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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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11 dict begin
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end readonly def
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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718 |
11 dict begin
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%%CreationDate: 1992 Apr 26 10:42:42
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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1100 |
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1102 |
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1110 |
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1177 |
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1178 |
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1179 |
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1180 |
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1181 |
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1216 |
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1219 |
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1220 |
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1221 |
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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
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1233 |
%%CreationDate: 1991 Aug 15 07:21:52
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|
1234 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
|
1235 |
11 dict begin
|
|
|
1236 |
/FontInfo 7 dict dup begin
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1237 |
/version (1.0) readonly def
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1238 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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|
1239 |
/FullName (CMSY7) readonly def
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|
|
1240 |
/FamilyName (Computer Modern) readonly def
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|
|
1241 |
/Weight (Medium) readonly def
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1242 |
/ItalicAngle -14.035 def
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1243 |
/isFixedPitch false def
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1244 |
end readonly def
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1245 |
/FontName /CMSY7 def
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|
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1246 |
/PaintType 0 def
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|
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1247 |
/FontType 1 def
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1248 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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1249 |
/Encoding 256 array
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1250 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
1251 |
dup 0 /.notdef put
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|
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1252 |
readonly def
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|
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1253 |
/FontBBox{-15 -951 1252 782}readonly def
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1254 |
/UniqueID 5000817 def
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1255 |
currentdict end
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1256 |
currentfile eexec
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1257 |
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1258 |
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1259 |
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1260 |
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1261 |
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1262 |
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1263 |
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1264 |
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1265 |
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1266 |
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1267 |
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1268 |
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1269 |
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1270 |
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|
1271 |
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|
1272 |
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1273 |
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1274 |
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1275 |
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1276 |
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1277 |
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|
1278 |
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1279 |
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|
1280 |
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|
|
1281 |
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|
1282 |
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|
1283 |
0000000000000000000000000000000000000000000000000000000000000000
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1284 |
0000000000000000000000000000000000000000000000000000000000000000
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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
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|
|
1292 |
%%EndFont
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|
|
1293 |
%%BeginFont: CMSY10
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|
|
