File size: 108,150 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 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 | {
"paper_id": "D09-1010",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T16:39:13.170804Z"
},
"title": "Efficient kernels for sentence pair classification",
"authors": [
{
"first": "Fabio",
"middle": [
"Massimo"
],
"last": "Zanzotto",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "DISP University of Rome \"Tor Vergata\"",
"location": {
"addrLine": "Via del Politecnico 1",
"postCode": "00133",
"settlement": "Roma",
"country": "Italy"
}
},
"email": ""
},
{
"first": "Lorenzo",
"middle": [],
"last": "Dell'arciprete",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "University of Rome \"",
"location": {
"addrLine": "Tor Vergata\" Via del Politecnico 1",
"postCode": "00133",
"settlement": "Roma",
"country": "Italy"
}
},
"email": "lorenzo.dellarciprete@gmail.com"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "In this paper, we propose a novel class of graphs, the tripartite directed acyclic graphs (tDAGs), to model first-order rule feature spaces for sentence pair classification. We introduce a novel algorithm for computing the similarity in first-order rewrite rule feature spaces. Our algorithm is extremely efficient and, as it computes the similarity of instances that can be represented in explicit feature spaces, it is a valid kernel function.",
"pdf_parse": {
"paper_id": "D09-1010",
"_pdf_hash": "",
"abstract": [
{
"text": "In this paper, we propose a novel class of graphs, the tripartite directed acyclic graphs (tDAGs), to model first-order rule feature spaces for sentence pair classification. We introduce a novel algorithm for computing the similarity in first-order rewrite rule feature spaces. Our algorithm is extremely efficient and, as it computes the similarity of instances that can be represented in explicit feature spaces, it is a valid kernel function.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Natural language processing models are generally positive combinations between linguistic models and automatically learnt classifiers. As trees are extremely important in many linguistic theories, a large amount of works exploiting machine learning algorithms for NLP tasks has been developed for this class of data structures (Collins and Duffy, 2002; Moschitti, 2004) . These works propose efficient algorithms for determining the similarity among two trees in tree fragment feature spaces.",
"cite_spans": [
{
"start": 327,
"end": 352,
"text": "(Collins and Duffy, 2002;",
"ref_id": "BIBREF2"
},
{
"start": 353,
"end": 369,
"text": "Moschitti, 2004)",
"ref_id": "BIBREF14"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Yet, some NLP tasks such as textual entailment recognition (Dagan and Glickman, 2004; and some linguistic theories such as HPSG (Pollard and Sag, 1994 ) require more general graphs and, then, more general algorithms for computing similarity among graphs. Unfortunately, algorithms for computing similarity among two general graphs in term of common subgraphs are still exponential (Ramon and G\u00e4rtner, 2003) . In these cases, approximated algorithms have been proposed. For example, the one proposed in (G\u00e4rtner, 2003) counts the number of subpaths in common. The same happens for the one proposed in (Suzuki et al., 2003) that is applicable to a particular class of graphs, i.e. the hierarchical directed acyclic graphs. These algorithms do not compute the number of subgraphs in common between two graphs. Then, these algorithms approximate the feature spaces we need in these NLP tasks. For computing similarities in these feature spaces, we have to investigate if we can define a particular class of graphs for the class of tasks we want to solve. Once we focused the class of graph, we can explore efficient similarity algorithms.",
"cite_spans": [
{
"start": 59,
"end": 85,
"text": "(Dagan and Glickman, 2004;",
"ref_id": "BIBREF4"
},
{
"start": 128,
"end": 150,
"text": "(Pollard and Sag, 1994",
"ref_id": "BIBREF15"
},
{
"start": 381,
"end": 406,
"text": "(Ramon and G\u00e4rtner, 2003)",
"ref_id": "BIBREF17"
},
{
"start": 502,
"end": 517,
"text": "(G\u00e4rtner, 2003)",
"ref_id": "BIBREF8"
},
{
"start": 600,
"end": 621,
"text": "(Suzuki et al., 2003)",
"ref_id": "BIBREF18"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "A very important class of graphs can be defined for tasks involving sentence pairs. In these cases, an important class of feature spaces is the one that represents first-order rewrite rules. For example, in textual entailment recognition , we need to determine whether a text T implies a hypothesis H, e.g., whether or not \"Farmers feed cows animal extracts\" entails \"Cows eat animal extracts\" (T 1 , H 1 ). If we want to learn textual entailment classifiers, we need to exploit first-order rules hidden in training instances. To positively exploit the training instance \"Pediatricians suggest women to feed newborns breast milk\" entails \"Pediatricians suggest that newborns eat breast milk\" (T 2 , H 2 ) for classifying the above example, learning algorithms should learn that the two instances hide the first-order rule \u03c1 = f eed Y Z \u2192 Y eat Z . The first-order rule feature space, introduced by (Zanzotto and Moschitti, 2006) , gives high performances in term of accuracy for textual entailment recognition with respect to other features spaces.",
"cite_spans": [
{
"start": 912,
"end": 928,
"text": "Moschitti, 2006)",
"ref_id": "BIBREF21"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In this paper, we propose a novel class of graphs, the tripartite directed acyclic graphs (tDAGs), that model first-order rule feature spaces and, using this class of graphs, we introduce a novel algorithm for computing the similarity in first-order rewrite rule feature spaces. The possibility of explicitly representing the first-order feature space as subgraphs of tDAGs makes the derived similarity function a valid kernel. With respect to the algorithm proposed in (Moschitti and Zanzotto, 2007) , our algorithm is more efficient and it is a valid kernel function.",
"cite_spans": [
{
"start": 470,
"end": 500,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The paper is organized as follows. In Sec. 2, we firstly describe tripartite directed acyclic graphs (tDAGs) to model first-order feature (FOR) spaces. In Sec. 3, we then present the related work. In Sec. 4, we introduce the similarity function for these FOR spaces. This can be used as kernel function in kernel-based machines (e.g., support vector machines (Cortes and Vapnik, 1995) ). We then introduce our efficient algorithm for computing the similarity among tDAGs. In Sec. 5, we analyze the computational efficiency of our algorithm showing that it is extremely more efficient than the algorithm proposed in (Moschitti and Zanzotto, 2007) . Finally, in Sec. 6, we draw conclusions and plan the future work.",
"cite_spans": [
{
"start": 359,
"end": 384,
"text": "(Cortes and Vapnik, 1995)",
"ref_id": "BIBREF3"
},
{
"start": 615,
