File size: 233,652 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 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380 2381 2382 2383 2384 2385 2386 2387 2388 2389 2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428 2429 2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448 2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 2472 2473 2474 2475 2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499 2500 2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533 2534 2535 2536 2537 2538 2539 2540 2541 2542 2543 2544 2545 2546 2547 2548 2549 2550 2551 2552 2553 2554 2555 2556 2557 2558 2559 2560 2561 2562 2563 2564 2565 2566 2567 2568 2569 2570 2571 2572 2573 2574 2575 2576 2577 2578 2579 2580 2581 2582 2583 2584 2585 2586 2587 2588 2589 2590 2591 2592 2593 2594 2595 2596 2597 2598 2599 2600 2601 2602 2603 2604 2605 2606 2607 2608 2609 2610 2611 2612 2613 2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 2748 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 2897 2898 2899 2900 2901 2902 2903 2904 2905 2906 2907 2908 2909 2910 2911 2912 2913 2914 2915 2916 2917 2918 2919 2920 2921 2922 2923 2924 2925 2926 2927 2928 2929 2930 2931 2932 2933 2934 2935 2936 2937 2938 2939 2940 2941 2942 2943 2944 2945 2946 2947 2948 2949 2950 2951 2952 2953 2954 2955 2956 2957 2958 2959 2960 2961 2962 2963 2964 2965 2966 2967 2968 2969 2970 2971 2972 2973 2974 2975 2976 2977 2978 2979 2980 2981 2982 2983 2984 2985 2986 2987 2988 2989 2990 2991 2992 2993 2994 2995 2996 2997 2998 2999 3000 3001 3002 3003 3004 3005 3006 3007 3008 3009 3010 3011 3012 3013 3014 3015 3016 3017 3018 3019 3020 3021 3022 3023 3024 3025 3026 3027 3028 3029 3030 3031 3032 3033 3034 3035 3036 3037 3038 3039 3040 3041 3042 3043 3044 3045 3046 3047 3048 3049 3050 3051 3052 3053 3054 3055 3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068 3069 3070 3071 3072 3073 3074 3075 3076 3077 3078 3079 3080 3081 3082 3083 3084 3085 3086 3087 3088 3089 3090 3091 3092 3093 3094 3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168 3169 3170 3171 3172 3173 3174 3175 3176 3177 3178 3179 3180 3181 3182 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289 3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328 3329 3330 3331 3332 3333 3334 3335 3336 3337 3338 3339 3340 3341 3342 3343 3344 3345 3346 3347 3348 3349 3350 3351 3352 3353 3354 3355 3356 3357 3358 3359 3360 3361 3362 3363 3364 3365 3366 3367 3368 3369 3370 3371 3372 3373 3374 3375 3376 3377 3378 3379 3380 3381 3382 3383 3384 3385 3386 3387 3388 3389 3390 3391 3392 3393 3394 3395 3396 3397 3398 3399 3400 3401 3402 3403 3404 3405 3406 3407 3408 3409 3410 3411 3412 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 3440 3441 3442 3443 3444 3445 3446 3447 3448 3449 3450 3451 3452 3453 3454 3455 3456 3457 3458 3459 3460 3461 3462 3463 3464 3465 3466 3467 3468 3469 3470 3471 3472 3473 3474 3475 3476 3477 3478 3479 3480 3481 3482 3483 3484 3485 3486 3487 3488 3489 3490 3491 3492 3493 3494 3495 3496 3497 3498 3499 3500 3501 3502 3503 3504 3505 3506 3507 3508 3509 3510 3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523 3524 3525 3526 3527 3528 3529 3530 3531 3532 3533 3534 3535 3536 3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 3549 3550 3551 3552 3553 3554 3555 3556 3557 3558 3559 3560 3561 3562 3563 3564 3565 3566 3567 3568 3569 3570 3571 3572 3573 3574 3575 3576 3577 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593 3594 3595 3596 3597 3598 3599 3600 3601 3602 3603 3604 3605 3606 3607 3608 3609 3610 3611 3612 3613 3614 3615 3616 3617 3618 3619 3620 3621 3622 3623 3624 3625 3626 3627 3628 3629 3630 3631 3632 3633 3634 3635 3636 3637 3638 3639 3640 3641 3642 3643 3644 3645 3646 3647 3648 3649 3650 3651 3652 3653 3654 3655 3656 3657 3658 3659 3660 3661 3662 3663 3664 3665 3666 3667 3668 3669 3670 3671 3672 3673 3674 3675 3676 3677 3678 3679 3680 3681 3682 3683 3684 3685 3686 3687 3688 3689 3690 3691 3692 3693 3694 3695 3696 3697 3698 3699 3700 3701 3702 3703 3704 3705 3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3721 