File size: 80,839 Bytes
6fa4bc9 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 | {
"paper_id": "P01-1018",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T09:29:41.014587Z"
},
"title": "Constraints on strong generative power",
"authors": [
{
"first": "David",
"middle": [],
"last": "Chiang",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "University of Pennsylvania",
"location": {
"addrLine": "200 S 33rd St Philadelphia",
"postCode": "19104",
"region": "PA",
"country": "USA"
}
},
"email": "dchiang@cis.upenn.edu"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "We consider the question \"How much strong generative power can be squeezed out of a formal system without increasing its weak generative power?\" and propose some theoretical and practical constraints on this problem. We then introduce a formalism which, under these constraints, maximally squeezes strong generative power out of context-free grammar. Finally, we generalize this result to formalisms beyond CFG.",
"pdf_parse": {
"paper_id": "P01-1018",
"_pdf_hash": "",
"abstract": [
{
"text": "We consider the question \"How much strong generative power can be squeezed out of a formal system without increasing its weak generative power?\" and propose some theoretical and practical constraints on this problem. We then introduce a formalism which, under these constraints, maximally squeezes strong generative power out of context-free grammar. Finally, we generalize this result to formalisms beyond CFG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "\"How much strong generative power can be squeezed out of a formal system without increasing its weak generative power?\" This question, posed by Joshi (2000) , is important for both linguistic description and natural language processing. The extension of tree adjoining grammar (TAG) to tree-local multicomponent TAG , or the extension of context free grammar (CFG) to tree insertion grammar (Schabes and Waters, 1993) or regular form TAG (Rogers, 1994) can be seen as steps toward answering this question. But this question is difficult to answer with much finality unless we pin its terms down more precisely.",
"cite_spans": [
{
"start": 144,
"end": 156,
"text": "Joshi (2000)",
"ref_id": "BIBREF7"
},
{
"start": 391,
"end": 417,
"text": "(Schabes and Waters, 1993)",
"ref_id": "BIBREF11"
},
{
"start": 438,
"end": 452,
"text": "(Rogers, 1994)",
"ref_id": "BIBREF10"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "First, what is meant by strong generative power? In the standard definition (Chomsky, 1965) a grammar G weakly generates a set of sentences L(G) and strongly generates a set of structural descriptions \u03a3(G); the strong generative capacity of a formalism F is then {\u03a3(G) | F provides G}. There is some vagueness in the literature, however, over what structural descriptions are and how they can reasonably be compared across theories (Miller (1999) gives a good synopsis). The approach that Vijay-Shanker et al. (1987) and Weir (1988) take, elaborated on by Becker et al. (1992) , is to identify a very general class of formalisms, which they call linear contextfree rewriting systems (CFRSs), and define for this class a large space of structural descriptions which serves as a common ground in which the strong generative capacities of these formalisms can be compared. Similarly, if we want to talk about squeezing strong generative power out of a formal system, we need to do so in the context of some larger space of structural descriptions.",
"cite_spans": [
{
"start": 76,
"end": 91,
"text": "(Chomsky, 1965)",
"ref_id": "BIBREF4"
},
{
"start": 432,
"end": 446,
"text": "(Miller (1999)",
"ref_id": "BIBREF8"
},
{
"start": 489,
"end": 516,
"text": "Vijay-Shanker et al. (1987)",
"ref_id": "BIBREF12"
},
{
"start": 521,
"end": 532,
"text": "Weir (1988)",
"ref_id": "BIBREF13"
},
{
"start": 556,
"end": 576,
"text": "Becker et al. (1992)",
"ref_id": "BIBREF3"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Second, why is preservation of weak generative power important? If we interpret this constraint to the letter, it is almost vacuous. For example, the class of all tree adjoining grammars which generate context-free languages includes the grammar shown in Figure 1a (which generates the language {a, b} * ). We can also add the tree shown in Figure 1b without increasing the grammar's weak generative capacity; indeed, we can add any trees we please, provided they yield only as and bs. Intuitively, the constraint of weak context-freeness has little force.",
"cite_spans": [],
"ref_spans": [
{
"start": 255,
"end": 264,
"text": "Figure 1a",
"ref_id": "FIGREF0"
},
{
"start": 341,
"end": 350,
"text": "Figure 1b",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "This intuition is verified if we consider that weak context-freeness is desirable for computational efficiency. Though a weakly context-free TAG might be recognizable in cubic time (if we know the equivalent CFG), it need not be parsable in cubic time-that is, given a string, to compute all its possible structural descriptions will take O(n 6 ) time in general. If we are interested in computing structural descriptions from strings, then we need a tighter constraint than preservation of weak generative power.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In Section 3 below we examine some restrictions on tree adjoining grammar which are weakly context-free, and observe that their parsers all work in the same way: though given a TAG G, they implicitly parse using a CFG G which derives the same strings as G, but also their corresponding structural descriptions under G, in such a way that preserves the dynamic-programming structure of the parsing algorithm.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Based on this observation, we replace the constraint of preservation of weak generative power with a constraint of simulability: essentially, a grammar G simulates another grammar G if it generates the same strings that G does, as well as their corresponding structural descriptions under G (see Figure 2 ).",
"cite_spans": [],
"ref_spans": [
{
"start": 296,
"end": 304,