"end": 645,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "As first step, we want to define the tripartite directed acyclic graphs (tDAGs). This is an extremely important class of graphs for the firstorder rule feature spaces we want to model. We want here to intuitively show that, if we model first-order rules and sentence pairs as tDAGs, determining whether or not a sentence pair can be unified with a first-order rewrite rule is a graph matching problem. This intuitive idea helps in determining our efficient algorithm for exploiting first-order rules in learning examples. To illustrate the above idea we will use an example based on the above rule \u03c1= f eed Y Z \u2192 Y eat Z and the above sentence pair (T 1 , H 1 ). The rule \u03c1 encodes the entailment relation of the verb to feed and the verb to eat. If represented over a syntactic interpretation, the rule has the following aspect:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "\u03c1 1 = VP VB feed NP Y NP Z \u2192 S NP Y VP VB eat NP Z",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "As in the case of feature structures (Carpenter, 1992) , we can observe this rule as a graph. As we are not interested in the variable names but we need to know the relation between the right hand side and the left hand side of the rule, we can substitute each variable with an unlabelled node. We then connect tree nodes having variables with the corresponding unlabelled node. The result is a graph as the one in Fig. 1 . The variables Y and Z are represented by the unlabelled nodes between the trees. In the same way we can represent the sentence pair (T 1 , H 1 ) using graph with explicit links between related words and nodes (see Fig. 2 ). We can link words using anchoring methods as in (Raina et al., 2005) . These links can then be propagated in the syntactic tree using semantic heads of the constituents (Pollard and Sag, 1994) . The rule \u03c1 1 matches over the pair (T 1 , H 1 ) if the graph \u03c1 1 is among the subgraphs of the graph in Fig. 2 .",
"cite_spans": [
{
"start": 37,
"end": 54,
"text": "(Carpenter, 1992)",
"ref_id": "BIBREF1"
},
{
"start": 696,
"end": 716,
"text": "(Raina et al., 2005)",
"ref_id": "BIBREF16"
},
{
"start": 817,
"end": 840,
"text": "(Pollard and Sag, 1994)",
"ref_id": "BIBREF15"
}
],
"ref_spans": [
{
"start": 415,
"end": 421,
"text": "Fig. 1",
"ref_id": null
},
{
"start": 638,
"end": 644,
"text": "Fig. 2",
"ref_id": null
},
{
"start": 947,
"end": 953,
"text": "Fig. 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "Both rules and sentence pairs are graphs of the same type. These graphs are basically two trees connected through an intermediate set of nodes representing variables in the rules and relations between nodes in the sentence pairs. We will hereafter call these graphs tripartite directed acyclic graphs (tDAGs). The formal definition follows.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "Definition tDAG: A tripartite directed acyclic graph is a graph G = (N, E) where",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "\u2022 the set of nodes N is partitioned in three sets N t , N g , and A",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "\u2022 the set of edges is partitioned in four sets E t , E g , E At , and E Ag such that t = (N t , E t ) and g = (N g , E g ) are two trees and E At = {(x, y)|x \u2208 N t and y \u2208 A} and E Ag = {(x, y)|x \u2208 N g and y \u2208 A} are the edges connecting the two trees.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "A tDAG is a partially labeled graph. The labeling function L only applies to the subsets of nodes related to the two trees, i.e., L :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "N t \u222a N g \u2192 L.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "Nodes in the set A are not labeled.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "The explicit representation of the tDAG in Fig. 2 has been useful to show that the unification of a rule and a sentence pair is a graph matching problem. Yet, it is complex to follow. We will then describe a tDAG with an alternative and more convenient representation. A tDAG G = (N, E) can be seen as pair G = (\u03c4, \u03b3) of extended trees \u03c4 and \u03b3 where \u03c4 = (N t \u222a A, E t \u222a E At ) and \u03b3 = (N g \u222a A, E g \u222a E Ag ). These are extended trees as each tree contains the relations with the other tree.",
"cite_spans": [],
"ref_spans": [
{
"start": 43,
"end": 49,
"text": "Fig. 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "As for the feature structures, we will graphically represent a (x, y) \u2208 E At and a (z, y) \u2208 E Ag as boxes y respectively on the node x and on the node z. These nodes will then appear as L(x) y and L(z) y , e.g., NP 1 . The name y is not a label but a placeholder representing an unlabelled node. This representation is used for rules and for sentence pairs. The sentence pair in Fig. 2 is then represented as reported in Fig. 3 .",
"cite_spans": [],
"ref_spans": [
{
"start": 379,
"end": 385,
"text": "Fig. 2",
"ref_id": null
},
{
"start": 421,
"end": 427,
"text": "Fig. 3",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Representing first-order rules and sentence pairs as tripartite directed acyclic graphs",
"sec_num": "2"
},
{
"text": "Automatically learning classifiers for sentence pairs is extremely important for applications like textual entailment recognition, question answering, and machine translation.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Related work",
"sec_num": "3"
},
{
"text": "In textual entailment recognition, it is not hard to see graphs similar to tripartite directed acyclic graphs as ways of extracting features from examples to feed automatic classifiers. Yet, these graphs are generally not tripartite in the sense described in the previous section and they are not used to extract features representing first-order rewrite rules. In (Raina et al., 2005; Hickl et al., 2006) , two connected graphs representing the two sentences s 1 and s 2 are used to compute distance features, i.e., features representing the distance between s 1 and s 2 . The underlying idea is that lexical, syntactic, and semantic similarities between sentences in a pair are relevant features to classify sentence pairs in classes such as entail and not-entail.",
"cite_spans": [
{
"start": 365,
"end": 385,
"text": "(Raina et al., 2005;",
"ref_id": "BIBREF16"
},
{
"start": 386,
"end": 405,
"text": "Hickl et al., 2006)",
"ref_id": "BIBREF10"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Related work",
"sec_num": "3"
},
{
"text": "In (de Marneffe et al., 2006) , first-order rewrite rule feature spaces have been explored. Yet, these spaces are extremely small. Only some features representing first-order rules have been explored. Pairs of graphs are used here to determine if a feature is active or not, i.e., the rule fires or not. A larger feature space of rewrite rules has been implicitly explored in (Wang and Neumann, 2007) but this work considers only ground rewrite rules.",
"cite_spans": [
{
"start": 3,
"end": 29,
"text": "(de Marneffe et al., 2006)",
"ref_id": "BIBREF6"
},
{
"start": 376,
"end": 400,
"text": "(Wang and Neumann, 2007)",
"ref_id": "BIBREF19"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Related work",
"sec_num": "3"
},
{
"text": "In (Zanzotto and Moschitti, 2006) , tripartite directed acyclic graphs are implicitly introduced and exploited to build first-order rule feature spaces. Yet, both in (Zanzotto and Moschitti, 2006) and in (Moschitti and Zanzotto, 2007) , the model proposed has two major limitations: it can represent rules with less than 7 variables and the proposed kernel is not a completely valid kernel as it uses the max function.",
"cite_spans": [
{
"start": 3,
"end": 33,
"text": "(Zanzotto and Moschitti, 2006)",
"ref_id": "BIBREF21"
},
{
"start": 180,
"end": 196,
"text": "Moschitti, 2006)",
"ref_id": "BIBREF21"
},
{
"start": 204,
"end": 234,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Related work",
"sec_num": "3"
},
{
"text": "In machine translation, some methods such as (Eisner, 2003) learn graph based rewrite rules for generative purposes. Yet, the method presented in (Eisner, 2003) can model first-order rewrite rules only with a very small amount of variables, i.e., two or three variables.",
"cite_spans": [
{
"start": 45,
"end": 59,
"text": "(Eisner, 2003)",
"ref_id": "BIBREF7"
},
{
"start": 146,