3722 3723 3724 3725 3726 3727 3728 3729 3730 3731 3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744 3745 3746 3747 3748 3749 3750 3751 3752 3753 3754 3755 3756 3757 3758 3759 3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770 3771 3772 3773 3774 3775 3776 3777 3778 3779 3780 3781 3782 3783 3784 3785 3786 3787 3788 3789 3790 3791 3792 3793 3794 3795 3796 3797 3798 3799 3800 3801 3802 3803 3804 3805 3806 3807 3808 3809 3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820 3821 3822 3823 3824 3825 3826 3827 3828 3829 3830 3831 3832 3833 3834 3835 3836 3837 3838 3839 3840 3841 3842 3843 3844 3845 3846 3847 3848 3849 3850 3851 3852 3853 3854 3855 3856 3857 3858 3859 3860 3861 3862 3863 3864 3865 3866 3867 3868 3869 3870 3871 3872 3873 3874 3875 3876 3877 3878 3879 3880 3881 3882 3883 3884 3885 3886 3887 3888 3889 3890 3891 3892 3893 3894 3895 3896 3897 3898 3899 3900 3901 3902 3903 3904 3905 3906 3907 3908 3909 3910 3911 3912 3913 3914 3915 3916 3917 3918 3919 3920 3921 3922 3923 3924 3925 3926 3927 3928 3929 3930 3931 3932 3933 3934 3935 3936 3937 3938 3939 3940 3941 3942 3943 3944 3945 3946 3947 3948 3949 3950 3951 3952 3953 3954 3955 3956 3957 3958 3959 3960 3961 3962 3963 3964 3965 3966 3967 3968 3969 3970 3971 3972 3973 3974 3975 3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999 4000 4001 4002 4003 4004 4005 4006 4007 4008 4009 4010 4011 4012 4013 4014 4015 4016 4017 4018 4019 4020 4021 4022 4023 4024 4025 4026 4027 4028 4029 4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047 4048 4049 4050 4051 4052 4053 4054 4055 4056 4057 4058 4059 4060 4061 4062 4063 4064 4065 4066 4067 4068 4069 4070 4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4081 4082 4083 4084 4085 4086 4087 4088 4089 4090 4091 4092 4093 4094 4095 4096 4097 4098 4099 4100 4101 4102 4103 4104 4105 4106 4107 4108 4109 4110 4111 4112 4113 4114 4115 4116 4117 4118 4119 4120 4121 4122 4123 4124 4125 4126 4127 4128 4129 4130 4131 4132 4133 4134 4135 4136 4137 4138 4139 4140 4141 4142 4143 4144 4145 4146 4147 4148 4149 4150 4151 4152 4153 4154 4155 4156 4157 4158 4159 4160 4161 4162 4163 4164 4165 4166 4167 4168 4169 4170 4171 4172 4173 4174 4175 4176 4177 4178 4179 4180 4181 4182 4183 4184 4185 4186 4187 4188 4189 4190 4191 4192 4193 4194 4195 4196 4197 4198 4199 4200 4201 4202 4203 4204 4205 4206 4207 4208 4209 4210 4211 4212 4213 4214 4215 4216 4217 4218 4219 4220 4221 4222 4223 4224 4225 4226 4227 4228 4229 4230 4231 4232 4233 4234 4235 4236 4237 4238 4239 4240 4241 4242 4243 4244 4245 4246 4247 4248 4249 4250 4251 4252 4253 4254 4255 4256 4257 4258 4259 4260 4261 4262 4263 4264 4265 4266 4267 4268 4269 4270 4271 4272 4273 4274 4275 4276 4277 4278 4279 4280 4281 4282 4283 4284 4285 4286 4287 4288 4289 4290 4291 4292 4293 4294 4295 4296 4297 4298 4299 4300 4301 4302 4303 4304 4305 4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 4337 4338 4339 4340 4341 4342 4343 4344 4345 4346 4347 4348 4349 4350 4351 4352 4353 4354 4355 4356 4357 4358 4359 4360 4361 4362 4363 4364 4365 4366 4367 4368 4369 4370 4371 4372 4373 4374 4375 4376 4377 4378 4379 4380 4381 4382 4383 4384 4385 4386 4387 4388 4389 4390 4391 4392 4393 4394 4395 4396 4397 4398 4399 4400 4401 4402 4403 4404 4405 4406 4407 4408 4409 4410 4411 4412 4413 4414 4415 4416 4417 4418 4419 4420 4421 4422 4423 4424 4425 4426 4427 4428 4429 4430 4431 4432 4433 4434 4435 4436 4437 4438 4439 4440 4441 4442 4443 4444 4445 4446 4447 4448 4449 4450 4451 4452 4453 4454 4455 4456 4457 4458 4459 4460 4461 4462 4463 4464 4465 4466 4467 4468 4469 4470 4471 4472 4473 4474 4475 4476 4477 4478 4479 4480 4481 4482 4483 4484 4485 4486 4487 4488 4489 4490 4491 4492 4493 4494 4495 4496 4497 4498 4499 4500 4501 4502 4503 4504 4505 4506 4507 4508 4509 4510 4511 4512 4513 4514 4515 4516 4517 4518 4519 4520 4521 4522 4523 4524 4525 4526 | {
"paper_id": "J13-2004",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T02:18:37.383976Z"
},
"title": "Mildly Non-Projective Dependency Grammar",
"authors": [