"text": "Figure 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "So then, within the class of context-free rewriting systems, how does this constraint of simulability limit strong generative power? In Section 4.1 we define a formalism called multicomponent multifoot TAG (MMTAG) which, when restricted to a regular form, characterizes precisely those CFRSs which are simulable by a CFG. Thus, in the sense we have set forth, this formalism can be said to squeeze as much strong generative power out of CFG as is possible. Finally, we generalize this result to formalisms beyond CFG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "First we define context-free rewriting systems. What these formalisms have in common is that their derivation sets are all local sets (that is, generable by a CFG). These derivations are taken as structural descriptions. The following definitions are adapted from Weir (1988) .",
"cite_spans": [
{
"start": 264,
"end": 275,
"text": "Weir (1988)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Characterizing structural descriptions",
"sec_num": "2"
},
{
"text": "Definition 1 A generalized context-free grammar G is a tuple V, S , F, P , where 1. V is a finite set of variables, 2. S \u2208 V is a distinguished start symbol, 3. F is a finite set of function symbols, and 4. P is a finite set of productions of the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Characterizing structural descriptions",
"sec_num": "2"
},
{
"text": "X Y X NA a X * d Y NA b Y * c S \u2192 \u03b1(X, Y) \u03b1( x 1 , x 2 , y 1 , y 2 ) = x 1 y 1 y 2 x 2 X \u2192 \u03b2 1 (X) \u03b2 1 ( x 1 , x 2 ) = ax 1 , x 2 d X \u2192 () () = , Y \u2192 \u03b2 2 (Y) \u03b2 2 ( y 1 , y 2 ) = by 1 , y 2 c Y \u2192 () () = ,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Characterizing structural descriptions",
"sec_num": "2"
},
{
"text": "A \u2192 f (A 1 , . . . , A n )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Characterizing structural descriptions",
"sec_num": "2"
},
{
"text": "where n \u2265 0, f \u2208 F, and A, A i \u2208 V. A generalized CFG G generates a set T (G) of terms, which are interpreted as derivations under some formalism. In this paper we require that G be free of spurious ambiguity, that is, that each term be uniquely generated.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Characterizing structural descriptions",
"sec_num": "2"
},
{
"text": "We say that a formalism F is a context-free rewriting system (CFRS) if its derivation sets can be characterized by generalized CFGs, and its derived structures are produced by a function \u2022 F from terms to strings such that for each function symbol f , there is a yield function f F such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 2",
"sec_num": null
},
{
"text": "f (t 1 , . . . , t n ) F = f F ( t 1 F , . . . , t n F )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 2",
"sec_num": null
},
{
"text": "(A linear CFRS is subject to further restrictions, which we do not make use of.)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 2",
"sec_num": null
},
{
"text": "As an example, Figure 3 shows a simple TAG with a corresponding GCFG and interpretation.",
"cite_spans": [],
"ref_spans": [
{
"start": 15,
"end": 23,
"text": "Figure 3",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Definition 2",
"sec_num": null
},
{
"text": "A nice property of CFRS is that any formalism which can be defined as a CFRS immediately lends itself to several extensions, which arise when we give additional interpretations to the function symbols. For example, we can interpret the functions as ranging over probabilities, creating a stochastic grammar; or we can interpret them as yield functions of another grammar, creating a synchronous grammar. Now we define strong generative capacity as the relationship between strings and structural descriptions. 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 2",
"sec_num": null
},
{
"text": "The strong generative capacity of a grammar G a CFRS F is the relation",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "{ t F , t | t \u2208 T (G)}.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "For example, the strong generative capacity of the grammar of Figure 3 is ",
"cite_spans": [],
"ref_spans": [
{
"start": 62,
"end": 70,
"text": "Figure 3",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "{ a m b n c n d m , \u03b1(\u03b2 m 1 ( ()), \u03b2 n 2 ( ())) }",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "{ a m b n c n d m , f m (g n (e())) }",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "That is, in a CFG the n bs and cs must appear later in the derivation than the m as and ds, whereas in our example they appear in parallel.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 3",
"sec_num": null
},
{
"text": "We now take a closer look at some examples of \"squeezed\" context-free formalisms to illustrate how a CFG can be used to simulate formalisms with greater strong generative power than CFG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulating structural descriptions",
"sec_num": "3"
},
{
"text": "Tree substitution grammar (TSG), tree insertion grammar (TIG), and regular-form TAG (RF-TAG) are all weakly context free formalisms which can additionally be parsed in cubic time (with a caveat for RF-TAG below). For each of these formalisms a CKY-style parser can be written whose items are of the form [X, i, j] and are combined in various ways, but always according to the schema",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Motivation",
"sec_num": "3.1"
},
{
"text": "[X, i, j] [Y, j, k] [Z, i, k]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Motivation",
"sec_num": "3.1"
},
{
"text": "just as in the CKY parser for CFG. In effect the parser dynamically converts the TSG, TIG, or RF-TAG into an equivalent CFG-each parser rule of the above form corresponds to the rule schema Z \u2192 XY. More importantly, given a grammar G and a string w, a parser can reconstruct all possible derivations of w under G by storing inside each chart item how that item was inferred. If we think of the parser as dynamically converting G into a CFG G , then this CFG is likewise able to compositionally reconstruct TSG, TIG, or RF-TAG derivations-we say that G simulates G.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Motivation",
"sec_num": "3.1"
},
{
"text": "Note that the parser specifies how to convert G into G , but G is not itself a parser. Thus these three formalisms have a special relationship to CFG that is independent of any particular parsing algorithm: for any TSG, TIG, or RF-TAG G, there is a CFG that simulates G. We make this notion more precise below.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Motivation",
"sec_num": "3.1"
},
{
"text": "Strictly speaking, the recognition algorithm Rogers gives cannot be extended to parsing; that is, it generates all possible derived trees for a given string, but not all possible derivations. It is correct, however, as a parser for a further restricted subclass of TAGs:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Excursus: regular form TAG",