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
|
|
|
1319 |
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1320 |
7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4
|
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1321 |
A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
|
|
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1322 |
E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
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1323 |
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1324 |
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1325 |
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1326 |
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1327 |
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1328 |
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1329 |
4ECC068A324D75B5F696B8688EEB2F17E5ED34CCD6D047A4E3806D000C199D7C
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1330 |
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1331 |
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1332 |
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1333 |
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|
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1334 |
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|
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1335 |
3F9AA7711D13BE9EC2FFB78F531449D3D4448DE9AA7038EF0052A9DC98006CFE
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1336 |
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1337 |
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1338 |
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1339 |
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1340 |
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1341 |
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1342 |
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1343 |
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1344 |
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1345 |
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1346 |
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1347 |
0FD0CF5D3781652B6DC4A9EA476E65AE7D11BC4FB528F7E24560BE8377F32F91
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1348 |
8A14BB90DE175FB40EB88549029951B44C1C4BDED68BDA35B0793A546AF53FCF
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1349 |
1A5180DEBEE45F1B7A0E54568A31BE62DAC40AFD4042072476EB3F9ED8043C4F
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1350 |
3F96D165441BB9CF646D1B8875C2547EB27CB618742BCFB379B038F7CE4A86AF
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1351 |
FE18C7EC138C478F53929724A408BF2C30D48DB5D68EA999A20B6F5AA9B49D78
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1352 |
00D973B90C69545DA23A554E54F07572455B1B2483F3FF305E1C6F95B5FF05D7
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1353 |
44E9742BB5E9C40962CD9D3D2FE6AA5BBFD4853B3BF0991E08DC1693E3CA0525
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1354 |
C742885310E33E727078304758C2EEDBD38B1651DD05EE33143BECC40D7BC362
|
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1355 |
FA9319D4B67F42754DA6885ED28A15FACF03DAFD59EBA937522FB4864F95D612
|
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1356 |
32561B4115563AEE927E1B039139C9FB192F76EBC23D0EF62D235BBB7409C0AD
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1357 |
0AF625D26D463013F5B1A94C2000AB634928A55FFEE750BEA2702BC3614EFB0E
|
|
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1358 |
39481E5E8F1BC79E1A8834810F006E33143C0BE4D60005893E97210DFF4DA528
|
|
|
1359 |
A34CA7B8D7D8C40A9532146B5635AED752B08F90D56C19377825C036E1C3A290
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1360 |
4C2606F0466629E2F43A66FD3665913B40D2C3F03D10498B62343A35163BFF73
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1361 |
0E5023D054D763A487D39C78C5988FEC3D1920BA8D7604F2837D5C95BFB75556
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1362 |
FA5509B6AEC4476A77DF78781658F25699870AB5062D469F4C1D7D8A113DBB77
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1363 |
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1364 |
9933F68822251EE2DB00FB8EE9EDB778007CF26037BCE2EB771BD8F2020893FB
|
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1365 |
17485A5F750A430CF79F77D0C47987B2007777F33D12F549E07234086F723967
|
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1366 |
032481AE20258AE41B808FCB37F5E217A068BAE19099AAEE344A325F4C4D67B9
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1367 |
7509A4235D3F173009BDC79A5752B7E328F99F4A9BC02FCE4A26BC469BB2327B
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1368 |
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1369 |
3203CB674C871F046425150561B9EEB28D8F132181EA5228FA66F9966AB37D19
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1370 |
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1371 |
ED017B195247C552389448E81CA5AB8EEAD286E2728459070EC065782E40B89F
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1372 |
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|
|
|
1373 |
29330B8F572D244F10A84B7C2E4C879C5E3C4387EE65C434687DDC2BAE653973
|
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|
1374 |
038FB52A585C56A8206F22C1472715E313B28C7112B53B73723A55DAEB8352DE
|
|
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1375 |
3D135D1CBC3781E1494F905F497162E34015907BA47E6550F2F9ACEAD52C3B77
|
|
|
1376 |
969997FFB3037447644F508597AF274E079300CCCE0737B9DCDC5E54AA02FE5F
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|
1377 |
1837C97D6A68921211A04894D6EE2CB149B218CB23B619007C1EA452DF40B751
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|