"end": 160,
"text": "(Eisner, 2003)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Related work",
"sec_num": "3"
},
{
"text": "In this section, we present our idea for an efficient algorithm for exploiting first-order rule feature spaces. In Sec. 4.1, we firstly define the similarity function, i.e., the kernel K(G 1 , G 2 ), that we need to determine for correctly using first-order rules feature spaces. This kernel is strongly based on the isomorphism between graphs. A relevant idea of this paper is the observation that we can define an efficient way to detect the isomorphism between the tDAGs (Sec. 4.2). This algorithm exploits the efficient algorithms of tree isomorphism as the one implicitly used in (Collins and Duffy, 2002) . After describing the isomorphism between tDAGs, We can present the idea of our efficient algorithm for computing K(G 1 , G 2 ) (Sec. 4.3). We introduce the algorithms to make it a viable solution (Sec. 4.4). Finally, in Sec. 4.5, we report the kernel computation we compare against presented by (Zanzotto and Moschitti, 2006; Moschitti and Zanzotto, 2007) .",
"cite_spans": [
{
"start": 585,
"end": 610,
"text": "(Collins and Duffy, 2002)",
"ref_id": "BIBREF2"
},
{
"start": 922,
"end": 938,
"text": "Moschitti, 2006;",
"ref_id": "BIBREF21"
},
{
"start": 939,
"end": 968,
"text": "Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "An efficient algorithm for computing the first-order rule space kernel",
"sec_num": "4"
},
{
"text": "The first-order rule feature space we want to model is huge. If we use kernel-based machine learning models such as SVM (Cortes and Vapnik, 1995) , we can implicitly define the space by defining its similarity functions, i.e., its kernel functions. We firstly introduce the first-order rule feature space and we then define the prototypical kernel function over this space.",
"cite_spans": [
{
"start": 120,
"end": 145,
"text": "(Cortes and Vapnik, 1995)",
"ref_id": "BIBREF3"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "The first-order rule feature space (F OR) is in general the space of all the possible first-order We want that these subgraphs represent first-order rules that can be matched with the pair G. Then, meaningful subgraphs of G = (\u03c4, \u03b3) are graphs as (t, g) where t and g are subtrees of \u03c4 and \u03b3. For example, the subgraphs of P 1 and P 2 in Fig. 3 are hereafter partially represented: In the FOR space, the kernel function K should then compute the number of subgraphs in common. The trivial way to describe the former kernel function is using the intersection operator, i.e., the kernel K(G 1 , G 2 ) is the following:",
"cite_spans": [],
"ref_spans": [
{
"start": 338,
"end": 344,
"text": "Fig. 3",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "S(P 1 ) = { S",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "K(G 1 , G 2 ) = |S(G 1 ) \u2229 S(G 2 )|",
"eq_num": "(1)"
}
],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "This is very simple to write and it is in principle correct. A graph g in the intersection S(G 1 ) \u2229 S(G 2 ) is a graph that belongs to both S(G 1 ) and S(G 2 ). Yet, this hides a very important fact: determining whether two graphs, g 1 and g 2 , are the same graph g 1 = g 2 is not trivial. For example, it is not sufficient to superficially compare graphs to determine that \u03c1 1 belongs both to S 1 and S 2 . We need to use the correct property for g 1 = g 2 , i.e., the isomorphism between two graphs. We can call the operator Iso(g 1 , g 2 ). When two graphs verify the property Iso(g 1 , g 2 ), both g 1 and g 2 can be taken as the graph g representing the two graphs. Detecting Iso(g 1 , g 2 ) has an exponential complexity (K\u00f6bler et al., 1993) . This complexity of the intersection operator between sets of graphs deserves a different way to represent the operation. We will use the same symbol but we will use the prefix notation. The operator is hereafter re-defined:",
"cite_spans": [
{
"start": 729,
"end": 750,
"text": "(K\u00f6bler et al., 1993)",
"ref_id": "BIBREF12"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "\u2229 (S(G 1 ), S(G 2 )) = = {g 1 |g 1 \u2208 S(G 1 ), \u2203g 2 \u2208 S(G 2 ), Iso(g 1 , g 2 )}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Kernel functions over first-order rule feature spaces",
"sec_num": "4.1"
},
{
"text": "As isomorphism between graphs is an essential activity for learning from structured data, we here review its definition and we adapt it to tDAGs.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "We then observe that isomorphism between two tDAGs can be divided in two sub-problems:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "\u2022 finding the isomorphism between two pairs of extended trees",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "\u2022 checking whether the partial isomorphism found between the two pairs of extended trees are compatible.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "In general, two tDAGs,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "G 1 = (N 1 , E 1 ) and G 2 = (N 2 , E 2 ) are isomorphic (or match) if |N 1 | = |N 2 |, |E 1 | = |E 2 |",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": ", and a bijective function f : N 1 \u2192 N 2 exists such that these properties hold:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "\u2022 for each node n \u2208 N 1 , L(f (n)) = L(n) \u2022 for each edge (n 1 , n 2 ) \u2208 E 1 an edge (f (n 1 ), f (n 2 )) is in E 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "The bijective function f is a member of the combinatorial set F of all the possible bijective functions between the two sets N 1 and N 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "The trivial algorithm for detecting if two graphs are isomorphic is exponential (K\u00f6bler et al., 1993) . It explores all the set F. It is still undetermined if the general graph isomorphism problem is NP-complete. Yet, we can use the fact that tDAGs are two extended trees for building a better algorithm. There is an efficient algorithm for computing isomorphism between trees (as the one implicitly used in (Collins and Duffy, 2002) ).",
"cite_spans": [
{
"start": 80,
"end": 101,
"text": "(K\u00f6bler et al., 1993)",
"ref_id": "BIBREF12"
},
{
"start": 408,
"end": 433,
"text": "(Collins and Duffy, 2002)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "Given two tDAGs G 1 = (\u03c4 1 , \u03b3 1 ) and G 2 = (\u03c4 2 , \u03b3 2 ) the isomorphism problem can be divided in detecting two properties:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "1. Partial isomorphism. Two tDAGs G 1 and G 2 are partially isomorphic, if \u03c4 1 and \u03c4 2 are isomorphic and if \u03b3 1 and \u03b3 2 are isomorphic. The partial isomorphism produces two bijective functions f \u03c4 and f \u03b3 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "2. Constraint compatibility. Two bijective functions f \u03c4 and f \u03b3 are compatible on the sets of nodes A 1 and A 2 , if for each n \u2208 A 1 , it happens that f \u03c4 (n) = f \u03b3 (n).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "We can rephrase the second property, i.e., the constraint compatibility, as follows. We define two constraints c(\u03c4 1 , \u03c4 2 ) and c(\u03b3 1 , \u03b3 2 ) representing the functions f \u03c4 and f \u03b3 on the sets A 1 and A 2 . The two constraints are defined as c(\u03c4 1 , \u03c4 2 ) = {(n, f \u03c4 (n))|n \u2208 A 1 } and c(\u03b3 1 , \u03b3 2 ) = {(n, f \u03b3 (n))|n \u2208 A 1 }. Two partially isomorphic tDAGs are isomorphic if the constraints match, i.e., c(\u03c4 1 , \u03c4 2 ) = c(\u03b3 1 , \u03b3 2 ). Figure 5 : Simple non-linguistic tDAGs For example, the third pair of S(P 1 ) and the second pair of S(P 2 ) are isomorphic as: (1) these are partially isomorphic, i.e., the right hand sides \u03c4 and the left hand sides \u03b3 are isomorphic; (2) both pairs of extended trees generate the constraint c 1 = {( 1 , 3 ), ( 3 , 4 )}. In the same way, the fourth pair of S(P 1 ) and the third pair of S(P 2 ) generate c 2 = {( 1 , 1 )}",
