y)o Fq(\))28 b(to)g(the)g(term)1075 614 y Ff(lam)22 |
|
3628 b(x)p Fi(:)14 b Ff(app)22 b(\()o(lam)g(x)p Fi(:)14 b |
|
3629 Ff(var)21 b(x\))o(\(app)h(\()o(var)g(y\))g(\(var)g(z\)\))523 |
|
3630 780 y Fq(giv)n(es)1071 880 y Ff(lam)g(y)p Fi(:)14 b Ff(app)21 |
|
3631 b(\(lam)g(y)p Fi(:)14 b Ff(var)22 b(y\))o(\()o(app)g(\(var)g(x\))g(\()o |
|
3632 (var)g(z\)\))g Fi(:)523 1019 y Fq(Unlik)n(e)28 b(renaming)g(via)g |
|
3633 (substitution,)g(sw)n(appings)g(preserv)n(e)e Fi(\013)p |
|
3634 Fq(-equiv)-5 b(alence.)28 b(F)-7 b(or)28 b(exam-)523 |
|
3635 1119 y(ple,)j(consider)f Ff(lam)22 b(y)p Fi(:)p Ff(var)g(x)31 |
|
3636 b Fq(and)g Ff(lam)21 b(z)p Fi(:)p Ff(var)h(x)o Fq(,)32 |
|
3637 b(whic)n(h)f(are)f Fi(\013)p Fq(-equiv)-5 b(alen)n(t.)31 |
|
3638 b(Sw)n(apping)f Ff(x)523 1219 y Fq(and)i Ff(y)g Fq(results)g(in)g |
|
3639 Ff(lam)22 b(x)p Fi(:)p Ff(var)g(y)32 b Fq(and)g Ff(lam)22 |
|
3640 b(z)p Fi(:)p Ff(var)f(y)p Fq(,)32 b(whic)n(h)g(are)g(still)g |
|
3641 Fi(\013)p Fq(-equiv)-5 b(alen)n(t,)32 b(but)523 1318 |
|
3642 y(na)-9 b(\177)-32 b(\020v)n(ely)22 b(substituting)j |
|
3643 Ff(y)f Fq(for)f Ff(x)h Fq(results)f(in)i(v)-5 b(ariable)23 |
|
3644 b(capture)g(in)i(the)f(\014rst)g(term.)g(Nominal)523 |
|
3645 1418 y(terms)j(ar)g(giv)n(en)g(b)n(y)g(the)h(grammar)1511 |
|
3646 1617 y Fi(t)23 b Fq(::=)f Fi(a)p Fh(j)p Fi(\031)g Fh(\001)d |
|
3647 Fi(X)7 b Fh(j)p Fi(a:t)p Fh(j)p Ff(f)14 b Fi(t)2182 1629 |
|
3648 y Fp(1)2231 1617 y Fi(:)g(:)g(:)g(t)2372 1629 y Fd(n)648 |
|
3649 1757 y Fq(Second,)26 b(uni\014cation)g(ma)n(y)f(generate)g(freshness)h |
|
3650 (constrain)n(ts)f(that)h(ensure)g(that)h(p)r(os-)523 |
|
3651 1856 y(sible)40 b(solutions)f(resp)r(ect)h Fi(\013)p |
|
3652 Fq(-equiv)-5 b(alence.)40 b(F)-7 b(or)39 b(example)h(if)h(one)e(wishes) |
|
3653 h(to)g(solv)n(e)f(the)523 1956 y(uni\014cation)28 b(problem)f |
|
3654 Ff(\(lam)42 b(x.t\))21 b Fh(\031)p Fq(?)i Ff(\(lam)66 |
|
3655 b(x'.t'\))p Fq(,)25 b(then)j(one)g(needs)f(to)g(solv)n(e)1369 |
|
3656 2122 y Ff(t)46 b Fh(\031)p Fq(?)f(\()p Ff(x)23 b(x')p |
|
3657 Fq(\))18 b Fh(\001)h Ff(t')82 b Fq(and)h Ff(x)22 b Fq(#?)i |
|
3658 Ff(t')523 2288 y Fq(where)h(the)i(\014rst)e(problem)g(asks)g(whether)h |
|
3659 Ff(t)f Fq(and)h(\()p Ff(x)d(x')o Fq(\))15 b Fh(\001)g |
|