"sec_num": "3.2"
},
{
"text": "Definition 4 We say that a TAG is in strict regular form if there exists some partial ordering over the nonterminal alphabet such that for every auxiliary tree \u03b2, if the root and foot of \u03b2 are labeled X, then for every node \u03b7 along \u03b2's spine where adjunction is allowed, X label(\u03b7), and X = label(\u03b7) only if \u03b7 is a foot node. (In this variant adjunction at foot nodes is permitted.)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Excursus: regular form TAG",
"sec_num": "3.2"
},
{
"text": "Thus the only kinds of adjunction which can occur to unbounded depth are off-spine adjunction and adjunction at foot nodes.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Excursus: regular form TAG",
"sec_num": "3.2"
},
{
"text": "This stricter definition still has greater strong generative capacity than CFG. For example, the TAG in Figure 3 is in strict regular form, because the only nodes along spines where adjunction is allowed are foot nodes.",
"cite_spans": [],
"ref_spans": [
{
"start": 104,
"end": 112,
"text": "Figure 3",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Excursus: regular form TAG",
"sec_num": "3.2"
},
{
"text": "So far we have not placed any restrictions on how these structural descriptions are computed. Even though we might imagine attaching arbitrary functions to the rules of a parser, an algorithm like CKY is only really capable of computing values of bounded size, or else structuresharing in the chart will be lost, increasing the complexity of the algorithm possibly to exponential complexity.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "For a parser to compute arbitrary-sized objects, such as the derivations themselves, it must use back-pointers, references to the values of subcomputations but not the values themselves. The only functions on a back-pointer the parser can compute online are the identity function (by copying the back-pointer) and constant functions (by replacing the back-pointer); any other function would have to dereference the back-pointer and destroy the structure of the algorithm. Therefore such functions must be computed offline.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "Definition 5 A simulating interpretation \u2022 is a bijection between two recognizable sets of terms such that 1. For each function symbol \u03c6, there is a function\u03c6 such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "\u03c6(t 1 , . . . , t n ) =\u03c6( t 1 , . . . , t n )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "2. Each\u03c6 is definable as:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "\u03c6( x 11 , . . . , x 1m 1 ) ), . . . , x n1 , . . . , x mn m ) = w 1 , . . . , w m",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "where each w i can take one of the following forms: (a) a variable x i j , or (b) a function application f (x i 1 j 1 , . . . x i n j n ), n \u2265 0 3. Furthermore, we require that for any recognizable set T , T is also a recognizable set. We say that \u2022 is trivial if every\u03c6 is definable as",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "\u03c6(x 1 , . . . x n ) = f (x \u03c0(1) , . . . x \u03c0(n) )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "where \u03c0 is a permutation of {1, . . . , n}. 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "The rationale for requirement (3) is that it should not be possible, simply by imposing local constraints on the simulating grammar, to produce a simulated grammar which does not even come from a CFRS. 3",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Simulability",
"sec_num": "3.3"
},
{
"text": "We say that a grammar G from a CFRS F is (trivially) simulable by a grammar G' from another CFRS F if there is a (trivial) simulating interpretation",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "\u2022 s : T (G ) \u2192 T (G) which satisfies t F = t s F for all t \u2208 T (G ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "As an example, a CFG which simulates the TAG of Figure 3 is shown in Figure 4 . Note that if we give additional interpretations to the simulated yield functions \u03b1, \u03b2 1 , and \u03b2 2 , this CFG can compute any probabilities, translations, etc., that the original TAG can.",
"cite_spans": [],
"ref_spans": [
{
"start": 48,
"end": 56,
"text": "Figure 3",
"ref_id": "FIGREF2"
},
{
"start": 69,
"end": 77,
"text": "Figure 4",
"ref_id": null
}
],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "Note that if G trivially simulates G, they are very nearly strongly equivalent, except that the yield functions of G might take their arguments in a different order than G, and there might be several yield functions of G which correspond to a single yield function of G used in several different contexts. In fact, for technical reasons we will use this notion instead of strong equivalence for testing the strong generative power of a formal system.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "Thus the original problem, which was, given a formalism F , to find a formalism that has as much strong generative power as possible but remains weakly equivalent to F , is now recast as Figure 4 : CFG which simulates the grammar of Figure 3 . Here we leave the yield functions anonymous; y \u2190 x denotes the function which maps x to y.",
"cite_spans": [],
"ref_spans": [
{
"start": 187,
"end": 195,
"text": "Figure 4",
"ref_id": null
},
{
"start": 233,
"end": 241,
"text": "Figure 3",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "S \u2192 \u03b1 0 \u2022 \u03b1(x 1 , x 2 ) \u2190 x 1 , x 2 \u03b1 0 \u2022 \u2192 \u03b1 0 \u2022 (), x 2 \u2190 \u2212, x 2 \u03b1 0 \u2022 \u2192 \u03b1 1 \u2022 \u2212, x 2 \u2190 \u2212, x 2 \u03b1 1 \u2022 \u2192 \u03b1 1 \u2022 \u2212, () \u2190 \u2212, \u2212 \u03b1 1 \u2022 \u2192 \u2212, \u2212 \u2190 \u2212, \u2212 \u03b1 0 \u2022 \u2192 \u03b2 0 1 [\u03b1 0 ] \u03b2 1 (x 1 ), x 2 \u2190 x 1 , x 2 \u03b2 0 1 [\u03b1 0 ] \u2192 a \u03b2 2 1 [\u03b1 0 ] d x 1 , x 2 \u2190 x 1 , x 2 \u03b2 2 1 [\u03b1 0 ] \u2192 \u03b2 0 1 [\u03b1 0 ] \u03b2 1 (x 1 ), x 2 \u2190 x 1 , x 2 \u03b2 2 1 [\u03b1 0 ] \u2192 \u03b1 0 \u2022 (), x 2 \u2190 \u2212, x 2 \u03b1 1 \u2022 \u2192 \u03b2 0 2 [\u03b1 1 ] \u2212, \u03b2 2 (x 2 ) \u2190 \u2212, x 2 \u03b2 0 2 [\u03b1 1 ] \u2192 b \u03b2 2 2 [\u03b1 1 ] c \u2212, x 2 \u2190 \u2212, x 2 \u03b2 2 2 [\u03b1 1 ] \u2192 \u03b2 1 2 [\u03b1 1 ] \u2212, \u03b2 2 (x 2 ) \u2190 \u2212, x 2 \u03b2 2 2 [\u03b1 1 ] \u2192 \u03b1 1 \u2022 \u2212, () \u2190 \u2212, \u2212",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "the following problem: find a formalism that trivially simulates as many grammars as possible but remains simulable by F .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 6",