"cite_spans": [],
"ref_spans": [
{
"start": 437,
"end": 445,
"text": "Figure 5",
"ref_id": null
}
],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "Pa = (\u03c4a, \u03b3a) = A 1 B 1 B 1 B 2 C 1 C 1 C 2 , I 1 M 1 M 2 M 1 N 1 N 2 N 1 P b = (\u03c4 b , \u03b3 b ) = A 1 B 1 B 1 B 2 C 1 C 1 C 3 , I 1 M 1 M 3 M 1 N 1 N 2 N 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Isomorphism between tDAGs",
"sec_num": "4.2"
},
{
"text": "As above discussed, two tDAGs are isomorphic if the two properties, the partial isomorphism and the constraint compatibility, hold. To compute the kernel function K(G 1 , G 2 ) defined in Sec. 4.1, we can exploit these properties in the reverse order. Given a constraint c, we can select all the graphs that meet the constraint c (constraint compatibility). Having the two set of all the tDAGs meeting the constraint, we can detect the partial isomorphism. We split each pair of tDAGs in the four extended trees and we determine if these extended trees are compatible. We introduce this innovative method to compute the kernel K(G 1 , G 2 ) in the FOR space in two steps. Firstly, we give an intuitive explanation and, secondly, we formally define the kernel.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "General idea for an efficient kernel function",
"sec_num": "4.3"
},
{
"text": "To give an intuition of the kernel computation, without loss of generality and for sake of simplicity, we use two non-linguistic tDAGs, P a and P b (see Fig. 5 ), and the subgraph function S(\u03b8). This latter is an approximated version of S(\u03b8) that generates tDAGs with subtrees rooted in the root of the initial trees of \u03b8.",
"cite_spans": [],
"ref_spans": [
{
"start": 153,
"end": 159,
"text": "Fig. 5",
"ref_id": null
}
],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "To exploit the constraint compatibility property, we define C as the set of all the relevant alternative constraints, i.e., the constraints c that are likely to be generated when detecting the partial isomorphism. For P a and P b , this set is ",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "C = {c 1 , c 2 } = \u2229( S(Pa), S(P b ))|c 1 = { A 1 B 1 C 1 , I 1 M 1 N 1 , A 1 B 1 B 1 B 2 C 1 , I 1 M 1 N 1 , A 1 B 1 B 1 B 2 C 1 , I 1 M 1 N 1 N 2 N 1 , A 1 B 1 C 1 , I 1 M 1 N 1 N 2 N 1 } = { A 1 B 1 C 1 , A 1 B 1 B 1 B 2 C 1 } \u00d7 { I 1 M 1 N 1 , I 1 M 1 N 1 N 2 N 1 } = = \u2229( S(\u03c4a), S(\u03c4 b ))|c 1 \u00d7 \u2229( S(\u03b3a), S(\u03b3 b ))|c 1 \u2229( S(Pa), S(P b ))|c 2 = { A 1 B 1 C 1 , I 1 M 1 N 1 , A 1 B 1 C 1 C 1 C 2 , I 1 M 1 N 1 , A 1 B 1 C 1 C 1 C 2 , I 1 M 1 M 2 M 1 N 1 , A 1 B 1 C 1 , I 1 M 1 M 2 M 1 N 1 } = { A 1 B 1 C 1 , A 1 B 1 C 1 C 1 C 2 } \u00d7 { I 1 M 1 N 1 , I 1 M 1 M 2 M 1 N 1 }= = \u2229( S(\u03c4a), S(\u03c4 b ))|c 2 \u00d7 \u2229( S(\u03b3a), S(\u03b3 b ))|c 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "{( 1 , 1 ), ( 2 , 2 )}, {( 1 , 1 ), ( 2 , 3 )} .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "We can then determine the kernel K(P a , P b ) as:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "K(Pa,P b )= |\u2229( S(Pa), S(P b ))|= = |\u2229( S(Pa), S(P b ))|c 1 \u2229( S(Pa), S(P b ))|c 2 |",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "where \u2229( S(P a ), S(P b ))| c are the common subgraphs that meet the constraint c. A tDAG g = (\u03c4 , \u03b3 ) in S(P a ) is in \u2229( S(P a ), S(P b ))| c if g = (\u03c4 , \u03b3 ) in S(P b ) exists, g is partially isomorphic to g , and c = c(\u03c4 , \u03c4 ) = c(\u03b3 , \u03b3 ) is covered by and compatible with the constraint c, i.e., c \u2286 c. For example in Fig. 4 , the first tDAG of the set \u2229( S(P a ), S(P b ))| c 1 belongs to the set as its constraint c = {( 1 , 1 )} is a subset of c 1 .",
"cite_spans": [],
"ref_spans": [
{
"start": 322,
"end": 328,
"text": "Fig. 4",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "Observing the kernel computation in this way is important. Elements in \u2229( S(P a ), S(P b ))| c already satisfy the property of constraint compatibility. We only need to determine if the partially isomorphic properties hold for elements in \u2229( S(P a ), S(P b ))| c . Then, we can write the following equivalence:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "\u2229( S(Pa), S(P b ))|c= =\u2229( S(\u03c4a), S(\u03c4 b ))|c\u00d7\u2229( S(\u03b3a), S(\u03b3 b ))|c",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "(2) Figure 4 reports this equivalence for the two sets derived using the constraints c 1 and c 2 . Note that this equivalence is not valid if a constraint is not applied, i.e., \u2229( S(P a ), S(P b )) = \u2229( S(\u03c4 a ), S(\u03c4 b )) \u00d7 \u2229( S(\u03b3 a ), S(\u03b3 b )). The pair P a itself does not belong to",
"cite_spans": [],
"ref_spans": [
{
"start": 4,
"end": 12,
"text": "Figure 4",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "\u2229( S(P a ), S(P b )) but it does belong to \u2229( S(\u03c4 a ), S(\u03c4 b )) \u00d7 \u2229( S(\u03b3 a ), S(\u03b3 b )).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "The equivalence (2) allows to compute the cardinality of \u2229( S(P a ), S(P b ))| c using the cardinalities of \u2229( S(\u03c4 a ), S(\u03c4 b ))| c and \u2229( S(\u03b3 a ), S(\u03b3 b ))| c . These latter sets contain only extended trees where the equivalences between unlabelled nodes are given by c. We can then compute the cardinalities of these two sets using methods developed for trees (e.g., the kernel function K S (\u03b8 1 , \u03b8 2 ) introduced in (Collins and Duffy, 2002) ).",
"cite_spans": [
{
"start": 420,
"end": 445,
"text": "(Collins and Duffy, 2002)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Intuitive explanation",
"sec_num": "4.3.1"
},
{
"text": "Given the idea of the previous section, it is easy to demonstrate that the kernel K(G 1 , G 2 ) can be written as follows:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "K(G 1 ,G 2 )=| c\u2208C \u2229(S(\u03c4 1 ),S(\u03c4 2 ))|c\u00d7\u2229(S(\u03b3 1 ),S(\u03b3 2 ))|c|",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "where C is set of alternative constraints and \u2229(S(\u03b8 1 ), S(\u03b8 2 ))| c are all the common extended trees compatible with the constraint c.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "We can compute the above kernel using the inclusion-exclusion property, i.e.,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "|A 1 \u222a \u2022 \u2022 \u2022 \u222a A n | = J\u22082 {1,...,n} (\u22121) |J|\u22121 |A J | (3)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "where 2 {1,...,n} is the set of all the subsets of {1, . . . , n} and A J = i\u2208J A i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "To describe the application of the inclusionexclusion model in our case, let firstly define:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "K S (\u03b8 1 , \u03b8 2 , c) = |\u2229(S(\u03b8 1 ), S(\u03b8 2 ))| c |",
"eq_num": "(4)"
}
],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "where \u03b8 1 can be both \u03c4 1 and \u03b3 1 and \u03b8 2 can be both \u03c4 2 and \u03b3 2 . Trivially, we can demonstrate that:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "K(G 1 , G 2 ) = = J \u22082 {1,...,|C|} (\u22121) |J |\u22121 K S (\u03c4 1 ,\u03c4 2 ,c(J))K S (\u03b3 1 ,\u03b3 2 ,c(J))",
"eq_num": "(5)"
}