3660 Ff(t')25 b Fq(unify)-7 b(,)27 b(where)e(\()p Ff(x)e(x')o |
|
3661 Fq(\))15 b Fh(\001)g Ff(t')25 b Fq(is)523 2387 y Ff(t')c |
|
3662 Fq(with)g(all)g(o)r(ccurrences)f(of)h Ff(x)g Fq(and)h |
|
3663 Ff(x')e Fq(sw)n(app)r(ed.)h(The)g(second)g(problem)g(asks)f(whether)h |
|
3664 Ff(x)523 2487 y Fq(is)i(fresh)h(for)f Ff(t')p Fq(,)f(i.e.)i |
|
3665 Ff(t')f Fq(cannot)g(con)n(tain)g(an)n(y)f(free)h(o)r(ccurrence)g(of)g |
|
3666 Ff(x)p Fq(.)g(P)n(osing)f(the)i(queries)517 2636 y(\(a\))41 |
|
3667 b Ff(type)h(nil)g(\(lam)g(x.\(lam)f(x.)i(\(var)f(x\)\)\))g(T)193 |
|
3668 b Fq(and)512 2733 y(\(b\))42 b Ff(type)g(nil)g(\(lam)g(x.\(lam)f(x.)i |
|
3669 (\(app)f(\(var)g(x\))h(\(var)f(x\)\)\))g(T')523 2885 |
|
3670 y Fq(to)20 b Fi(\013)p Fq(Prolog)f(pro)r(duces)g(the)i(unique)g(answ)n |
|
3671 (er)d Ff(T)44 b(->)e(arrow)g(A)h(\(arrow)e(B)i(B\))20 |
|
3672 b Fq(for)g(\(a\),)g(and)523 2985 y(no)27 b(answ)n(er)f(for)i(\(b\).)648 |
|
3673 3085 y(One)19 b(imp)r(ortan)n(t)h(p)r(oin)n(t)h(in)f(using)g(nominal)g |
|
3674 (uni\014cation)g(is)g(that)h(binders)f(ha)n(v)n(e)f(explicit)523 |
|
3675 3184 y(names.)33 b(This)g(has)g(the)h(pleasan)n(t)e(consequence)g(that) |
|
3676 i(manipulations)e(in)n(v)n(olving)g(terms)523 3284 y(with)e(binders)f |
|
3677 (can)h(b)r(e)g(expressed)e(v)n(ery)h(elegan)n(tly)f(in)i |
|
3678 Fi(\013)p Fq(Prolog.)e(Consider)g(for)h(example)523 3383 |
|
3679 y(t)n(w)n(o)k(of)g(the)h(transformations)d(required)h(for)h |
|
3680 (calculating)g(a)g(prenex-normal)e(form)i(of)g(a)523 |
|
3681 3483 y(form)n(ula)800 3649 y Fh(8)p Fi(x:P)e Fh(^)19 |
|
3682 b(8)p Fi(x:Q)k Fq(=)-14 b Fh(\))23 b(8)p Fi(x:)p Fq(\()p |
|
3683 Fi(P)31 b Fh(^)19 b Fi(Q)p Fq(\))83 b(and)g(\()p Fh(8)p |
|
3684 Fi(x:P)12 b Fq(\))19 b Fh(^)g Fi(Q)k Fq(=)-14 b Fh(\))23 |
|
3685 b(8)p Fi(x:)p Fq(\()p Fi(P)31 b Fh(^)19 b Fi(Q)p Fq(\))523 |
|
3686 3815 y(where)30 b(in)g(the)h(second)f(transformation)e |
|
3687 Fi(Q)i Fq(m)n(ust)h(not)f(con)n(tain)g(an)n(y)f(free)h(o)r(ccurrence)f |
|
3688 (of)523 3915 y Fi(x)p Fq(.)f(These)g(transformations)d(can)i(b)r(e)h |
|
3689 (implemen)n(ted)h(as)e(the)h(follo)n(wing)e(clauses.)656 |
|
3690 4062 y Ff(prenex)42 b(\(and)f(\(forall)g(x.P\))h(\(forall)f(x.Q\)\))g |
|
3691 (\(forall)g(x.\(and)g(P)i(Q\)\).)656 4162 y(prenex)f(\(and)f(\(forall)g |