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Uppsala University",
"location": {}
},
"email": "marco.kuhlmann@lingfil.uu.se.submission"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "Syntactic representations based on word-to-word dependencies have a long-standing tradition in descriptive linguistics, and receive considerable interest in many applications. Nevertheless, dependency syntax has remained something of an island from a formal point of view. Moreover, most formalisms available for dependency grammar are restricted to projective analyses, and thus not able to support natural accounts of phenomena such as wh-movement and cross-serial dependencies. In this article we present a formalism for non-projective dependency grammar in the framework of linear context-free rewriting systems. A characteristic property of our formalism is a close correspondence between the non-projectivity of the dependency trees admitted by a grammar on the one hand, and the parsing complexity of the grammar on the other. We show that parsing with unrestricted grammars is intractable. We therefore study two constraints on non-projectivity, block-degree and well-nestedness. Jointly, these two constraints define a class of \"mildly\" non-projective dependency grammars that can be parsed in polynomial time. An evaluation on five dependency treebanks shows that these grammars have a good coverage of empirical data.",
"pdf_parse": {
"paper_id": "J13-2004",
"_pdf_hash": "",
"abstract": [
{
"text": "Syntactic representations based on word-to-word dependencies have a long-standing tradition in descriptive linguistics, and receive considerable interest in many applications. Nevertheless, dependency syntax has remained something of an island from a formal point of view. Moreover, most formalisms available for dependency grammar are restricted to projective analyses, and thus not able to support natural accounts of phenomena such as wh-movement and cross-serial dependencies. In this article we present a formalism for non-projective dependency grammar in the framework of linear context-free rewriting systems. A characteristic property of our formalism is a close correspondence between the non-projectivity of the dependency trees admitted by a grammar on the one hand, and the parsing complexity of the grammar on the other. We show that parsing with unrestricted grammars is intractable. We therefore study two constraints on non-projectivity, block-degree and well-nestedness. Jointly, these two constraints define a class of \"mildly\" non-projective dependency grammars that can be parsed in polynomial time. An evaluation on five dependency treebanks shows that these grammars have a good coverage of empirical data.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Syntactic representations based on word-to-word dependencies have a long-standing tradition in descriptive linguistics. Since the seminal work of Tesni\u00e8re (1959) , they have become the basis for several linguistic theories, such as Functional Generative Description (Sgall, Haji\u010dov\u00e1, and Panevov\u00e1 1986) , Meaning-Text Theory (Mel'\u010duk 1988) , and Word Grammar (Hudson 2007) . In recent years they have also been used for a wide range of practical applications, such as information extraction, machine translation, and question answering. We ascribe the widespread interest in dependency structures to their intuitive appeal, their conceptual simplicity, and in particular to the availability of accurate and efficient dependency parsers for a wide range of languages (Buchholz and Marsi 2006; Nivre et al. 2007) .",
"cite_spans": [
{
"start": 146,
"end": 161,
"text": "Tesni\u00e8re (1959)",
"ref_id": "BIBREF50"
},
{
"start": 266,
"end": 302,
"text": "(Sgall, Haji\u010dov\u00e1, and Panevov\u00e1 1986)",
"ref_id": "BIBREF46"
},
{
"start": 305,
"end": 339,
"text": "Meaning-Text Theory (Mel'\u010duk 1988)",
"ref_id": null
},
{
"start": 359,
"end": 372,
"text": "(Hudson 2007)",
"ref_id": "BIBREF20"
},
{
"start": 766,
"end": 791,
"text": "(Buchholz and Marsi 2006;",