],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "where c(J) = i\u2208J c i . Given the nature of the constraint set C, we can compute efficiently the previous equation as it often happens that two different J 1 and J 2 in 2 {1,...,|C|} generate the same c, i.e.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "c = i\u2208J 1 c i = i\u2208J 2 c i (6)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "Then, we can define C * as the set of all intersections of constraints in C, i.e. C * = {c(J)|J \u2208 2 {1,...,|C|} }. We can rewrite the equation as:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "K(G 1 , G 2 ) = = c\u2208C * K S (\u03c4 1 , \u03c4 2 , c)K S (\u03b3 1 , \u03b3 2 , c)N (c) (7)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "where",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "N (c) = J\u22082 {1,...,|C|} c=c(J) (\u22121) |J|\u22121",
"eq_num": "(8)"
}
],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "The complexity of the above kernel strongly depends on the cardinality of C and the related cardinality of C * . The worst-case computational complexity is still exponential with respect to the size of A 1 and A 2 . Yet, the average case complexity (Wang, 1997) is promising.",
"cite_spans": [
{
"start": 249,
"end": 261,
"text": "(Wang, 1997)",
"ref_id": "BIBREF20"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "The set C is generally very small with respect to the worst case. If F (A 1 ,A 2 ) are all the possible correspondences between the nodes A 1 and A 2 , it happens that |C| <<",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "|F (A 1 ,A 2 ) | where |F (A 1 ,A 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "| is the worst case. For example, in the case of P 1 and P 2 , the cardinality of",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "C = {( 1 , 1 )}, {( 1 , 3 ), ( 3 , 4 ), ( 2 , 5",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": ")} is extremely smaller than the one of {( 1 , 3 ) ,( 2 , 4 ),( 3 , 5 )}}. In Sec. 4.5 we argue that the algorithm presented in (Moschitti and Zanzotto, 2007) has the worst-case complexity.",
"cite_spans": [
{
"start": 128,
"end": 158,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [
{
"start": 40,
"end": 50,
"text": "{( 1 , 3 )",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "F (A 1 ,A 2 ) = {{( 1 , 1 ),( 2 , 2 ),( 3 , 3 )}, {( 1 , 2 ),( 2 , 1 ),( 3 , 3 )}, {( 1 , 2 ),( 2 , 3 ),( 3 , 1 )}, ...,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "Moreover, the set C * is extremely smaller than 2 {1,...,|C|} due to the above property (6).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "We will analyze the average-case complexity with respect to the worst-case complexity in Sec. 5.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formal definition",
"sec_num": "4.3.2"
},
{
"text": "The above idea for computing the kernel function is extremely interesting. Yet, we need to make it viable by describing the way we can determine efficiently the three main parts of the equation 7: 1) the set of alternative constraints C (Sec. 4.4.1); 2) the set C * of all the possible intersections of constraints in C (Sec. 4.4.2) ; and, finally, 3) the numbers N (c) (Sec. 4.4.3) .",
"cite_spans": [],
"ref_spans": [
{
"start": 320,
"end": 332,
"text": "(Sec. 4.4.2)",
"ref_id": "FIGREF3"
},
{
"start": 370,
"end": 382,
"text": "(Sec. 4.4.3)",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Enabling the efficient kernel function",
"sec_num": "4.4"
},
{
"text": "The first step of equation 7is to determine the alternative constraints C. We can here strongly use the possibility of dividing tDAGs in two trees. We build C as C \u03c4 \u222a C \u03b3 where: 1) C \u03c4 are the constraints obtained from pairs of isomorphic extended trees t 1 \u2208 S(\u03c4 1 ) and t 2 \u2208 S(\u03c4 2 ); 2) C \u03b3 are the constraints obtained from pairs of isomorphic extended trees t 1 \u2208 S(\u03b3 1 ) and t 2 \u2208 S(\u03b3 2 ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the set of alternative constraints",
"sec_num": "4.4.1"
},
{
"text": "The idea for an efficient algorithm is that we can compute the C without explicitly looking at all the subgraphs involved. We instead use and combine the constraints derived comparing the productions of the extended trees. We can compute then C \u03c4 with the productions of \u03c4 1 and \u03c4 2 and C \u03b3 with the productions of \u03b3 1 and \u03b3 2 . For example (see Fig. 3 ), focusing on the \u03c4 , the rule N P 3 \u2192 N N 2 N N S 3 of G 1 and N P 4 \u2192 N N 5 N N S 4 of G 2 generates the constraint c = {( 3 , 4 ), ( 2 , 5 )}.",
"cite_spans": [],
"ref_spans": [
{
"start": 346,
"end": 352,
"text": "Fig. 3",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Determining the set of alternative constraints",
"sec_num": "4.4.1"
},
{
"text": "Using the above intuition it is possible to define an algorithm that builds an alternative constraint set C with the following two properties:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the set of alternative constraints",
"sec_num": "4.4.1"
},
{
"text": "1. for each common subtree according to a set of constraints c, \u2203c \u2208 C such that c \u2286 c ;",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the set of alternative constraints",
"sec_num": "4.4.1"
},
{
"text": "2. c , c \u2208 C such that c \u2282 c and c = \u2205.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the set of alternative constraints",
"sec_num": "4.4.1"
},
{
"text": "The set C * is defined as the set of all possible intersections of alternative constraints in C. Figure 6 presents the algorithm determining C * . Due to the property (6) discussed in Sec. 4.3, we can empirically demonstrate that the average complexity of the algorithm is not bigger than O(|C| 2 ). Yet, again, the worst case complexity is exponential.",
"cite_spans": [],
"ref_spans": [
{
"start": 97,
"end": 106,
"text": "Figure 6",
"ref_id": null
}
],
"eq_spans": [],
"section": "Determining the set C *",
"sec_num": "4.4.2"
},
{
"text": "The multiplier N (c) (Eq. 8) represents the number of times the constraint c is considered in the sum of equation 5, keeping into account the sign of Algorithm Build the set C * from the set C Figure 6 : Algorithm for computing C * the corresponding addend. It is possible to demonstrate that:",
"cite_spans": [],
"ref_spans": [
{
"start": 193,
"end": 201,
"text": "Figure 6",
"ref_id": null
}
],
"eq_spans": [],
"section": "Determining the values of N (c)",
"sec_num": "4.4.3"
},
{
"text": "C + \u2190 C ; C 1 \u2190 C ; C 2 \u2190 \u2205 WHILE |C 1 | > 1 FORALL c \u2208 C 1 FORALL c \u2208 C 1 such that c = c c \u2190 c \u2229 c IF c / \u2208 C + add c to C 2 C + \u2190 C + \u222a C 2 ; C 1 \u2190 C 2 ; C 2 \u2190 \u2205 C * \u2190 C \u222a C + \u222a {\u2205}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the values of N (c)",
"sec_num": "4.4.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "N (c) = 1 \u2212 c \u2208C * c \u2283c N c",
"eq_num": "(9)"
}
],
"section": "Determining the values of N (c)",
"sec_num": "4.4.3"
},
{
"text": "This recursive formulation of the equation allows us to easily determine the value of N (c) for every c belonging to C * . It is possible to prove this property using set properties and the binomial theorem. The proof is omitted for lack of space.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Determining the values of N (c)",
"sec_num": "4.4.3"
},
{
"text": "To understand if ours is an efficient algorithm, we compare it with the algorithm presented by (Moschitti and Zanzotto, 2007) . We will hereafter call this algorithm K max . The K max algorithm and kernel is an approximation of what is a kernel needed for a FOR space as it is not difficult to demonstrate that",
"cite_spans": [
{
"start": 95,