|
3692 (x.P\))h(Q\))h(\(forall)e(x.\(and)g(P)i(Q\)\))f(/)h(x#Q.)523 |
|
3693 4310 y Fq(In)34 b(the)g(\014rst)g(clause,)f(the)h(use)g(of)g(nominal)f |
|
3694 (uni\014cation)h(allo)n(ws,)e(roughly)h(sp)r(eaking,)g(to)523 |
|
3695 4410 y(sync)n(hronize)21 b(the)h(t)n(w)n(o)f(binders.)h(In)g(e\013ect)h |
|
3696 (this)f(clause)f(is)h(applicable)g(for)f(an)n(y)h(term)g(of)g(the)523 |
|
3697 4510 y(form)j Ff(\(and)41 b(\(forall)g(y....\))g(\(forall)g(z....\))p |
|
3698 Fq(.)22 b(What)k(is)e(pleasan)n(t)g(ab)r(out)h Fi(\013)p |
|
3699 Fq(Prolog)523 4609 y(is)31 b(that)h(in)f(this)h(clause)f(no)g(explicit) |
|
3700 g(renaming)g(of)g(the)h(binders)f(is)g(required:)f(it)i(will)g(b)r(e) |
|
3701 523 4709 y(done)g(implicitly)h(b)n(y)f(the)h(sw)n(apping)e(op)r |
|
3702 (eration)g(of)h(the)h(uni\014cation)f(algorithm.)f(In)i(the)523 |
|
3703 4809 y(second)f(clause,)h(the)g(freshness)f(constrain)n(t)g |
|
3704 Ff(x#Q)f Fq(mak)n(es)h(sure)g(that)h(the)h(side-condition)523 |
|
3705 4908 y(placed)27 b(up)r(on)h(the)g(second)f(transformation)f(holds.)p |
|
3706 eop end |
|
3707 %%Page: 4 4 |
|
3708 TeXDict begin 4 3 bop 648 448 a Fq(Another)27 b(example)h(is)g(giv)n |
|
3709 (en)f(b)r(elo)n(w.)h(It)g(implemen)n(ts)h(the)f(capture-a)n(v)n(oiding) |
|
3710 d(substi-)523 548 y(tution)j(of)g(a)f(v)-5 b(ariable)26 |
|
3711 b(in)i(a)f Fi(\025)p Fq(-term.)1683 518 y Fp(3)656 692 |
|
3712 y Ff(id)43 b(X)g(X.)656 791 y(subst)f(\(var)g(X\))h(X)g(T)g(T.)656 |
|
3713 891 y(subst)f(\(var)g(Y\))h(X)g(T)g(\(var)f(Y\))h(:-)f(not\(id)g(X)h |
|
3714 (Y\).)656 991 y(subst)f(\(app)g(M)h(N\))g(X)g(T)g(\(app)f(M')h(N'\))f |
|
3715 (:-)h(subst)e(M)j(X)f(T)g(M',)2182 1090 y(subst)e(N)j(X)f(T)g(N'.)656 |
|
3716 1190 y(subst)f(\(lam)g(y.M\))g(X)h(T)g(\(lam)f(y.M'\))g(/)h(y#T,)f(y#X) |
|
3717 g(:-)h(subst)e(M)i(X)h(T)f(M'.)523 1352 y Fq(Giv)n(en)28 |
|
3718 b(this)h(program,)e(p)r(osing)h(the)h(query)g Ff(subst)42 |
|
3719 b(\(lam)f(\(x.)i(var)f(y\)\))g(y)i(\(var)e(x\))g(R)523 |
|
3720 1451 y Fq(yields)29 b(the)h(result)f Ff(R)43 b(->)g(lam)f(x43.var)f(x) |
|
3721 29 b Fq(\(the)g(index)h Ff(43)e Fq(arises)g(during)h(pro)r(of)g(searc)n |
|
3722 (h)523 1551 y(where)e(clauses)g(are)f(`freshened';)i(see)f(b)r(elo)n |
|