|
1378 |
F1509CA2E103B905BB7B5CF26F86737A21A54DDE2FF6F41498BF3113D4448C46
|
|
|
1379 |
7CB05F5A800F33246ED6084D0A9F62B19792E822718E0D54C9E88A43C64F457C
|
|
|
1380 |
98B795F029E7507ED6C3201B477B59A430F0036D527AC9A4C1706C9B0B2AB4C6
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|
|
1381 |
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|
1382 |
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|
1383 |
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1384 |
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|
|
|
1385 |
BE39BDDC2DED6DC02F80ED35C03EF6BDA93E309A3975F76006F5D49066A5DA80
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|
|
1386 |
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|
|
|
1387 |
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|
|
|
1388 |
30FB74489499D209A0530B8BAC6C2F33ACCA476F00B003728E2C66F5697F820A
|
|
|
1389 |
AE3B96F1680DEEA425511CC41BA4ABFD91980D2C26453B353FE20859A0B53D1A
|
|
|
1390 |
26D24FFDCBC9639F6EB3789273B157B98346CD771BE046E96CCFDCE70D2557AC
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|
|
1391 |
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|
|
|
1392 |
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|
1393 |
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|
|
1394 |
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|
|
|
1395 |
318DFA42F0F06F29E47AAE43972F01A9367E4DD036E2F0B91780F4094461B94D
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|
|
1396 |
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|
|
|
1397 |
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|
|
|
1398 |
3BEC5E5B56CAA7E790AEAC98B1862D2B0367A45D98A73412E4E2EAA4934B50DC
|
|
|
1399 |
F8A757A72D7FAD879FA7F60891F58EF35B98DA961C86E3B93AE3F2D172F8F596
|
|
|
1400 |
1F66B1D39CA6B043CC272FC30BC8DF89CB2C719311924683A90C68B687482277
|
|
|
1401 |
7E168678B15334300A40BE616F066E6FCC9D0922D66500ECF8E35E1FDD8C866E
|
|
|
1402 |
37F48E6FCBE68E1198F97BBD8730B49512F8482142072FD0C24A58CBF4AE3872
|
|
|
1403 |
CDB02CBE5121129C60026AB862C3D2D82DFE09C65455EFC8A10E20A040A2A59B
|
|
|
1404 |
399273595AF42EB76991A863CE6C7C355D098134CE8FCE7AB801024BC0A220B6
|
|
|
1405 |
E47BEBEB664940FAF2B70EAB3632B428A73A9767F91481BFB714F7503A76FA15
|
|
|
1406 |
893FFB90DCE8B1DA32ED036CD8B9581BA1841678DDF26947AC573302A159AFBC
|
|
|
1407 |
3BB82ACA1C
|
|
|
1408 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1409 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1410 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1411 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1412 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1413 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1414 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1415 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
1416 |
cleartomark
|
|
|
1417 |
%%EndFont
|
|
|
1418 |
%%BeginFont: CMMI10
|
|
|
1419 |
%!PS-AdobeFont-1.1: CMMI10 1.100
|
|
|
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
|
|
|
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
|
|
|
1441 |
/UniqueID 5087385 def
|
|
|
1442 |
currentdict end
|
|
|
1443 |
currentfile eexec
|
|
|
1444 |
D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
|
|
|
1445 |
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|
|
|
1446 |
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|
|
1447 |
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|
|
1448 |
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|
|
1449 |
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|
|
|
1450 |
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|
|
|
1451 |
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|
|
|
1452 |
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|
|
|
1453 |
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|
|
|
1454 |
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|
|
|
1455 |
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|
|
1456 |
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|
|
|
1457 |
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|
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|
1458 |
1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE
|
|
|
1459 |
03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909
|
|
|
1460 |
95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1
|
|
|
1461 |
74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2
|
|
|
1462 |
3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8
|
|
|
1463 |
47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19
|
|
|
1464 |
AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8
|
|
|
1465 |
42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8
|
|
|
1466 |
40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837
|
|
|
1467 |
B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53
|
|
|
1468 |
9560176676BD77D63B504843E6B9DB3EC18FCC62037E5445C91EEF09D5A887EF
|
|
|
1469 |
67857D586AC94874708708F31667D7AACFFCC684D03633EE405D082344402C01
|
|
|
1470 |
52567CE7223C74BD324068E226BD9B5740D56E8510ABD9DD445A29498A0119BD
|
|
|
1471 |
08271B87B9C2EF2028F18D1E235AE8BB547382368BCF6A36D0704E8BA8483FE2
|
|
|
1472 |
B0F8343897E439E077B42C2DE9760948241E311ACE714EE97930804AC81A06C7
|
|
|
1473 |
D149B16B3AB02B03A9B2BF6611E6ECB11F1223471A05CEC0024BFC71562CCBBD
|
|
|
1474 |
33E532D9F2E11C14C7CBBEC889BF98AA6E9797D057DC637DE0D677E1AC98BC9B
|
|
|
1475 |
B1FDAA62DEC5D0A2333D161E0B77984538F706D6A9AC234D8E29192EEBB3050F
|
|
|
1476 |
88463CE2F1C4D811E5530CD32F766E9C56E2550DC2CB435A19CF404219136A01
|
|
|
1477 |
12D13C0D1018CE1E71A4B33293F7D98B90D70E302AD5055DE1073DF73D754E93
|
|
|
1478 |
9AD30D0466071185A193D4BD7C09A1A924EF6ADC9564CD8CA20762E8C4D60FE0
|
|
|
1479 |