"ref_id": "BIBREF7"
},
{
"start": 792,
"end": 810,
"text": "Nivre et al. 2007)",
"ref_id": "BIBREF39"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "Although there exist both a considerable practical interest and an extensive linguistic literature, dependency syntax has remained something of an island from a formal point of view. In particular, there are relatively few results that bridge between dependency syntax and other traditions, such as phrase structure or categorial syntax. This makes it hard to gauge the similarities and differences between the paradigms, and hampers the exchange of linguistic resources and computational methods. An overarching goal of this article is to bring dependency grammar closer to the mainland of formal study.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "One of the few bridging results for dependency grammar is thanks to Gaifman (1965) , who studied a formalism that we will refer to as Hays-Gaifman grammar, and proved it to be weakly equivalent to context-free phrase structure grammar. Although this result is of fundamental importance from a theoretical point of view, its practical usefulness is limited. In particular, Hays-Gaifman grammar is restricted to projective dependency structures, which is similar to the familiar restriction to contiguous constituents. Yet, non-projective dependencies naturally arise in the analysis of natural language. One classic example of this is the phenomenon of cross-serial dependencies in Dutch. In this language, the nominal arguments of verbs that also select an infinitival complement occur in the same order as the verbs themselves: (i) dat Jan 1 Piet 2 Marie 3 zag 1 helpen 2 lezen 3 (Dutch) that Jan Piet Marie saw help read 'that Jan saw Piet help Marie read'",
"cite_spans": [
{
"start": 68,
"end": 82,
"text": "Gaifman (1965)",
"ref_id": "BIBREF12"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "In German, the order of the nominal arguments instead inverts the verb order:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "(ii) dass Jan 1 Piet 2 Marie 3 lesen 3 helfen 2 sah 1 (German) that Jan Piet Marie read help saw Figure 1 shows dependency trees for the two examples. 1 The German linearization gives rise to a projective structure, where the verb-argument dependencies are nested within each other, whereas the Dutch linearization induces a non-projective structure with crossing edges. To account for such structures we need to turn to formalisms more expressive than Hays-Gaifman grammars. In this article we present a formalism for non-projective dependency grammar based on linear context-free rewriting systems (LCFRSs) (Vijay-Shanker, Weir, and Joshi 1987; Weir 1988) . This framework was introduced to facilitate the comparison of various grammar formalisms, including standard context-free grammar, tree-adjoining grammar (Joshi and Schabes 1997) , and combinatory categorial grammar (Steedman and Baldridge 2011) . It also comprises, among others, multiple context-free grammars (Seki et al. 1991) , minimalist grammars (Michaelis 1998) , and simple range concatenation grammars (Boullier 2004) .",
"cite_spans": [
{
"start": 151,
"end": 152,
"text": "1",
"ref_id": null
},
{
"start": 453,
"end": 475,
"text": "Hays-Gaifman grammars.",
"ref_id": null
},
{
"start": 625,
"end": 646,
"text": "Weir, and Joshi 1987;",
"ref_id": "BIBREF51"
},
{
"start": 647,
"end": 657,
"text": "Weir 1988)",
"ref_id": "BIBREF53"
},
{
"start": 814,
"end": 838,
"text": "(Joshi and Schabes 1997)",
"ref_id": "BIBREF22"
},
{
"start": 876,
"end": 905,
"text": "(Steedman and Baldridge 2011)",
"ref_id": "BIBREF49"
},
{
"start": 972,
"end": 990,
"text": "(Seki et al. 1991)",
"ref_id": null
},
{
"start": 1013,
"end": 1029,
"text": "(Michaelis 1998)",
"ref_id": null
},
{
"start": 1072,
"end": 1087,
"text": "(Boullier 2004)",
"ref_id": "BIBREF5"
}
],
"ref_spans": [
{
"start": 97,
"end": 105,