3723 (w\).)523 1816 y Fj(2)112 b(Syn)m(tax)523 2015 y Fq(Figure)29 |
|
3724 b(1)h(summarizes)e(the)i(abstract)f(syn)n(tax)g(of)h |
|
3725 Fi(\013)p Fq(Prolog.)e(In)i Fi(\013)p Fq(Prolog,)d(the)k(syn)n(tactic) |
|
3726 523 2114 y(en)n(tities)23 b(that)g(can)f(b)r(e)h(b)r(ound)g(and)f(sw)n |
|
3727 (app)r(ed)g(are)g(called)g Fe(atoms)p Fq(,)h(as)f(opp)r(osed)g(to)h |
|
3728 (\(logical\))523 2214 y Fe(variables)p Fq(,)30 b(whic)n(h)d(can)h(b)r |
|
3729 (e)g(instan)n(tiated)f(with)h(other)f(terms)g(via)g(substitution.)648 |
|
3730 2314 y(The)18 b(terms)h(include)g(constan)n(ts,)e(function)j |
|
3731 (applications,)e(tuples,)h(atoms)f Fi(a)g Fq(and)h(atom-)523 |
|
3732 2413 y(abstractions)j Fi(a:t)p Fq(.)i(Goal)e(form)n(ulae)h(include)h |
|
3733 Fh(>)f Fq(\(true\),)h(conjunctions,)f(disjunctions,)h(and)523 |
|
3734 2513 y(atomic)i(prop)r(ositions)f(\(i.e.)i(predicates)f(together)f |
|
3735 (with)i(a)f(term\).)h(The)g(program)d(clauses)523 2612 |
|
3736 y(are)31 b(of)g(the)i(form)e Fi(p)f(t)f(=)h Fh(r)g Fq(:)p |
|
3737 Fh(\000)g Fi(G)p Fq(,)i(where)f Fi(G)h Fq(is)g(a)f(goal,)f |
|
3738 Fi(p)g(t)i Fq(is)f(an)h(atomic)f(prop)r(osition,)523 |
|
3739 2712 y(and)f Fh(r)g Fq(is)g(a)g(set)g(of)g Fe(fr)l(eshness)j(c)l(onstr) |
|
3740 l(aints)j Fi(a)28 b Fq(#)f Fi(X)36 b Fq(whic)n(h)31 b(m)n(ust)f(b)r(e)g |
|
3741 (satis\014ed)g(in)g(order)523 2812 y(for)d(the)g(rule)g(to)f(apply)-7 |
|
3742 b(.)27 b(If)h Fh(r)f Fq(is)g(empt)n(y)-7 b(,)27 b(w)n(e)g(omit)g(it)g |
|
3743 (and)g(the)h Fi(=)e Fq(sym)n(b)r(ol;)h(similarly)-7 b(,)26 |
|
3744 b(if)i Fi(G)523 2911 y Fq(is)g Fh(>)p Fq(,)f(w)n(e)g(omit)h(it)g(and)f |
|
3745 (the)h(:)p Fh(\000)g Fq(sym)n(b)r(ol.)648 3011 y(As)39 |
|
3746 b(usual,)g(clauses)g(are)f(implicitly)i(univ)n(ersally)e(quan)n |
|
3747 (ti\014ed)h(o)n(v)n(er)f(all)h(free)g(logical)523 3111 |
|
3748 y(v)-5 b(ariables)30 b Ff(X)p Fq(.)h(V)-7 b(ariables)31 |
|
3749 b(can)g(b)r(e)h(used)f(in)h(place)f(of)g(atom)g(o)r(ccurrences)f(but)i |
|
3750 (not)g(in)g(ab-)523 3210 y(stractions:)20 b(i.e.,)i Ff(var)43 |
|
3751 b(X)21 b Fq(is)g(a)h(legal)e(term)i(whereas)e Ff(lam)43 |
|
3752 b(X.\(var)e(X\))21 b Fq(is)g(not.)h(Instead,)f(one)523 |
|
3753 3310 y(needs)e(to)h(write)f(in)g Fi(\013)p Fq(Prolog)f |
|