91BC901CB110EA2FA01FC9FD66FDC67A1D15E6792C8E891E21A19FCAE34D3EC0
|
|
|
1480 |
794458B4DFBF90DDA23B09CBFF957F1829843FD35BDE83518770D898DAF09D16
|
|
|
1481 |
9560538664BBC25EF4A8E70A9F87D2F37279B03A0B5A574ACA9F10D56A39F80F
|
|
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1805 |
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1806 |
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1807 |
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1808 |
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1809 |
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1810 |
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1811 |
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1812 |
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1813 |
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1814 |
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1815 |
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1816 |
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1817 |
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1818 |
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1819 |
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1820 |
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1821 |
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1822 |
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1823 |
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1824 |
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1825 |
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1826 |
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1827 |
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1828 |
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1829 |
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1830 |
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1831 |
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1832 |
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1833 |
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1834 |
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1835 |
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1836 |
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1837 |
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1838 |
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1839 |
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1840 |
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1841 |
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1842 |
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1843 |
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1844 |
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1845 |
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1846 |
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1847 |
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1848 |
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1849 |
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1850 |
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1851 |
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1852 |
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1853 |
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1854 |
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1855 |
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1856 |
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1857 |
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|
1858 |
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|
|
1859 |
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|
|
1860 |
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|
1861 |
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|
1862 |
0000000000000000000000000000000000000000000000000000000000000000
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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
|
|
|
1871 |
%%EndFont
|
|
|
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
|
|
|
1896 |
currentdict end
|
|
|
1897 |
currentfile eexec
|
|
|
1898 |
D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
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|
|
1899 |
3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
|
|
|
1900 |
532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
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|
|
1901 |
B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
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|
|
1902 |
986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE
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|
|
1903 |
D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958
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|
|
1904 |
9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935
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|
|
1905 |
86F9CFDB57732045913A6422AD4206418610C81D882EE493DE9523CC1BFE1505
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1906 |
DD1390B19BC1947A01B93BC668BE9B2A0E69A968554239B88C00AF9FBDF09CCD
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|
|
1907 |
67D3B2094C11A04762FE8CC1E91D020A28B3C122D24BEAACF82313F4604F2FEF
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|
1908 |
6E176D730A879BE45DD0D4996EF0247AEB1CA0AB08FF374D99F06D47B36F9554
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|
1909 |
FAD9A2D3CE451B7791C3709D8A1DDDEFBD840C1B42AB824D5A0DFF0E0F15B0B7
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|
1910 |
22AEEB877FF489581DA6FA8DA64944555101EB16F7AB0B717E148B7B98D8DBFD
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|
|
1911 |
730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F
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|