"text": "Figure 1",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "The article is structured as follows. In Section 2 we provide the technical background to our work; in particular, we introduce our terminology and notation for linear context-free rewriting systems. An LCFRS generates a set of terms (formal expressions) which are interpreted as derivation trees of objects from some domain. Each term also has a secondary interpretation under which it denotes a tuple of strings, representing the string yield of the derived object. In Section 3 we introduce the central notion of a lexicalized linear context-free rewriting system, which is an LCFRS in which each rule of the grammar is associated with an overt lexical item, representing a syntactic head (cf. Schabes, Abeill\u00e9, and Joshi 1988 and Schabes 1990) . We show that this property gives rise to an additional interpretation under which each term denotes a dependency tree on its yield. With this interpretation, lexicalized LCFRSs can be used as dependency grammars.",
"cite_spans": [
{
"start": 697,
"end": 718,
"text": "Schabes, Abeill\u00e9, and",
"ref_id": "BIBREF45"
},
{
"start": 719,
"end": 733,
"text": "Joshi 1988 and",
"ref_id": "BIBREF45"
},
{
"start": 734,
"end": 747,
"text": "Schabes 1990)",
"ref_id": "BIBREF45"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "In Section 4 we show how to acquire lexicalized LCFRSs from dependency treebanks. This works in much the same way as the extraction of context-free grammars from phrase structure treebanks (cf. Charniak 1996) , except that the derivation trees of dependency trees are not immediately accessible in the treebank. We therefore present an efficient algorithm for computing a canonical derivation tree for an input dependency tree; from this derivation tree, the rules of the grammar can be extracted in a straightforward way. The algorithm was originally published by Kuhlmann and Satta (2009) . It produces a restricted type of lexicalized LCFRS that we call \"canonical.\" In Section 5 we provide a declarative characterization of this class of grammars, and show that every lexicalized LCFRS is (strongly) equivalent to a canonical one, in the sense that it induces the same set of dependency trees.",
"cite_spans": [
{
"start": 194,
"end": 208,
"text": "Charniak 1996)",
"ref_id": "BIBREF9"
},
{
"start": 565,
"end": 590,
"text": "Kuhlmann and Satta (2009)",
"ref_id": "BIBREF30"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "In Section 6 we present a simple parsing algorithm for LCFRSs. Although the runtime of this algorithm is polynomial in the length of the sentence, the degree of the polynomial depends on two grammar-specific measures called fan-out and rank. We show that even in the restricted case of canonical grammars, parsing is an NPhard problem. It is important therefore to keep the fan-out and the rank of a grammar as low as possible, and much of the recent work on LCFRSs has been devoted to the development of techniques that optimize parsing complexity in various scenarios G\u00f3mez-Rodr\u00edguez and Satta 2009; G\u00f3mez-Rodr\u00edguez et al. 2009; Kuhlmann and Satta 2009; Gildea 2010; G\u00f3mez-Rodr\u00edguez, Kuhlmann, and Satta 2010; Sagot and Satta 2010; and Crescenzi et al. 2011) .",
"cite_spans": [
{
"start": 570,
"end": 601,
"text": "G\u00f3mez-Rodr\u00edguez and Satta 2009;",
"ref_id": "BIBREF15"
},
{
"start": 602,
"end": 630,
"text": "G\u00f3mez-Rodr\u00edguez et al. 2009;",
"ref_id": "BIBREF15"
},
{
"start": 631,
"end": 655,
"text": "Kuhlmann and Satta 2009;",
"ref_id": "BIBREF30"
},
{
"start": 656,
"end": 668,
"text": "Gildea 2010;",
"ref_id": "BIBREF13"
},
{
"start": 669,
"end": 711,
"text": "G\u00f3mez-Rodr\u00edguez, Kuhlmann, and Satta 2010;",
"ref_id": "BIBREF15"
},
{
"start": 712,
"end": 733,
"text": "Sagot and Satta 2010;",
"ref_id": "BIBREF43"
},
{
"start": 734,
"end": 760,