3754 Ff(lam)42 b(x.\(var)f(x\))p Fq(,)19 b(whic)n(h)g(will)h(unify)g(with)g |
|
3755 (an)n(y)e Fi(\013)p Fq(-equiv)-5 b(alen)n(t)523 3409 |
|
3756 y Fi(\025)p Fq(-term.)523 3675 y Fj(3)112 b(Pro)s(of)37 |
|
3757 b(searc)m(h)523 3873 y Fq(The)j(main)f(di\013erences)h(b)r(et)n(w)n |
|
3758 (een)f Fi(\013)p Fq(Prolog)f(and)i(standard)e(logic)h(programming)f |
|
3759 (lan-)523 3973 y(guages)25 b(are)h(\(1\))h(it)g(uses)f(nominal)h |
|
3760 (uni\014cation)f(to)h(decide)g(whether)f(a)g(goal)g(and)g(a)h(clause) |
|
3761 523 4072 y(head)c(are)f(uni\014able,)h(\(2\))g(it)h(in)n(terprets)e |
|
3762 (freshness)g(constrain)n(ts)g(in)h(clauses)f(as)g(obligations)523 |
|
3763 4172 y(to)i(generate)e(fresh)i(atoms,)f(and)g(\(3\))h(it)g(answ)n(ers)e |
|
3764 (queries)h(with)h(a)g(substitution)g(plus)g(a)f(set)523 |
|
3765 4272 y(of)30 b(freshness)f(constrain)n(ts)f(on)h(some)g(v)-5 |
|
3766 b(ariables)29 b(in)h(the)g(substitution.)g(W)-7 b(e)30 |
|
3767 b(shall)g(presen)n(t)523 4371 y(a)d(nondeterministic)h(transition)f |
|
3768 (seman)n(tics)g(of)g(pro)r(of)g(searc)n(h)f(in)i Fi(\013)p |
|
3769 Fq(Prolog.)648 4471 y(A)21 b(program)f(state)h(is)g(a)g(tuple)h(\()p |
|
3770 Fi(\011)5 b(;)14 b(G;)g Fh(r)p Fi(;)g(\022)r Fq(\),)22 |
|
3771 b(where)f Fi(\011)30 b Fq(is)21 b(a)g(set)h(of)f(program)e(clauses,)i |
|
3772 Fi(G)523 4571 y Fq(a)j(goal,)g Fh(r)h Fq(a)f(set)h(of)g(freshness)f |
|
3773 (constrain)n(ts,)f(and)i Fi(\022)i Fq(a)d(substitution.)i(The)e(result) |
|
3774 h(of)g(pro)r(of)523 4670 y(searc)n(h)i(is)i(a)f(pair)g |
|
3775 Fh(h)p Fi(\022)r(;)14 b Fh(ri)p Fq(,)30 b(where)e Fi(\022)j |
|
3776 Fq(is)d(a)g(substitution)i(instan)n(tiating)e(some)g(v)-5 |
|
3777 b(ariables)27 b(of)523 4770 y Fi(G)h Fq(to)f(terms,)h(and)f |
|
3778 Fh(r)h Fq(is)g(a)f(set)g(of)h(freshness)f(constrain)n(ts)f(of)i(the)g |
|
3779 (form)f Fi(a)c Fq(#)g Fi(X)7 b Fq(.)p 523 4839 473 4 |
|
3780 v 546 4893 a Fo(3)606 4924 y Fm(not)26 b Fn(is)h(the)e(usual)h |
|
3781 (negation-b)n(y-failure.)p eop end |
|
3782 %%Page: 5 5 |
|
3783 TeXDict begin 5 4 bop 908 429 a Fn(Nominal)26 b(terms:)226 |
|
3784 b Fk(t)23 b Fn(::=)g Fk(c)f Fc(j)f Fk(f)30 b(t)2020 437 |
|
3785 y Fo(1)2067 429 y Fc(\001)13 b(\001)h(\001)f Fk(t)2198 |
|
3786 437 y Fb(n)2261 429 y Fc(j)22 b Fn(\()p Fk(t)2362 437 |
|