|
1912 |
449963D0AD2FAFDE33BA3D77BCA9D1DF878DDAFCA2E22CC4BACD542B282164C7
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|
|
1913 |
97C2BDE318AF9D501CA21F6E662E7AAB75A5F24D2C182E598D175D44E88AB19A
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|
|
1914 |
E7CD59584F95B389183EE21B525BF52A3F23C0FE5383A5565A19361D716F508C
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|
|
1915 |
AAB78411CA5A4D27552CC1C435760D5A89D535B71C593E755C616661363308DA
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|
|
1916 |
A683F54ED0C23FB2C225A008392B0B719F66F11A946A090B7C00B662A3C69599
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|
|
1917 |
B4ECB0CC70C85C4BBBF207E0026F6C7A19F2ACFB7A60804FC98A4BFFD7BFFF2B
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|
|
1918 |
9529E6D9D4238002BBC255BC62959D6F3381FE06E0621B879D5FE5B541D45A1E
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|
|
1919 |
759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3
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|
|
1920 |
5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2
|
|
|
1921 |
BAC23C984D506D8F7D614BCB6B244F5F41321549BB0BD041FBF3053307168680
|
|
|
1922 |
3435E9C947D629FED7A794EB4593FB10152C6A33840B3AEE2500C40CB2A3E768
|
|
|
1923 |
C1C3314500CAECF86223D90B80EC1B983575E27F081B8CA6D780B3B52AA71F25
|
|
|
1924 |
A7B15DD3559B5B3F752AF0A96154D4EB5B7E4ACED277394B00F1846EBBDF44EF
|
|
|
1925 |
660208EFA8A81C314E51201D486E7019755D9B887CC66D30EBED419D6E051600
|
|
|
1926 |
C0387AF1BD052A30C47B739F554877E27CA50CA54C49FF165B7A58499F0B2188
|
|
|
1927 |
468F0656D849A8EFCD5F25A21E33018CD8CDADBFA5E8E9119304F9266B47972F
|
|
|
1928 |
39320A9F45CB11CACFE85644C10014A6C5F078615D06F4FB86FD07495DBAA493
|
|
|
1929 |
DE737E02D2390DF19D2897E974708DFBDA3C802A1E9407CBA645D5A2AFE34142
|
|
|
1930 |
27EC84D1F43E00A555AEA748BFDD663D1D6CAC5F1C79086AEAE7EF0ECA537F79
|
|
|
1931 |
8B976706C23A57E6F7ECD38C8B1BDE1E6C180F1AD1413F7AE5739674318A0843
|
|
|
1932 |
DF6FDC96B95E9A6DEE658A1137AA3847C674839F630653DFDE99652DC0ED1C68
|
|
|
1933 |
1EF74A588654D4ED9339488D7BF24C38866D1591A6C1E19F675D221B55384C8D
|
|
|
1934 |
61C58385B406A786D4C5CCDF34DCCEF62F49928760E5036237C4BE105B5AF62F
|
|
|
1935 |
14C2C6129B04BA48420DDC47A7C7CD24A49E3A1C596DA213F151AA25CCAB9E4F
|
|
|
1936 |
C6116FA2965428375E54F7EE2BEE31132741E69ED2EC6C5E91037931C6ABE035
|
|
|
1937 |
F225349511B5A044F7A9B776A5B906B974B3B1086958BEACAA21576DA3E01D65
|
|
|
1938 |
A8352C36FC2C1B0C95A36C7F36409B730C5E455B4C9B31CCFB1C89F7D34410EC
|
|
|
1939 |
61C462176E8A126BE83D0D8FE0214B559D03D8F2EAFD1B9A9EFEC0E51C39318F
|
|
|
1940 |
B986612C4D1FCBB5828D6CD3C06662C268813EE181B6A0B3BA674474379979B8
|
|
|
1941 |
846FE4D02953FBEC269885B1BCB9B596751BCF0479233264DEB0D6D101A9848C
|
|
|
1942 |
D3D81DCD773053F92AB20A86E320953AF16CE56C59BC80E06408EAB2CC3D6FEF
|
|
|
1943 |
579CD48E05FB6842F2E77B6E2CB0921307747D350757CC56ADFAE41F096B56B4
|
|
|
1944 |
48DA9EDD036B17A6CD3EA367135774025A66F3EF8F95C7F24F760D20A8F137AA
|
|
|
1945 |
4C841E0B3DC090B372D7F97C2C5C55C354209B7FBB23EE682B3F9D6150978B1F
|
|
|
1946 |
D7BD91C4BA734D860DDEFDD63FC82481B8A44CCFDA0A4979150528CCBC6477AA
|
|
|
1947 |
818E01F8E123743F8A48F8BBFA80560A58A147B9F772AACF16D6724453D4C02F
|
|
|
1948 |
700DE56AE001F7313AFEEDC030C1CFD457C505C2B50EAC8385067A67B4DF08B0
|
|
|
1949 |
6F3991B9F979B0F58D299C74124990DA1CD0029C996ED4067A6B728670FD48E1
|
|
|
1950 |
56FA01AC81D418F3AE636938EA1C65B9D9102CEFA74CD16C85FECC964FD0244B
|
|
|
1951 |
7C041B3B260F0144363D74A8E6FBEE772751AA7CB1D6E646F3E0722E8E64288D
|
|
|
1952 |
9AF535786D64A075035066C7085B5281BBEBA2BFA7D71A5512F98195BD5E15EE
|
|
|
1953 |
E31A616136C7A7411EF9C63AFEBFDBFE7962AF66EA87C6D3FD7E4A8960350CBE
|
|
|
1954 |
46E32368F95FFEB188B9984832A6D830B2BE2FBDB372DD5E3C4C6B91617587A8
|
|
|
1955 |
606CA54BE2B664A26F1F0E1E8A828B8724EBCD17856AC8315BADC731559CD039
|
|
|
1956 |
E8F5D5DCB42B82B370CCE02FE7AADED2287E298364A28868319F9D6A31588831
|
|
|
1957 |
1DCE249D67492816398D3624221663ED2D0A5FBA9172F301E056696CE7D5D251
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|
|
1958 |
FCCA1D1E8F6D383955E3B7C7828DC369A798F09EFD23E8B20618D32ED3B4ECBC
|
|
|
1959 |
D04BCACF543408A075B5304029B7D841C371E14DA00A70E52EFCB83BC5C6C05E
|
|
|
1960 |
349D22A5C7616C6264BDDB389BF53D9E9ED7A0B7CB4219FAD2E25688294D326B
|
|
|
1961 |
6F76CF3C7BEF571CACF5062175FD5A44F00E268717FAB3C65E26B97F312DB8B2
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|
|
1962 |
15ED78A0653C5931DDBE4A4E70313C22381E1B7175CBA606C7591783121C11EE
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|
|
1963 |
2132B342B1DA483553D182414CB1EA846954F8E6EDAB13B4A7C228DFA891EA98
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|
|
1964 |
21C5E26B4C2F96B03A83CB90E5C4BA2DB4FE81EB7E431B97A54692EEDEAC1DB6
|
|
|
1965 |
E7937CBF197D606763F7ED5A04C53325904668FD074E2A00F4145C57ED7792BC
|
|
|
1966 |
A1CC59C884A23F93CD8A48411468B1FBD0CE459CF4E650A4D5586087856953B9
|
|
|
1967 |
A1D7998E5B28986126DDC2EC592E3947D8CE0F9062EBE382793A2B46AFD4DF44
|
|
|
1968 |
23684DFC31FB21BE34A6C28BCE6BEA73CCF5A0E0B2F7B2EAEE59C1BDAAF26BBC
|
|
|
1969 |
90066C9B0D4ACC751DDE5E1E3F9209BE5DA4525188E952F2B12A66D1F3D21750
|
|
|
1970 |
C0945241D15E6187774549E07CF0B527B49DCA270B3AEBD51EC9101A3B71B735
|
|
|
1971 |
7815235211EC6ED743B057BC3F5D98C02387324416D3719FA0E0AD74DB5EA410
|
|
|
1972 |
CDEBE56033E0710FDECBD44CEE5B0509752C2B5C1EAB86D282955F768840B660
|
|
|
1973 |
7DEB472FA6158DCC450FBABEDB556951A27512EEE1409464ED9158F4C84334C1
|
|
|
1974 |
82BD407AD813F25696D290227ACF299E791883B96B7581121B22FDA482CB79A1
|
|
|
1975 |
07C06B7CC39D2BE2644524A0907DE0E83CDE8A0D782E5539CA7E0812F804E563
|
|
|
1976 |
E799B42F58225734723830BE778343D3A85A7F1FFD3B865021B52B14EB352A55
|
|
|
1977 |