"sec_num": null
},
{
"text": "The following is easy to show:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "Proposition 1 Simulability is reflexive and transitive.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "Because of transitivity, it is impossible that a formalism which is simulable by F could simulate a grammar that is not simulable by F . So we are looking for a formalism that can trivially simulate exactly those grammars that F can. In Section 4.1 we define a formalism called multicomponent multifoot TAG (MMTAG), and then in Section 4.2 we prove the following result:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "Proposition 2 A grammar G from a CFRS is simulable by a CFG if and only if it is trivially simulable by an MMTAG in regular form.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "The \"if\" direction (\u21d0) implies (because simulability is reflexive) that RF-MMTAG is simulable by a CFG, and therefore cubic-time parsable. (The proof below does give an effective procedure for constructing a simulating CFG for any RF-MMTAG.) The \"only if\" direction (\u21d2) shows that, in the sense we have defined, RF-MMTAG is the most powerful such formalism.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "We can generalize this result using the notion of a meta-level grammar (Dras, 1999) .",
"cite_spans": [
{
"start": 71,
"end": 83,
"text": "(Dras, 1999)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "Definition 7 If F 1 and F 2 are two CFRSs, F 2 \u2022 F 1 is the CFRS characterized by the interpretation function",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "\u2022 F 2 \u2022F 1 = \u2022 F 2 \u2022 \u2022 F 1 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "F 1 is the meta-level formalism, which generates derivations for F 2 . Obviously F 1 must be a treerewriting system. Proposition 3 For any CFRS F , a grammar G from a (possibly different) CFRS is simulable by a grammar in F if and only if it is trivially simulable by a grammar in F \u2022 RF-MMTAG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "The \"only if\" direction (\u21d2) follows from the fact that the MMTAG constructed in the proof of Proposition 2 generates the same derived trees as the CFG. The \"if\" direction (\u21d0) is a little trickier because the constructed CFG inserts and relabels nodes.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "3.4"
},
{
"text": "MMTAG resembles a cross between set-local multicomponent TAG and ranked node rewriting grammar (Abe, 1988) , a variant of TAG in which auxiliary trees may have multiple foot nodes. It also has much in common with dtree substitution grammar (Rambow et al., 1995) .",
"cite_spans": [
{
"start": 95,
"end": 106,
"text": "(Abe, 1988)",
"ref_id": "BIBREF0"
},
{
"start": 240,
"end": 261,
"text": "(Rambow et al., 1995)",
"ref_id": "BIBREF9"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Definitions",
"sec_num": "4.1"
},
{
"text": "Definition 8 An elementary tree set \u03b1 is a finite set of trees (called the components of \u03b1) with the following properties:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definitions",
"sec_num": "4.1"
},
{
"text": "1. Zero or more frontier nodes are designated foot nodes, which lack labels (following Abe), but are marked with the diacritic * ; 2. Zero or more (non-foot) nodes are designated adjunction nodes, which are partitioned into one or more disjoint sets called adjunction sites. We notate this by assigning an index i to each adjunction site and marking each node of site i with the diacritic i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definitions",
"sec_num": "4.1"
},
{
"text": "bol called its type. This is analogous to the left-hand side of a CFG rule (again, following Abe). 4. The components of \u03b1 are connected by dedges from foot nodes to root nodes (notated by dotted lines) to form a single tree structure. A single foot node may have multiple d-children, and their order is significant. (See Figure 5 for an example.)",
"cite_spans": [],
"ref_spans": [
{
"start": 321,
"end": 329,
"text": "Figure 5",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "A multicomponent multifoot tree adjoining grammar is a tuple \u03a3, P, S , where: 1. \u03a3 is a finite alphabet; 2. P is a finite set of tree sets; and 3. S \u2208 \u03a3 is a distinguished start symbol.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "A * X 1 Y 2 * X 1 * X 1 * A * X 1 Y 3 * X 1 * X 1 * A * A Y 3 X 1 Y 2 * X 1 * X 1 *",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "Definition 9 A component \u03b1 is adjoinable at a node \u03b7 if \u03b7 is an adjunction node and the type of \u03b1 equals the label of \u03b7.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "The result of adjoining a component \u03b1 at a node \u03b7 is the tree set formed by separating \u03b7 from its children, replacing \u03b7 with the root of \u03b1, and replacing the ith foot node of \u03b1 with the ith child of \u03b7. (Thus adjunction of a one-foot component is analogous to TAG adjunction, and adjunction of a zero-foot component is analogous to substitution.)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "A tree set \u03b1 is adjoinable at an adjunction site \u03b7 if there is a way to adjoin each component of \u03b1 at a different node of \u03b7 (with no nodes left over) such that the dominance and precedence relations within \u03b1 are preserved. (See Figure 5 for an example.)",
"cite_spans": [],
"ref_spans": [
{
"start": 228,
"end": 236,
"text": "Figure 5",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "We now define a regular form for MMTAG that is analogous to strict regular form for TAG. A spine is the path from the root to a foot of a single component. Whenever adjunction takes place, several spines are inserted inside or concatenated with other spines. To ensure that unbounded insertion does not take place, we impose an ordering on spines, by means of functions \u03c1 i that map the type of a component to the rank of that component's ith spine.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Each component is associated with a sym-",
"sec_num": "3."