"end": 125,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "K max (G 1 , G 2 ) \u2264 K(G 1 , G 2 ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "The K max approximation is based on maximization over the set of possible correspondences of the placeholders. Following our formulation, this kernel appears as:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "K max (G 1 , G 2 ) = = max c\u2208F (A 1 ,A 2 ) K S (\u03c4 1 , \u03c4 2 , c)K S (\u03b3 1 , \u03b3 2 , c) (10)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "where F (A 1 ,A 2 ) are all the possible correspondences between the nodes A 1 and A 2 of the two tDAGs as the one presented in Sec. 4.3. This formulation of the kernel has the worst case complexity of our formulation, i.e., Eq. 7. For computing the basic kernel for the extended trees, i.e. K S (\u03b8 1 , \u03b8 2 , c) we use the model algorithm presented by (Zanzotto and Moschitti, 2006) and refined by (Moschitti and Zanzotto, 2007) based on the algorithm for tree fragment feature Figure 7 : Mean execution time in milliseconds (ms) of the two algorithms wrt. n \u00d7 m where n and m are the number of placeholders of the two tDAGs spaces (Collins and Duffy, 2002) . As we are using the same basic kernel, we can empirically compare the two methods.",
"cite_spans": [
{
"start": 366,
"end": 382,
"text": "Moschitti, 2006)",
"ref_id": "BIBREF21"
},
{
"start": 398,
"end": 428,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
},
{
"start": 632,
"end": 657,
"text": "(Collins and Duffy, 2002)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [
{
"start": 478,
"end": 486,
"text": "Figure 7",
"ref_id": null
}
],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "K(G 1 , G 2 ) K max (G 1 , G 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Reviewing the strictly related work",
"sec_num": "4.5"
},
{
"text": "In this section we want to empirically estimate the benefits on the computational cost of our novel algorithm with respect to the algorithm proposed by (Moschitti and Zanzotto, 2007) . Our algorithm is in principle exponential with respect to the set of alternative constraints C. Yet, due to what presented in Sec. 4.4 and as the set C * is usually very small, the average complexity is extremely low. Following the theory on the average-cost computational complexity (Wang, 1997) , we estimated the behavior of the algorithms on a large distribution of cases. We then compared the computing times of the two algorithms. Finally, as K and K max compute slightly different kernels, we compare the accuracy of the two methods. We implemented both algorithms K(G 1 , G 2 ) and K max (G 1 , G 2 ) in support vector machine classifier (Joachims, 1999) and we experimented with both implementations on the same machine. We hereafter analyze the results in term of execution time (Sec. 5.1) and in term of accuracy (Sec. 5.2).",
"cite_spans": [
{
"start": 152,
"end": 182,
"text": "(Moschitti and Zanzotto, 2007)",
"ref_id": "BIBREF13"
},
{
"start": 469,
"end": 481,
"text": "(Wang, 1997)",
"ref_id": "BIBREF20"
},
{
"start": 831,
"end": 847,
"text": "(Joachims, 1999)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Experimental evaluation",
"sec_num": "5"
},
{
"text": "For this first set of experiments, the source of examples is the one of the recognizing textual entailment challenge, i.e., RTE2 (Bar-Haim et al., Figure 8 : Total execution time in seconds (s) of the training phase on RTE2 wrt. different numbers of allowed placeholders 2006). The dataset of the challenge has 1,600 sentence pairs.",
"cite_spans": [],
"ref_spans": [
{
"start": 147,
"end": 155,
"text": "Figure 8",
"ref_id": null
}
],
"eq_spans": [],
"section": "Average computing time analysis",
"sec_num": "5.1"
},
{
"text": "K(G 1 , G 2 ) K max (G 1 , G 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Average computing time analysis",
"sec_num": "5.1"
},
{
"text": "The computational cost of both K(G 1 , G 2 ) and K max (G 1 , G 2 ) depends on the number of placeholders n = |A 1 | of G 1 and on m = |A 2 | the number of placeholders of G 2 . Then, in the first experiment we want to determine the relation between the computational time and the factor n\u00d7m. Results are reported in Fig. 7 where the computation times are plotted with respect to n \u00d7 m. Each point in the curve represents the average execution time for the pairs of instances having n \u00d7 m placeholders. As expected, the computation of the function K is more efficient than the computation K max . The difference between the two execution times increases with n \u00d7 m.",
"cite_spans": [],
"ref_spans": [
{
"start": 317,
"end": 323,
"text": "Fig. 7",
"ref_id": null
}
],
"eq_spans": [],
"section": "Average computing time analysis",
"sec_num": "5.1"
},
{
"text": "We then performed a second experiment that wants to determine the relation of the total execution with the maximum number of placeholders in the examples. This is useful to estimate the behavior of the algorithm with respect to its application in learning models. Using the RTE2 data, we artificially build different versions with increasing number of placeholders. We then have RTE2 with 1 placeholder at most in each pair, RTE2 with 2 placeholders, etc. The number of pairs in each set is the same. What changes is the maximal number of placeholders. Results are reported in Fig. 8 where the execution time of the training phase in seconds (s) is plotted for each different set. We see that the computation of K max is exponential with respect to the number of placeholders and it becomes intractable after 7 placeholders. The computation of K is instead more flat. This can be explained as the computation of K is related to the real alternative constraints that appears in the dataset. The computation of the kernel K then outperforms the computation of the kernel K max .",
"cite_spans": [],
"ref_spans": [
{
"start": 577,
"end": 583,
"text": "Fig. 8",
"ref_id": null
}
],
"eq_spans": [],
"section": "Average computing time analysis",
"sec_num": "5.1"
},
{
"text": "As K max that has been demonstrated very effective in term of accuracy for RTE and K compute a slightly different similarity function, we want to show that the performance of our more computationally efficient K is comparable, and even better, to the performances of K max . We then performed an experiment taking as training all the data derived from RTE1, RTE2, and RTE3, (i.e., 4567 training examples) and taking as testing RTE-4 (i.e., 1000 testing examples). The results are reported in Tab. 1. As the table shows, the accuracy of K is higher than the accuracy of K max . There are two main reasons. The first is that K max is an approximation of K. The second is that we can now consider sentence pairs with more than 7 placeholders. Then, we can use the complete training set as the third column of the table shows.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Accuracy analysis",
"sec_num": "5.2"
},
{
"text": "We presented an interpretation of first order rule feature spaces as tripartite directed acyclic graphs (tDAGs). This view on the problem gave us the possibility of defining a novel and efficient algorithm for computing the kernel function for first order rule feature spaces. Moreover, the resulting algorithm is a valid kernel as it can be written as dot product in the explicit space of the tDAG fragments. We demonstrated that our algorithm outperforms in term of average complexity the previous algorithm and it yields to better accuracies for the final task. We are investigating if this is a valid algorithm for two general directed acyclic graphs.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusions and future work",
"sec_num": "6"
}
],
"back_matter": [],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"title": "The second pascal recognising textual entailment challenge",