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1978 |
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1979 |
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1980 |
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1981 |
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1982 |
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1983 |
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1984 |
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1985 |
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1986 |
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1987 |
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1988 |
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1989 |
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1990 |
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cleartomark
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%%EndFont
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2009 |
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2010 |
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2011 |
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2012 |
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2013 |
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2016 |
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2017 |
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2018 |
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2019 |
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2414 |
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2415 |
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2416 |
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2417 |
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2418 |
0000000000000000000000000000000000000000000000000000000000000000
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2419 |
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2420 |
0000000000000000000000000000000000000000000000000000000000000000
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2421 |
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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
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2428 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
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|
2429 |
11 dict begin
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|
|
2430 |
/FontInfo 7 dict dup begin
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|
|
2431 |
/version (1.0) readonly def
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|
|
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
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|
|
2439 |
/FontName /CMR9 def
|
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2440 |
/PaintType 0 def
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|
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2441 |
/FontType 1 def
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|
|
2442 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
|
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|
2443 |
/Encoding 256 array
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|
|
2444 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
2445 |
dup 0 /.notdef put
|
|
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2446 |
readonly def
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|
|
2447 |
/FontBBox{-39 -250 1036 750}readonly def
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|
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2448 |
/UniqueID 5000792 def
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|
2449 |
currentdict end
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|
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2450 |
currentfile eexec
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2491 |
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2659 |
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2660 |
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2666 |
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2667 |
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2668 |
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2696 |
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2697 |
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2705 |
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2706 |
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2707 |
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2708 |
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2709 |
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2710 |
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2711 |
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2712 |
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2713 |
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2714 |
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2715 |
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2716 |
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2717 |
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2718 |
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2719 |
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2720 |
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2721 |
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2722 |
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2723 |
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2724 |
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2725 |
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2726 |
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2727 |
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2728 |