},
{
"text": "We say that an adjunction node \u03b7 \u2208 \u03b7 is safe in a spine if it is the lowest node (except the foot) in that spine, and if each component under that spine consists only of a member of \u03b7 and zero or more foot nodes.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 10",
"sec_num": null
},
{
"text": "We say that an MMTAG G is in regular form if there are functions \u03c1 i from \u03a3 into the domain of some partial ordering such that for each component \u03b1 of type X, for each adjunction node \u03b7 \u2208 \u03b1, if the jth child of \u03b7 dominates the ith foot node of \u03b1 (that is, another component's jth spine would adjoin into the ith spine), then \u03c1 i (X) \u03c1 j (label(\u03b7)), and \u03c1 i (X) = \u03c1 j (label(\u03b7)) only if \u03b7 is safe in the ith spine.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 10",
"sec_num": null
},
{
"text": "Thus the only kinds of adjunction which can occur to unbounded depth are off-spine adjunction and safe adjunction. The adjunction shown in Figure 5 is an example of safe adjunction.",
"cite_spans": [],
"ref_spans": [
{
"start": 139,
"end": 147,
"text": "Figure 5",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Definition 10",
"sec_num": null
},
{
"text": "(\u21d0) First we describe how to construct a simulating CFG for any RF-MMTAG; then this direction of the proof follows from the transitivity of simulability.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "When a CFG simulates a regular form TAG, each nonterminal must encapsulate a stack (of bounded depth) to keep track of adjunctions. In the multicomponent case, these stacks must be generalized to trees (again, of bounded size).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "So the nonterminals of G are of the form [\u03b7, t] , where t is a derivation fragment of G with a dot (\u2022) at exactly one node \u03b1, and \u03b7 is a node of \u03b1. Let \u03b7 be the node in the derived tree where \u03b7 ends up.",
"cite_spans": [
{
"start": 41,
"end": 47,
"text": "[\u03b7, t]",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "A fragment t can be put into a normal form as follows:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "1. For every \u03b1 above the dot, if\u03b7 does not lie along a spine of \u03b1, delete everything above \u03b1.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "2. For every \u03b1 not above or at the dot, if\u03b7 does not lie along a d-edge of \u03b1, delete \u03b1 and everything below and replace it with if\u03b7 dominates \u03b1; otherwise replace it with \u22a5.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "3. If there are two nodes \u03b1 1 and \u03b1 2 along a path which name the same tree set and\u03b7 lies along the same spine or same d-edge in both of them, collapse \u03b1 1 and \u03b1 2 , deleting everything in between.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Basically this process removes all unboundedly long paths, so that the set of normal forms is finite. In the rule schemata below, the terms in the lefthand sides range over normalized terms, and their corresponding right-hand sides are renormalized. Let up(t) denote the tree that results from moving the dot in t up one step. The value of a subderivation t of G under \u2022 s is a tuple of partial derivations of G, one for each symbol in the root label of t , in order. Where we do not define a yield function for a production below, the identity function is understood.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "For every set \u03b1 with a single, S -type component rooted by \u03b7, add the rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "S \u2192 [\u03b7, \u2022 \u03b1( , . . . , )] \u03b1(x 1 , . . . , x n ) \u2190 x 1 , . . . , x n",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "For every non-adjunction, non-foot node \u03b7 with children \u03b7 1 , . . . , \u03b7 n (n \u2265 0),",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "[\u03b7, t] \u2192 [\u03b7 1 , t] \u2022 \u2022 \u2022 [\u03b7 n , t]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "For every component with root \u03b7 that is adjoinable at \u03b7, [\u03b7, up(t) ",
"cite_spans": [
{
"start": 57,
"end": 66,
"text": "[\u03b7, up(t)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "] \u2192 [\u03b7 , t]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "If \u03b7 is the root of the whole set \u03b1 , this rule rewrites a to several symbols; the corresponding yield function is then",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": ". . . , \u03b1 (x 1 , . . . , x n ), . . . \u2190 . . . , x 1 , . . . , x n , . . .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "For every component with ith foot \u03b7 i that is adjoinable at a node with ith child \u03b7 i ,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "[\u03b7 i , t] \u2192 [\u03b7 i , up(t)]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "This last rule skips over deleted parts of the derivation tree, but this is harmless in a regular form MMTAG, because all the skipped adjunctions are safe.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "(\u21d2) First we describe how to decompose any given derivation t of G into a set of elementary tree sets.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Let t = t s . (Note the convention that primed variables always pertain to the simulating grammar, unprimed variables to the simulated grammar.) If, during the computation of t, a node \u03b7 creates the node \u03b7, we say that \u03b7 is productive and produces \u03b7. Without loss of generality, let us assume that there is a one-to-one correspondence between productive nodes and nodes of t. 4 To start, let \u03b7 be the root of t, and \u03b7 1 , . . . , \u03b7 n its children.",