"authors": [
{
"first": "Roy",
"middle": [],
"last": "Bar-Haim",
"suffix": ""
},
{
"first": "Ido",
"middle": [],
"last": "Dagan",
"suffix": ""
},
{
"first": "Bill",
"middle": [],
"last": "Dolan",
"suffix": ""
},
{
"first": "Lisa",
"middle": [],
"last": "Ferro",
"suffix": ""
},
{
"first": "Danilo",
"middle": [],
"last": "Giampiccolo",
"suffix": ""
},
{
"first": "Idan",
"middle": [],
"last": "Magnini",
"suffix": ""
},
{
"first": "Bernardo",
"middle": [],
"last": "Szpektor",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the Second PASCAL Challenges Workshop on Recognising Textual Entailment",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Roy Bar-Haim, Ido Dagan, Bill Dolan, Lisa Ferro, Danilo Giampiccolo, and Idan Magnini, Bernardo Szpektor. 2006. The second pascal recog- nising textual entailment challenge. In Proceedings of the Second PASCAL Challenges Workshop on Recognising Textual Entailment. Venice, Italy.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "The Logic of Typed Feature Structures",
"authors": [
{
"first": "Bob",
"middle": [],
"last": "Carpenter",
"suffix": ""
}
],
"year": 1992,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Bob Carpenter. 1992. The Logic of Typed Fea- ture Structures. Cambridge University Press, Cam- bridge, England.",
"links": null
},
"BIBREF2": {
"ref_id": "b2",
"title": "New ranking algorithms for parsing and tagging: Kernels over discrete structures, and the voted perceptron",
"authors": [
{
"first": "Michael",
"middle": [],
"last": "Collins",
"suffix": ""
},
{
"first": "Nigel",
"middle": [],
"last": "Duffy",
"suffix": ""
}
],
"year": 2002,
"venue": "Proceedings of ACL02",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Michael Collins and Nigel Duffy. 2002. New rank- ing algorithms for parsing and tagging: Kernels over discrete structures, and the voted perceptron. In Pro- ceedings of ACL02.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "Support vector networks",
"authors": [
{
"first": "C",
"middle": [],
"last": "Cortes",
"suffix": ""
},
{
"first": "V",
"middle": [],
"last": "Vapnik",
"suffix": ""
}
],
"year": 1995,
"venue": "Machine Learning",
"volume": "20",
"issue": "",
"pages": "1--25",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "C. Cortes and V. Vapnik. 1995. Support vector net- works. Machine Learning, 20:1-25.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Probabilistic textual entailment: Generic applied modeling of language variability",
"authors": [
{
"first": "Ido",
"middle": [],
"last": "Dagan",
"suffix": ""
},
{
"first": "Oren",
"middle": [],
"last": "Glickman",
"suffix": ""
}
],
"year": 2004,
"venue": "Proceedings of the Workshop on Learning Methods for Text Understanding and Mining",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Ido Dagan and Oren Glickman. 2004. Probabilistic textual entailment: Generic applied modeling of lan- guage variability. In Proceedings of the Workshop on Learning Methods for Text Understanding and Mining, Grenoble, France.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "The pascal recognising textual entailment challenge",
"authors": [
{
"first": "Oren",
"middle": [],
"last": "Ido Dagan",
"suffix": ""
},
{
"first": "Bernardo",
"middle": [],
"last": "Glickman",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Magnini",
"suffix": ""
}
],
"year": 2005,
"venue": "",
"volume": "3944",
"issue": "",
"pages": "177--190",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Ido Dagan, Oren Glickman, and Bernardo Magnini. 2006. The pascal recognising textual entailment challenge. In Quionero-Candela et al., editor, LNAI 3944: MLCW 2005, pages 177-190, Milan, Italy. Springer-Verlag.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "Learning to distinguish valid textual entailments",
"authors": [
{
"first": "Marie-Catherine",
"middle": [],
"last": "De Marneffe",
"suffix": ""
},
{
"first": "Bill",
"middle": [],
"last": "Maccartney",
"suffix": ""
},
{
"first": "Trond",
"middle": [],
"last": "Grenager",
"suffix": ""
},
{
"first": "Daniel",
"middle": [],
"last": "Cer",
"suffix": ""
},
{
"first": "Anna",
"middle": [],
"last": "Rafferty",
"suffix": ""
},
{
"first": "Christopher",
"middle": [
"D"
],
"last": "Manning",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the Second PASCAL Challenges Workshop on Recognising Textual Entailment",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Marie-Catherine de Marneffe, Bill MacCartney, Trond Grenager, Daniel Cer, Anna Rafferty, and Christo- pher D. Manning. 2006. Learning to distinguish valid textual entailments. In Proceedings of the Sec- ond PASCAL Challenges Workshop on Recognising Textual Entailment, Venice, Italy.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "Learning non-isomorphic tree mappings for machine translation",
"authors": [
{
"first": "Jason",
"middle": [],
"last": "Eisner",
"suffix": ""
}
],
"year": 2003,
"venue": "Proceedings of the 41st Annual Meeting of the Association for Computational Linguistics (ACL), Companion Volume",
"volume": "",
"issue": "",
"pages": "205--208",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jason Eisner. 2003. Learning non-isomorphic tree mappings for machine translation. In Proceedings of the 41st Annual Meeting of the Association for Computational Linguistics (ACL), Companion Vol- ume, pages 205-208, Sapporo, July.",
"links": null
},
"BIBREF8": {
"ref_id": "b8",
"title": "A survey of kernels for structured data",
"authors": [
{
"first": "Thomas",
"middle": [],
"last": "G\u00e4rtner",
"suffix": ""
}
],
"year": 2003,
"venue": "SIGKDD Explorations",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Thomas G\u00e4rtner. 2003. A survey of kernels for struc- tured data. SIGKDD Explorations.",
"links": null
},
"BIBREF9": {
"ref_id": "b9",
"title": "Robust textual inference via graph matching",
"authors": [
{
"first": "D",
"middle": [],
"last": "Aria",
"suffix": ""
},
{
"first": "Andrew",
"middle": [
"Y"
],
"last": "Haghighi",
"suffix": ""
},
{
"first": "Christopher",
"middle": [
"D"
],
"last": "Ng",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Manning",
"suffix": ""
}
],
"year": 2005,
"venue": "HLT '05: Proceedings of the conference on Human Language Technology and Empirical Methods in Natural Language Processing",
"volume": "",
"issue": "",
"pages": "387--394",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Aria D. Haghighi, Andrew Y. Ng, and Christopher D. Manning. 2005. Robust textual inference via graph matching. In HLT '05: Proceedings of the con- ference on Human Language Technology and Em- pirical Methods in Natural Language Processing, pages 387-394, Morristown, NJ, USA. Association for Computational Linguistics.",
"links": null
},
"BIBREF10": {
"ref_id": "b10",
"title": "Recognizing textual entailment with LCCs GROUND-HOG system",
"authors": [
{
"first": "Andrew",
"middle": [],
"last": "Hickl",
"suffix": ""
},
{
"first": "John",
"middle": [],
"last": "Williams",
"suffix": ""
},
{
"first": "Jeremy",
"middle": [],
"last": "Bensley",
"suffix": ""
},
{
"first": "Kirk",
"middle": [],
"last": "Roberts",
"suffix": ""
},
{
"first": "Bryan",
"middle": [],
"last": "Rink",
"suffix": ""
},
{
"first": "Ying",
"middle": [],
"last": "Shi",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the Second PASCAL Recognizing Textual Entailment Challenge",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andrew Hickl, John Williams, Jeremy Bensley, Kirk Roberts, Bryan Rink, and Ying Shi. 2006. Rec- ognizing textual entailment with LCCs GROUND- HOG system. In Bernardo Magnini and Ido Dagan, editors, Proceedings of the Second PASCAL Recog- nizing Textual Entailment Challenge, Venice, Italy. Springer-Verlag.",
"links": null
},
"BIBREF11": {
"ref_id": "b11",