3787 y Fo(1)2396 429 y Fk(;)13 b(:::;)h(t)2556 437 y Fb(n)2599 |
|
3788 429 y Fn(\))21 b Fc(j)g Fk(X)28 b Fc(j)22 b Fk(a)f Fc(j)g |
|
3789 Fk(a:t)908 521 y Fn(Goal)27 b(form)n(ulae:)217 b Fk(G)24 |
|
3790 b Fn(::=)f Fc(>)e(j)h Fk(G)17 b Fc(^)g Fk(G)22 b Fc(j)g |
|
3791 Fk(G)17 b Fc(_)g Fk(G)k Fc(j)h Fk(p)f(t)908 612 y Fn(Program)27 |
|
3792 b(clauses:)143 b Fk(C)29 b Fn(::=)23 b Fk(p)e(t)g(=)h |
|
3793 Fc(r)f Fn(:)q Fc(\000)g Fk(G)1410 842 y Fl(Fig.)14 b(1.)25 |
|
3794 b Fk(\013)p Fn(Prolog)j(syn)n(tax)d(summary)648 1099 |
|
3795 y Fq(If)38 b Fi(G)f Fq(is)h(a)f(non-atomic)g(form)n(ula,)f(then)i(pro)r |
|
3796 (of)f(searc)n(h)f(is)i(relativ)n(ely)e(standard,)h(as)523 |
|
3797 1199 y(sho)n(wn)27 b(b)r(elo)n(w:)558 1329 y(1.)41 b(\()p |
|
3798 Fi(\011)5 b(;)14 b Fh(>)p Fi(;)g(\022)r(;)g Fh(r)p Fq(\))28 |
|
3799 b(succeeds)f(with)h(answ)n(er)e Fh(h)p Fi(\022)r(;)14 |
|
3800 b Fh(ri)p Fq(.)558 1421 y(2.)41 b(\()p Fi(\011)5 b(;)14 |
|
3801 b(G)854 1433 y Fp(1)898 1421 y Fh(^)7 b Fi(G)1025 1433 |
|
3802 y Fp(2)1064 1421 y Fi(;)14 b(\022)r(;)g Fh(r)p Fq(\))22 |
|
3803 b(steps)g(to)g(\()p Fi(\011)5 b(;)14 b(\022)1767 1391 |
|
3804 y Fg(0)1790 1421 y Fi(G)1855 1433 y Fp(2)1893 1421 y |
|
3805 Fi(;)g(\022)1971 1391 y Fg(0)2001 1421 y Fh(\016)7 b |
|
3806 Fi(\022)r(;)14 b Fh(r)2197 1391 y Fg(0)2221 1421 y Fq(\),)22 |
|
3807 b(pro)n(vided)f(\()p Fi(\011)5 b(;)14 b(G)2824 1433 y |
|
3808 Fp(1)2861 1421 y Fi(;)g(\022)r(;)g Fh(r)p Fq(\))23 b(succeeds)664 |
|
3809 1521 y(with)28 b(answ)n(er)e Fh(h)p Fi(\022)1201 1491 |
|
3810 y Fg(0)1225 1521 y Fi(;)14 b Fh(r)1331 1491 y Fg(0)1355 |
|
3811 1521 y Fh(i)p Fq(.)558 1613 y(3.)41 b(\()p Fi(\011)5 |
|
3812 b(;)14 b(G)854 1625 y Fp(1)910 1613 y Fh(_)k Fi(G)1048 |
|
3813 1625 y Fp(2)1086 1613 y Fi(;)c(\022)r(;)g Fh(r)p Fq(\))28 |
|
3814 b(steps)g(to)f(\()p Fi(\011)5 b(;)14 b(G)1830 1625 y |
|
3815 Fd(i)1858 1613 y Fi(;)g(\022)r(;)g Fh(r)p Fq(\))28 b(for)f |
|
3816 Fi(i)c Fh(2)g(f)p Fq(1)p Fi(;)14 b Fq(2)p Fh(g)p Fq(.)523 |
|
3817 1750 y(Otherwise,)27 b(if)h(the)g(the)g(goal)e(form)n(ula)h(is)g |
|
3818 (atomic)g(\(that)i(is)e(of)h(the)g(form)f Fi(p)c(t)p |
|
3819 Fq(\),)28 b(then:)558 1880 y(1.)41 b(Let)28 b Fi(p)23 |
|
3820 b(t)908 1850 y Fg(0)954 1880 y Fi(=)g Fh(r)1088 1850 |
|