"cite_spans": [
{
"start": 376,
"end": 377,
"text": "4",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Define the domain of \u03b7 i as follows: any node in t that produces \u03b7 i or any of its descendants is in the domain of \u03b7 i , and any non-productive node whose parent is in the domain of \u03b7 i is also in the domain of \u03b7 i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "For each \u03b7 i , excise each connected component of the domain of \u03b7 i . This operation is the reverse of adjunction (see Figure 6 ): each component gets Figure 6 : Example derivation (left) of the grammar of Figure 4 , and first step of decomposition. Non-adjunction nodes are shown with the placeholder \u2022 (because the yield functions in the original grammar were anonymous), the Greek letters indicating what is produced by each node. Adjunction nodes are shown with labels Q i in place of the (very long) true labels. Figure 3 ). Each components' type is written to its left. foot nodes to replace its lost children, and the components are connected by d-edges according to their original configuration.",
"cite_spans": [],
"ref_spans": [
{
"start": 119,
"end": 127,
"text": "Figure 6",
"ref_id": null
},
{
"start": 151,
"end": 159,
"text": "Figure 6",
"ref_id": null
},
{
"start": 206,
"end": 214,
"text": "Figure 4",
"ref_id": null
},
{
"start": 518,
"end": 526,
"text": "Figure 3",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "\u2022 \u03b1 \u2022 \u03b2 1 \u2022 a \u2022 \u2022 d Q 1 : \u2022 \u03b2 1 \u2022 a \u2022 * d \u2022 \u03b1 Q 1 1 \u2022",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "S : \u2022 Q 1 1 Q 2 2 Q 1 : \u2022 \u2022 a Q 1 1 * d Q 1 : \u2022 * Q 2 : \u2022 \u2022 b Q 2 2 c Q 2 : \u2022",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Meanwhile an adjunction node is created in place of each component. This node is given a label (which also becomes the type of the excised component) whose job is to make sure the final grammar does not overgenerate; we describe how the label is chosen below. The adjunction nodes are partitioned such that the ith site contains all the adjunction nodes created when removing \u03b7 i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "The tree set that is left behind is the elementary tree set corresponding to \u03b7 (rather, the function symbol that labels \u03b7); this process is repeated recursively on the children of \u03b7, if any.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Thus any derivation of G can be decomposed into elementary tree sets. Let\u011c be the union of the decompositions of all possible derivations of G (see Figure 7 for an example).",
"cite_spans": [],
"ref_spans": [
{
"start": 148,
"end": 156,
"text": "Figure 7",
"ref_id": "FIGREF4"
}
],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Labeling adjunction nodes For any node \u03b7 , and any list of nodes \u03b7 1 , . . . , \u03b7 n , let the signature of \u03b7 with respect to \u03b7 1 , . . . , \u03b7 n be A, a 1 , . . . , a m , where A is the left-hand side of the GCFG production that generated \u03b7 , and a i = j, k if \u03b7 gets its ith field from the kth field of \u03b7 j , or * if \u03b7 produces a function symbol in its ith field.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "So when we excise the domain of \u03b7 i , the label of the node left behind by a component \u03b1 is s, s 1 , . . . , s n , where s is the signature of the root of \u03b1 with respect to the foot nodes and s 1 , . . . , s n are the signatures of the foot nodes with respect to their d-children. Note that the number of possible adjunction labels is finite, though large.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "G trivially simulates G. Since each tree of\u011c corresponds to a function symbol (though not necessarily one-to-one), it is easy to write a trivial simulating interpretation \u2022 : T (\u011c) \u2192 T (G). To see that\u011c does not overgenerate, observe that the nonterminal labels inside the signatures ensure that every derivation of\u011c corresponds to a valid derivation of G , and therefore G. To see that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "\u2022 is one-to-one, observe that the adjunction labels keep track of how G constructed its simulated derivations, ensuring that for any derivation t of\u011c, the decomposition of the derived tree oft ist itself. Therefore two derivations of\u011c cannot correspond to the same derivation of G , nor of G. G is finite. Briefly, suppose that the number of components per tree set is unbounded. Then it is possible, by intersecting G with a recognizable set, to obtain a grammar whose simulated derivation set is non-recognizable. The idea is that multicomponent tree sets give rise to dependent paths in the derivation set, so if there is no bound on the number of components in a tree set, neither is there a bound on the length of dependent paths. This contradicts the requirement that a simulating interpretation map recognizable sets to recognizable sets.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "Suppose that the number of nodes per component is unbounded. If the number of components per tree set is bounded, so must the number of adjunction nodes per component; then it is possible, again by intersecting G with a recognizable set, to obtain a grammar which is infinitely ambiguous with respect to simulated derivations, which contradicts the requirement that simulating interpretations be bijective. \u011c is in regular form. A component of\u011c corresponds to a derivation fragment of G which takes fields from several subderivations and processes them, combining some into a larger structure and copying some straight through to the root. Let \u03c1 i (X) be the number of fields that a component of type X copies from its ith foot up to its root. This information is encoded in X, in the signature of the root. Then\u011c satisfies the regular form constraint, because when adjunction inserts one spine into another spine, the the inserted spine must copy at least as many fields as the outer one. Furthermore, if the adjunction site is not safe, then the inserted spine must additionally copy the value produced by some lower node.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof of Proposition 2",