"title": "Making large-scale svm learning practical",
"authors": [
{
"first": "Thorsten",
"middle": [],
"last": "Joachims",
"suffix": ""
}
],
"year": 1999,
"venue": "Advances in Kernel Methods-Support Vector Learning",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Thorsten Joachims. 1999. Making large-scale svm learning practical. In B. Schlkopf, C. Burges, and A. Smola, editors, Advances in Kernel Methods- Support Vector Learning. MIT Press.",
"links": null
},
"BIBREF12": {
"ref_id": "b12",
"title": "The graph isomorphism problem: its structural complexity",
"authors": [
{
"first": "Johannes",
"middle": [],
"last": "K\u00f6bler",
"suffix": ""
},
{
"first": "Uwe",
"middle": [],
"last": "Sch\u00f6ning",
"suffix": ""
},
{
"first": "Jacobo",
"middle": [],
"last": "Tor\u00e1n",
"suffix": ""
}
],
"year": 1993,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Johannes K\u00f6bler, Uwe Sch\u00f6ning, and Jacobo Tor\u00e1n. 1993. The graph isomorphism problem: its struc- tural complexity. Birkhauser Verlag, Basel, Switzer- land, Switzerland.",
"links": null
},
"BIBREF13": {
"ref_id": "b13",
"title": "Fast and effective kernels for relational learning from texts",
"authors": [
{
"first": "Alessandro",
"middle": [],
"last": "Moschitti",
"suffix": ""
},
{
"first": "Fabio",
"middle": [
"Massimo"
],
"last": "Zanzotto",
"suffix": ""
}
],
"year": 2007,
"venue": "Proceedings of the International Conference of Machine Learning (ICML)",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Alessandro Moschitti and Fabio Massimo Zanzotto. 2007. Fast and effective kernels for relational learn- ing from texts. In Proceedings of the International Conference of Machine Learning (ICML). Corvallis, Oregon.",
"links": null
},
"BIBREF14": {
"ref_id": "b14",
"title": "A study on convolution kernels for shallow semantic parsing",
"authors": [
{
"first": "Alessandro",
"middle": [],
"last": "Moschitti",
"suffix": ""
}
],
"year": 2004,
"venue": "proceedings of the ACL",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Alessandro Moschitti. 2004. A study on convolution kernels for shallow semantic parsing. In proceed- ings of the ACL, Barcelona, Spain.",
"links": null
},
"BIBREF15": {
"ref_id": "b15",
"title": "Head-driven Phrase Structured Grammar",
"authors": [
{
"first": "C",
"middle": [],
"last": "Pollard",
"suffix": ""
},
{
"first": "I",
"middle": [
"A"
],
"last": "Sag",
"suffix": ""
}
],
"year": 1994,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "C. Pollard and I.A. Sag. 1994. Head-driven Phrase Structured Grammar. Chicago CSLI, Stanford.",
"links": null
},
"BIBREF16": {
"ref_id": "b16",
"title": "Robust textual inference using diverse knowledge sources",
"authors": [
{
"first": "Rajat",
"middle": [],
"last": "Raina",
"suffix": ""
},
{
"first": "Aria",
"middle": [],
"last": "Haghighi",
"suffix": ""
},
{
"first": "Christopher",
"middle": [],
"last": "Cox",
"suffix": ""
},
{
"first": "Jenny",
"middle": [],
"last": "Finkel",
"suffix": ""
},
{
"first": "Jeff",
"middle": [],
"last": "Michels",
"suffix": ""
},
{
"first": "Kristina",
"middle": [],
"last": "Toutanova",
"suffix": ""
},
{
"first": "Bill",
"middle": [],
"last": "Mac-Cartney",
"suffix": ""
},
{
"first": "Marie-Catherine",
"middle": [],
"last": "De Marneffe",
"suffix": ""
},
{
"first": "Manning",
"middle": [],
"last": "Christopher",
"suffix": ""
},
{
"first": "Andrew",
"middle": [
"Y"
],
"last": "Ng",
"suffix": ""
}
],
"year": 2005,
"venue": "Proceedings of the 1st Pascal Challenge Workshop",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Rajat Raina, Aria Haghighi, Christopher Cox, Jenny Finkel, Jeff Michels, Kristina Toutanova, Bill Mac- Cartney, Marie-Catherine de Marneffe, Manning Christopher, and Andrew Y. Ng. 2005. Robust tex- tual inference using diverse knowledge sources. In Proceedings of the 1st Pascal Challenge Workshop, Southampton, UK.",
"links": null
},
"BIBREF17": {
"ref_id": "b17",
"title": "Expressivity versus efficiency of graph kernels",
"authors": [
{
"first": "Jan",
"middle": [],
"last": "Ramon",
"suffix": ""
},
{
"first": "Thomas",
"middle": [],
"last": "G\u00e4rtner",
"suffix": ""
}
],
"year": 2003,
"venue": "First International Workshop on Mining Graphs, Trees and Sequences",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jan Ramon and Thomas G\u00e4rtner. 2003. Expressivity versus efficiency of graph kernels. In First Interna- tional Workshop on Mining Graphs, Trees and Se- quences.",
"links": null
},
"BIBREF18": {
"ref_id": "b18",
"title": "Hierarchical directed acyclic graph kernel: Methods for structured natural language data",
"authors": [
{
"first": "Jun",
"middle": [],
"last": "Suzuki",
"suffix": ""
},
{
"first": "Tsutomu",
"middle": [],
"last": "Hirao",
"suffix": ""
},
{
"first": "Yutaka",
"middle": [],
"last": "Sasaki",
"suffix": ""
},
{
"first": "Eisaku",
"middle": [],
"last": "Maeda",
"suffix": ""
}
],
"year": 2003,
"venue": "Proceedings of the 41st Annual Meeting of the Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "32--39",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jun Suzuki, Tsutomu Hirao, Yutaka Sasaki, and Eisaku Maeda. 2003. Hierarchical directed acyclic graph kernel: Methods for structured natural language data. In In Proceedings of the 41st Annual Meet- ing of the Association for Computational Linguis- tics, pages 32-39.",
"links": null
},
"BIBREF19": {
"ref_id": "b19",
"title": "Recognizing textual entailment using a subsequence kernel method",
"authors": [
{
"first": "Rui",
"middle": [],
"last": "Wang",
"suffix": ""
},
{
"first": "G\u00fcnter",
"middle": [],
"last": "Neumann",
"suffix": ""
}
],
"year": 2007,
"venue": "Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence (AAAI-07)",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Rui Wang and G\u00fcnter Neumann. 2007. Recog- nizing textual entailment using a subsequence ker- nel method. In Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence (AAAI- 07), July 22-26, Vancouver, Canada.",
"links": null
},
"BIBREF20": {
"ref_id": "b20",
"title": "Average-case computational complexity theory",
"authors": [
{
"first": "Jie",
"middle": [],
"last": "Wang",
"suffix": ""
}
],
"year": 1997,
"venue": "",
"volume": "",
"issue": "",
"pages": "295--328",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jie Wang. 1997. Average-case computational com- plexity theory. pages 295-328.",
"links": null
},
"BIBREF21": {
"ref_id": "b21",
"title": "Automatic learning of textual entailments with cross-pair similarities",
"authors": [
{
"first": "Fabio",
"middle": [],
"last": "Massimo Zanzotto",
"suffix": ""
},
{
"first": "Alessandro",
"middle": [],
"last": "Moschitti",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the 21st Coling and 44th ACL",
"volume": "",
"issue": "",
"pages": "401--408",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Fabio Massimo Zanzotto and Alessandro Moschitti. 2006. Automatic learning of textual entailments with cross-pair similarities. In Proceedings of the 21st Coling and 44th ACL, pages 401-408. Sydney, Australia, July.",
"links": null
}
},
"ref_entries": {
"FIGREF0": {
"text": "Figure 1: A simple rewrite rule seen as a graph S",
"uris": null,
"type_str": "figure",
"num": null
},
"FIGREF1": {
"text": "Two tripartite DAGs rules defined as tDAGs. Within this space it is possible to define the function S(G) that determines all the possible active features of the tDAG G in F OR. The function S(G) determines all the possible and meaningful subgraphs of G.",
"uris": null,
"type_str": "figure",
"num": null
},
"FIGREF3": {
"text": "Intuitive idea for the kernel computation",
"uris": null,
"type_str": "figure",
"num": null
},
"TABREF1": {
"num": null,
"content": "<table/>",
"text": "Comparative performances of Kmax and K",
"html": null,
"type_str": "table"
}
}
}
} |