"text": "and Crescenzi et al. 2011)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "In this article we explore the impact of non-projectivity on parsing complexity. In Section 7 we present the structural correspondent of the fan-out of a lexicalized LCFRS, a measure called block-degree (or gap-degree) (Holan et al. 1998) . Although there is no theoretical upper bound on the block-degree of the dependency trees needed for linguistic analysis, we provide evidence from several dependency treebanks showing that, from a practical point of view, this upper bound can be put at a value of as low as 2. In Section 8 we study a second constraint on non-projectivity called well-nestedness (Bodirsky, Kuhlmann, and M\u00f6hl 2005) , and show that its presence facilitates tractable parsing. This comes at the cost of a small loss in coverage on treebank data. Bounded block-degree and well-nestedness jointly define a class of \"mildly\" non-projective dependency grammars that can be parsed in polynomial time.",
"cite_spans": [
{
"start": 219,
"end": 238,
"text": "(Holan et al. 1998)",
"ref_id": "BIBREF19"
},
{
"start": 602,
"end": 637,
"text": "(Bodirsky, Kuhlmann, and M\u00f6hl 2005)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "Section 9 summarizes our main contributions and concludes the article.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1."
},
{
"text": "We assume basic familiarity with linear context-free rewriting systems (see, e.g., Vijay-Shanker, Weir, and Joshi 1987 and Weir 1988) and only review the terminology and notation that we use in this article. A linear context-free rewriting system (LCFRS) is a structure G = (N, \u03a3, P, S) where N is a set of nonterminals, \u03a3 is a set of function symbols, P is a finite set of production rules, and S \u2208 N is a distinguished start symbol. Rules take the form",
"cite_spans": [
{
"start": 98,
"end": 107,
"text": "Weir, and",
"ref_id": "BIBREF53"
},
{
"start": 108,
"end": 122,
"text": "Joshi 1987 and",
"ref_id": "BIBREF51"
},
{
"start": 123,
"end": 133,
"text": "Weir 1988)",
"ref_id": "BIBREF53"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "A 0 \u2192 f (A 1 , . . . , A m )",
"eq_num": "( 1 )"
}
],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "where f is a function symbol and the A i are nonterminals. Rules are used for rewriting in the same way as in a context-free grammar, with the function symbols acting as terminals. The outcome of the rewriting process is a set T(G) of terms, tree-formed expressions built from function symbols. Each term is then associated with a string yield, more specifically a tuple of strings. For this, every function symbol f comes with a yield function that specifies how to compute the yield of a term f (t 1 , . . . , t m ) from the yields of its subterms t i . Yield functions are defined by equations",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "f ( x 1,1 , . . . , x 1,k 1 , . . . , x m,1 , . . . , x m,k m ) = \u03b1 1 , . . . , \u03b1 k 0 (2)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "where the tuple on the right-hand side consists of strings over the variables on the left-hand side and some given alphabet of yield symbols, and contains exactly one occurrence of each variable. For a yield function f defined by an equation of this form, we say that f is of type",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "k 1 \u2022 \u2022 \u2022 k m \u2192 k 0 , denoted by f : k 1 \u2022 \u2022 \u2022 k m \u2192 k 0 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "To guarantee that the string yield of a term is well-defined, each nonterminal A is associated with a fan-out \u03d5(A) \u2265 1, and it is required that for every rule (1),",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "f : \u03d5(A 1 ) \u2022 \u2022 \u2022 \u03d5(A m ) \u2192 \u03d5(A 0 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "In Equation 2, the values m and k 0 are called the rank and the fan-out of f , respectively. The rank and the fan-out of an LCFRS are the maximal rank and fan-out of its yield functions. Figure 2 shows an example of an LCFRS for the language { a n b n c n d",
"cite_spans": [],
"ref_spans": [
{
"start": 187,
"end": 195,
"text": "Figure 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Technical Background",
"sec_num": "2."