"sec_num": "4.2"
},
{
"text": "We have proposed a more constrained version of Joshi's question, \"How much strong generative power can be squeezed out of a formal system without increasing its weak generative power,\" and shown that within these constraints, a variant of TAG called MMTAG characterizes the limit of how much strong generative power can be squeezed out of CFG. Moreover, using the notion of a meta-level grammar, this result is extended to formalisms beyond CFG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Discussion",
"sec_num": "5"
},
{
"text": "It remains to be seen whether RF-MMTAG, whether used directly or for specifying meta-level grammars, provides further practical benefits on top of existing \"squeezed\" grammar formalisms like tree-local MCTAG, tree insertion grammar, or regular form TAG.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Discussion",
"sec_num": "5"
},
{
"text": "This way of approaching Joshi's question is by no means the only way, but we hope that this work will contribute to a better understanding of the strong generative capacity of constrained grammar formalisms as well as reveal more powerful formalisms for linguistic analysis and natural language processing.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Discussion",
"sec_num": "5"
},
{
"text": "This is similar in spirit, but not the same as, the notion of derivational generative capacity(Becker et al., 1992).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "Simulating interpretations and trivial simulating interpretations are similar to the generalized and \"ungeneralized\" syntax-directed translations, respectively, ofAho and Ullman (1969;1971).3 Without this requirement, there are certain pathological cases that cause the construction of Section 4.2 to produce infinite MM-TAGs.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "If G does not have this property, it can be modified so that it does. This may change the derived trees slightly, which makes the proof of Proposition 3 trickier.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
}
],
"back_matter": [
{
"text": "This research is supported in part by NSF grant SBR-89-20230-15. Thanks to Mark Dras, William Schuler, Anoop Sarkar, Aravind Joshi, and the anonymous reviewers for their valuable help. S. D. G.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Acknowledgments",
"sec_num": null
}
],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"title": "Feasible learnability of formal grammars and the theory of natural language acquisition",
"authors": [
{
"first": "Naoki",
"middle": [],
"last": "Abe",
"suffix": ""
}
],
"year": 1988,
"venue": "Proceedings of the Twelfth International Conference on Computational Linguistics (COLING-88)",
"volume": "",
"issue": "",
"pages": "1--6",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Naoki Abe. 1988. Feasible learnability of formal grammars and the theory of natural language ac- quisition. In Proceedings of the Twelfth Inter- national Conference on Computational Linguistics (COLING-88), pages 1-6, Budapest.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "Syntax directed translations and the pushdown assembler",
"authors": [
{
"first": "A",
"middle": [
"V"
],
"last": "Aho",
"suffix": ""
},
{
"first": "J",
"middle": [
"D"
],
"last": "Ullman",
"suffix": ""
}
],
"year": 1969,
"venue": "J. Comp. Sys. Sci",
"volume": "3",
"issue": "",
"pages": "37--56",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "A. V. Aho and J. D. Ullman. 1969. Syntax directed translations and the pushdown assembler. J. Comp. Sys. Sci, 3:37-56.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "The derivational generative power of formal systems, or, Scrambling is beyond LCFRS",
"authors": [
{
"first": "Tilman",
"middle": [],
"last": "Becker",
"suffix": ""
},
{
"first": "Owen",
"middle": [],
"last": "Rambow",
"suffix": ""
},
{
"first": "Michael",
"middle": [],
"last": "Niv",
"suffix": ""
}
],
"year": 1992,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Tilman Becker, Owen Rambow, and Michael Niv. 1992. The derivational generative power of formal systems, or, Scrambling is beyond LCFRS. Tech- nical Report IRCS-92-38, Institute for Research in Cognitive Science, University of Pennsylvania.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Aspects of the Theory of Syntax",
"authors": [
{
"first": "Noam",
"middle": [],
"last": "Chomsky",
"suffix": ""
}
],
"year": 1965,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Noam Chomsky. 1965. Aspects of the Theory of Syn- tax. MIT Press, Cambridge, MA.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "A meta-level grammar: redefining synchronous TAG for translation and paraphrase",
"authors": [
{
"first": "Mark",
"middle": [],
"last": "Dras",
"suffix": ""
}
],
"year": 1999,
"venue": "Proceedings of the 37th Annual Meeting of the Assocation for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "80--87",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Mark Dras. 1999. A meta-level grammar: redefining synchronous TAG for translation and paraphrase. In Proceedings of the 37th Annual Meeting of the Assocation for Computational Linguistics, pages 80-87, College Park, MD.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "An introduction to tree adjoining grammars",