3821 y Fg(0)1134 1880 y Fq(:)p Fh(\000)g Fi(G)1310 1850 y |
|
3822 Fg(0)1361 1880 y Fq(b)r(e)28 b(an)f(elemen)n(t)h(of)g |
|
3823 Fi(\011)9 b Fq(.)558 1972 y(2.)41 b(Let)33 b Fi(p)f(t)922 |
|
3824 1942 y Fg(\003)991 1972 y Fi(=)g Fh(r)1134 1942 y Fg(\003)1204 |
|
3825 1972 y Fq(:)p Fh(\000)f Fi(G)1388 1942 y Fg(\003)1459 |
|
3826 1972 y Fq(b)r(e)i(the)h(`freshened')e(v)n(ersion)f(of)i |
|
3827 Fi(p)f(t)2635 1942 y Fg(0)2690 1972 y Fi(=)f Fh(r)2832 |
|
3828 1942 y Fg(0)2887 1972 y Fq(:)p Fh(\000)g Fi(G)3071 1942 |
|
3829 y Fg(0)3095 1972 y Fq(,)i(i.e.)g(the)664 2072 y(clause)c(where)f(all)h |
|
3830 (atoms)f(men)n(tioned)h(in)g Fh(r)2087 2042 y Fg(0)2140 |
|
3831 2072 y Fq(and)g(all)f(v)-5 b(ariables)28 b(are)g(renamed)g(with)664 |
|
3832 2171 y(new)g(names)f(\(the)h(latter)f(is)h(standard)f(in)g(logic)g |
|
3833 (programming\))558 2264 y(3.)41 b(If)28 b Fi(t)777 2234 |
|
3834 y Fg(\003)842 2264 y Fq(uni\014es)f(with)h Fi(t)f Fq(sub)5 |
|
3835 b(ject)28 b(to)f(the)g(constrain)n(ts)f Fh(r)2370 2234 |
|
3836 y Fg(\003)2408 2264 y Fq(,)h(resulting)g(in)g Fh(h)p |
|
3837 Fi(\022)2969 2234 y Fg(00)3012 2264 y Fi(;)14 b Fh(r)3118 |
|
3838 2234 y Fg(00)3161 2264 y Fh(i)p Fq(,)27 b(then)664 2363 |
|
3839 y(\()p Fi(\011)5 b(;)14 b(p)23 b(t;)14 b(\022)r(;)g Fh(r)p |
|
3840 Fq(\))28 b(steps)f(to)h(\()p Fi(\011)5 b(;)14 b(\022)1604 |
|
3841 2333 y Fg(00)1646 2363 y Fi(G)1711 2333 y Fg(\003)1749 |
|
3842 2363 y Fi(;)g(\022)1827 2333 y Fg(00)1888 2363 y Fh(\016)k |
|
3843 Fi(\022)r(;)c Fh(r)2095 2333 y Fg(00)2138 2363 y Fq(\).)523 |
|
3844 2500 y(That)33 b(is,)g(to)f(solv)n(e)g(an)h(atomic)f(goal)g(b)n(y)g |
|
3845 (resolving)f(it)j(with)f(a)g(clause,)f(w)n(e)g(\014rst)h(replace)523 |
|
3846 2600 y(atoms)24 b(men)n(tioned)g(in)g Fh(r)h Fq(with)f(fresh)g(atoms,)g |
|
3847 (then)h(unify)f(the)h(goal)e(and)h(the)h(head)e(of)i(the)523 |
|
3848 2699 y(clause,)g(while)g(also)g(c)n(hec)n(king)f(that)h(all)g(the)h |
|
3849 (constrain)n(ts)e(are)g(satis\014ed.)h(Uni\014cation)h(ma)n(y)523 |
|
3850 2799 y(pro)r(duce)f(new)h(constrain)n(ts)f(that)h(cannot)f(b)r(e)h |
|
3851 (resolv)n(ed)f(y)n(et;)g(if)i(so,)e(these)h(constrain)n(ts)e(are)523 |
|
3852 2899 y(propagated)i(through)h(pro)r(of)g(searc)n(h)f(un)n(til)i(they)g |
|