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2729 |
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2730 |
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2731 |
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2732 |
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2733 |
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2734 |
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2735 |
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2736 |
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2737 |
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2738 |
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|
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|
2739 |
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2740 |
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2741 |
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2742 |
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|
2743 |
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|
|
|
2744 |
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|
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|
2745 |
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|
|
|
2746 |
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|
|
|
2747 |
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|
2748 |
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|
2749 |
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|
2750 |
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|
2751 |
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|
|
|
2752 |
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|
|
|
2753 |
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|
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|
2754 |
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|
|
|
2755 |
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|
|
|
2756 |
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|
|
|
2757 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2758 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2759 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2760 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2761 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2762 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2763 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
2764 |
0000000000000000000000000000000000000000000000000000000000000000
|
|
|
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
|
|
|
2793 |
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|
|
|
2794 |
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|
|
|
2795 |
9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F
|
|
|
2796 |
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|
|
|
2797 |
469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8
|
|
|
2798 |
2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C
|
|
|
2799 |
68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361
|
|
|
2800 |
3645B82392D5CAE11A7CB49D7E2E82DCD485CBA17D1AFFF95F4224CF7ECEE45C
|
|
|
2801 |
BFB7C8C77C22A01C345078D28D3ECBF804CDC2FE5025FA0D05CCC5EFC0C4F87E
|
|
|
2802 |
CBED13DDDF8F34E404F471C6DD2E43331D73E89BBC71E7BF889F6293793FEF5A
|
|
|
2803 |
C9DD3792F032E37A364C70914843F7AA314413D022AE3238730B420A7E9D0CF5
|
|
|
2804 |
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|
|
|
2805 |
23131C09ED104C052054520268A4FA3C6338BA6CF14C3DE3BAF2EA35296EE3D8
|
|
|
2806 |
D6496277E11DFF6076FE64C8A8C3419FA774473D63223FFA41CBAE609C3D976B
|
|
|
2807 |
93DFB4079ADC7C4EF07303F93808DDA9F651F61BCCF79555059A44CBAF84A711
|
|
|
2808 |
6D98083CEF58230D54AD486C74C4A257FC703ACF918219D0A597A5F680B606E4
|
|
|
2809 |
EF94ADF8BF91A5096A806DB64EC96636A98397D22A74932EB7346A9C4B5EE953
|
|
|
2810 |
CB3C80AA634BFC28AA938C704BDA8DC4D13551CCFE2B2784BE8BF54502EBA9AF
|
|
|
2811 |
D49B79237B9C56310550BC30E9108BB06EAC755D6AA4E688EFE2A0AAB17F20FE
|
|
|
2812 |
00CD0BFF1B9CB6BDA0FA3A29A3117388B6686657A150CE6421FD5D420F4F7FB5
|
|
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2813 |
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2931 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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2932 |
11 dict begin
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2933 |
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TeXDict begin 39158280 55380996 1000 600 600 (aprolog.dvi)
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@start /Fa 134[37 3[41 25 31 32 1[39 39 43 63 20 35 1[24
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3406 |
39 35 24 35 39 35 35 39 12[55 43 2[52 1[57 1[48 1[40
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30 57 4[55 1[57 14[39 39 39 1[24 1[24 31[43 12[{
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rf /Fb 145[38 110[{ TeXaae443f0Encoding ReEncodeFont }1
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3[79 58 58 1[43 2[52 60 58 70 3[28 58 60 2[59 55 1[58
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3[85 62 62 60 46 2[57 65 62 76 52 1[43 30 62 65 54 2[60
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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
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|
|
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
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|
|
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
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|
|
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
|
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|
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
|
|
|
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
|