"authors": [
{
"first": "K",
"middle": [],
"last": "Aravind",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Joshi",
"suffix": ""
}
],
"year": 1987,
"venue": "Mathematics of Language. John Benjamins",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Aravind K. Joshi. 1987. An introduction to tree ad- joining grammars. In Alexis Manaster-Ramer, ed- itor, Mathematics of Language. John Benjamins, Amsterdam.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "Relationship between strong and weak generative power of formal systems",
"authors": [
{
"first": "K",
"middle": [],
"last": "Aravind",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Joshi",
"suffix": ""
}
],
"year": 2000,
"venue": "Proceedings of the Fifth International Workshop on TAG and Related Formalisms (TAG+5)",
"volume": "",
"issue": "",
"pages": "107--113",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Aravind K. Joshi. 2000. Relationship between strong and weak generative power of formal systems. In Proceedings of the Fifth International Workshop on TAG and Related Formalisms (TAG+5), pages 107- 113.",
"links": null
},
"BIBREF8": {
"ref_id": "b8",
"title": "Strong Generative Capacity: The Semantics of Linguistic Formalism. Number 103 in CSLI lecture notes",
"authors": [
{
"first": "H",
"middle": [],
"last": "Philip",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Miller",
"suffix": ""
}
],
"year": 1999,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Philip H. Miller. 1999. Strong Generative Capacity: The Semantics of Linguistic Formalism. Number 103 in CSLI lecture notes. CSLI Publications, Stan- ford.",
"links": null
},
"BIBREF9": {
"ref_id": "b9",
"title": "D-tree grammars",
"authors": [
{
"first": "Owen",
"middle": [],
"last": "Rambow",
"suffix": ""
},
{
"first": "K",
"middle": [],
"last": "Vijay-Shanker",
"suffix": ""
},
{
"first": "David",
"middle": [],
"last": "Weir",
"suffix": ""
}
],
"year": 1995,
"venue": "Proceedings of the 33rd Annual Meeting of the Assocation for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "151--158",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Owen Rambow, K. Vijay-Shanker, and David Weir. 1995. D-tree grammars. In Proceedings of the 33rd Annual Meeting of the Assocation for Com- putational Linguistics, pages 151-158, Cambridge, MA.",
"links": null
},
"BIBREF10": {
"ref_id": "b10",
"title": "Capturing CFLs with tree adjoining grammars",
"authors": [
{
"first": "James",
"middle": [],
"last": "Rogers",
"suffix": ""
}
],
"year": 1994,
"venue": "Proceedings of the 32nd Annual Meeting of the Assocation for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "155--162",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "James Rogers. 1994. Capturing CFLs with tree ad- joining grammars. In Proceedings of the 32nd An- nual Meeting of the Assocation for Computational Linguistics, pages 155-162, Las Cruces, NM.",
"links": null
},
"BIBREF11": {
"ref_id": "b11",
"title": "Lexicalized context-free grammars",
"authors": [
{
"first": "Yves",
"middle": [],
"last": "Schabes",
"suffix": ""
},
{
"first": "Richard",
"middle": [
"C"
],
"last": "Waters",
"suffix": ""
}
],
"year": 1993,
"venue": "Proceedings of the 31st Annual Meeting of the Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "121--129",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Yves Schabes and Richard C. Waters. 1993. Lexical- ized context-free grammars. In Proceedings of the 31st Annual Meeting of the Association for Com- putational Linguistics, pages 121-129, Columbus, OH.",
"links": null
},
"BIBREF12": {
"ref_id": "b12",
"title": "Characterizing structural descriptions produced by various grammatical formalisms",
"authors": [
{
"first": "K",
"middle": [],
"last": "Vijay-Shanker",
"suffix": ""
},
{
"first": "David",
"middle": [],
"last": "Weir",
"suffix": ""
},
{
"first": "Aravind",
"middle": [],
"last": "Joshi",
"suffix": ""
}
],
"year": 1987,
"venue": "Proceedings of the 25th Annual Meeting of the Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "104--111",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "K. Vijay-Shanker, David Weir, and Aravind Joshi. 1987. Characterizing structural descriptions pro- duced by various grammatical formalisms. In Pro- ceedings of the 25th Annual Meeting of the Associa- tion for Computational Linguistics, pages 104-111, Stanford, CA.",
"links": null
},
"BIBREF13": {
"ref_id": "b13",
"title": "Characterizing Mildly Context-Sensitive Grammar Formalisms",
"authors": [
{
"first": "David",
"middle": [
"J"
],
"last": "Weir",
"suffix": ""
}
],
"year": 1988,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "David J. Weir. 1988. Characterizing Mildly Context- Sensitive Grammar Formalisms. Ph.D. thesis, Univ. of Pennsylvania.",
"links": null
}
},
"ref_entries": {
"FIGREF0": {
"uris": null,
"num": null,
"type_str": "figure",
"text": "Example of weakly context-free TAG."
},
"FIGREF1": {
"uris": null,
"num": null,
"type_str": "figure",
"text": "Figure 2: Simulation: structural descriptions as derived structures."
},
"FIGREF2": {
"uris": null,
"num": null,
"type_str": "figure",
"text": "Example of TAG with corresponding GCFG and interpretation. Here adjunction at foot nodes is allowed."
},
"FIGREF3": {
"uris": null,
"num": null,
"type_str": "figure",
"text": "Example of MMTAG adjunction. The types of the components, not shown in the figure, are all X."
},
"FIGREF4": {
"uris": null,
"num": null,
"type_str": "figure",
"text": "MMTAG converted from CFG of Figure 4 (cf. the original TAG in"
}
}
}
} |