3853 (can)f(b)r(e)h(resolv)n(ed.)523 3135 y Fj(4)112 b(Conclusions)523 |
|
3854 3297 y Fq(W)-7 b(e)38 b(ha)n(v)n(e)f(dev)n(elop)r(ed)g(a)h(pro)r |
|
3855 (of-of-concept)e(implemen)n(tation)i(of)g Fi(\013)p Fq(Prolog)e(and)i |
|
3856 (are)f(in)523 3389 y(the)28 b(pro)r(cess)f(of)h(scaling)f(it)h(up)g(to) |
|
3857 g(realistic)f(examples.)h(There)f(are)g(man)n(y)g(directions)h(for)523 |
|
3858 3480 y(future)i(w)n(ork.)f(F)-7 b(or)30 b(example,)f(the)i(complexit)n |
|
3859 (y)e(of)h(nominal)g(uni\014cation)g(remains)f(to)h(b)r(e)523 |
|
3860 3571 y(in)n(v)n(estigated.)22 b(On)i(the)g(theoretical)e(side,)i(w)n(e) |
|
3861 f(in)n(tend)h(to)f(relate)g(pro)r(of)g(searc)n(h)f(in)i |
|
3862 Fi(\013)p Fq(Prolog)523 3662 y(to)k(a)f(suitable)g(notion)g(of)h |
|
3863 (uniform)g(pro)r(ofs)e(in)i(Nominal)g(Logic)e([2].)523 |
|
3864 3754 y Fl(Ac)n(kno)n(wledgmen)n(ts:)f Fn(Dominik)i(W)-6 |
|
3865 b(ee)28 b(implemen)n(ted)f(a)h(preliminary)g(v)n(ersion)g(of)h |
|
3866 Fk(\013)p Fn(Prolog)g(as)523 3845 y(a)d(thesis)g(pro)t(ject.)523 |
|
3867 4082 y Fj(References)523 4236 y Fn(1.)42 b(G.)25 b(Nadath)n(ur)e(and)g |
|
3868 (D.)h(Miller.)33 b(Higher-order)24 b(logic)h(programming.)33 |
|
3869 b(In)23 b Fa(Handb)l(o)l(ok)28 b(of)d(L)l(o)l(gic)624 |
|
3870 4328 y(in)31 b(A)n(rti\014cial)f(Intel)t(ligenc)l(e)i(and)f(L)l(o)l |
|
3871 (gic)g(Pr)l(o)l(gr)l(amming)p Fn(,)g(v)n(olume)e(5,)h(c)n(hapter)e(8,)i |
|
3872 (pages)g(499{)624 4419 y(590.)d(Oxford)f(Univ)n(ersit)n(y)e(Press,)j |
|
3873 (1998.)523 4503 y(2.)42 b(A.)26 b(M.)h(Pitts.)36 b(Nominal)26 |
|
3874 b(logic:)i(A)e(\014rst)g(order)g(theory)g(of)g(names)h(and)e(binding.) |
|
3875 36 b(In)25 b Fa(Pr)l(o)l(c.)j(of)624 4594 y(the)36 b(International)f |
|
3876 (Symp)l(osium)g(T)-6 b(A)n(CS)35 b(2001,)f(Sendai,)h(Jap)l(an)p |
|
3877 Fn(,)f(v)n(olume)g(2215)g(of)g Fa(LNCS)p Fn(,)624 4686 |
|
3878 y(pages)27 b(219{242.)h(Springer-V)-6 b(erlag,)26 b(2001.)523 |
|
3879 4770 y(3.)42 b(C.)50 b(Urban,)e(A.)g(M.)i(Pitts,)f(and)g(M.)g(Gabba)n |
|
3880 (y)-6 b(.)102 b(Nominal)50 b(uni\014cation.)103 b(Submitted.)624 |
|
3881 4861 y Fm(http://www.cl.cam.ac.uk/~cu2)q(00/U)q(nific)q(atio)q(n/)p |
|
3882 Fn(.)p eop end |
|
3883 %%Trailer |
|
3884 |
|
3885 userdict /end-hook known{end-hook}if |
|
3886 %%EOF |