},
{
"text": "n | n \u2265 0 }.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 1",
"sec_num": null
},
{
"text": "Equation 2is uniquely determined by the tuple on the right-hand side of the equation. We call this tuple the template of the yield function f , and use it as the canonical function symbol for f . This gives rise to a compact notation for LCFRSs,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 1",
"sec_num": null
},
{
"text": "An LCFRS that generates the yield language { a n b n c n d n | n \u2265 0 }. illustrated in the right column of Figure 2 . In this notation, to save some subscripts, we use the following shorthands for variables: x and x 1 for x 1,1 ; x 2 for x 1,2 ; x 3 for x 1,3 ; y and y 1 for x 2,1 ; y 2 for x 2,2 ; y 3 for x 2,3 .",
"cite_spans": [],
"ref_spans": [
{
"start": 107,
"end": 115,
"text": "Figure 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Figure 2",
"sec_num": null
},
{
"text": "Recall the following examples for verb-argument dependencies in German and Dutch from Section 1: (iii) dass Jan 1 Piet 2 Marie 3 lesen 3 helfen 2 sah 1 (German) that Jan Piet Marie read help saw (iv) dat Jan 1 Piet 2 Marie 3 zag 1 helpen 2 lezen 3 (Dutch) that Jan Piet Marie saw help read 'that Jan saw Piet help Marie read' Figure 3 shows the production rules of two linear context-free rewriting systems (one for German, one for Dutch) that generate these examples. The grammars are lexicalized in the sense that each of their yield functions is associated with a lexical item, such as sah or zag (cf. Schabes, Abeill\u00e9, and Joshi 1988 and Schabes 1990) . Productions with lexicalized yield functions can be read as dependency rules. For example, the rules",
"cite_spans": [
{
"start": 605,
"end": 626,
"text": "Schabes, Abeill\u00e9, and",
"ref_id": "BIBREF45"
},
{
"start": 627,
"end": 641,
"text": "Joshi 1988 and",
"ref_id": "BIBREF45"
},
{
"start": 642,
"end": 655,
"text": "Schabes 1990)",
"ref_id": "BIBREF45"
}
],
"ref_spans": [
{
"start": 326,
"end": 334,
"text": "Figure 3",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Lexicalized LCFRSs as Dependency Grammars",
"sec_num": "3."
},
{
"text": "V \u2192 x y sah (N, V) (German) V \u2192 x y 1 zag y 2 (N, V) (Dutch)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lexicalized LCFRSs as Dependency Grammars",
"sec_num": "3."
},
{
"text": "can be read as stating that the verb to see requires two dependents, one noun (N) and one verb (V). Based on this reading, every term generated by a lexicalized LCFRS does not only yield a tuple of strings, but also induces a dependency tree on these strings: Each parent-child relation in the term represents a dependency between the associated lexical items (cf. Rambow and Joshi 1997) . Thus every lexicalized LCFRS can be reinterpreted as a dependency grammar. To illustrate the idea, Figure 4 shows (the tree representations of) two terms generated by the grammars G 1 and G 2 , together with the dependency trees induced by them. Note that these are the same trees that we gave for (iii) and (iv) in Figure 1 . Our goal for the remainder of this section is to make the notion of induction formally precise. To this end we will reinterpret the yield functions of lexicalized LCFRSs as operations on dependency trees. ",
"cite_spans": [
{
"start": 365,
"end": 387,
"text": "Rambow and Joshi 1997)",
"ref_id": "BIBREF41"
}
],
"ref_spans": [
{
"start": 489,
"end": 497,
"text": "Figure 4",
"ref_id": "FIGREF2"
},
{
"start": 706,
"end": 714,
"text": "Figure 1",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Lexicalized LCFRSs as Dependency Grammars",
"sec_num": "3."
},
{
"text": "By a dependency tree, we mean a pair ( w, D), where w is a tuple of strings, and D is a tree-shaped graph whose nodes correspond to the occurrences of symbols in w, and whose edges represent dependency relations between these occurrences. We identify occurrences in w by pairs (i, j) of integers, where i indexes the component of w that contains the occurrence, and j specifies the linear position of the occurrence within that component. We can then formally define a dependency graph for a tuple of strings",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "w = a 1,1 \u2022 \u2022 \u2022 a 1,n 1 , . . . , a k,1 \u2022 \u2022 \u2022 a k,n k as a directed graph G = (V, E) where V = { (i, j) | 1 \u2264 i \u2264 k, 1 \u2264 j \u2264 n i } and E \u2286 V \u00d7 V",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "We use u and v as variables for nodes, and denote edges (u, v) as u \u2192 v. A dependency tree D for w is a dependency graph for w in which there exists a root node r such that for any node u, there is exactly one directed path from r to u. A dependency tree is called simple if w consists of a single string w. In this case, we write the dependency tree as (w, D), and identify occurrences by their linear positions j in w, with 1 \u2264 j \u2264 |w|.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "Example 2 Figure 5 shows examples of dependency trees. In pictures of such structures we use dashed boxes to group nodes that correspond to occurrences from the same tuple component; however, we usually omit the box when there is only one component.",
"cite_spans": [],
"ref_spans": [
{
"start": 10,
"end": 18,
"text": "Figure 5",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "Writing D i as D i = (V i , E i )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "we have:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "V 1 = {(1, 1)} E 1 = {} V 2 = {(1, 1), (1, 2)} E 2 = {(1, 1) \u2192 (1, 2)} V 3 = {(1, 1), (2, 1)} E 3 = {(1, 1) \u2192 (2, 1)} V 4 = {(1, 1), (3, 1)} E 4 = {(1, 1) \u2192 (3, 1)}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "We use standard terminology from graph theory for dependency trees and the relations between their nodes. In particular, for a node u, the set of descendants of u, which we denote by u , is the set of nodes that can be reached from u by following a directed path consisting of zero or more edges. We write u < v to express that the node u precedes the node v when reading the yield from left to right. Formally, precedence is the lexicographical order on occurrences:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "(i 1 , j 1 ) < (i 2 , j 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "if and only if either",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "i 1 < i 2 or (i 1 = i 2 and j 1 < j 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dependency Trees",
"sec_num": "3.1"
},
{
"text": "A yield function f is called lexicalized if its template contains exactly one yield symbol, representing a lexical item; this symbol is then called the anchor of f . With every lexicalized yield function f we associate an operation f on dependency trees as follows. Let w 1 , . . . , w m , w be tuples of strings such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Operations on Dependency Trees",
"sec_num": "3.2"
},
{
"text": "f ( w 1 , . . . , w m ) = w",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Operations on Dependency Trees",
"sec_num": "3.2"
},
{
"text": "and let D i be a dependency tree for w i . By the definition of yield functions, every occurrence u in an input tuple w i corresponds to exactly one occurrence in the output tuple w; we denote this occurrence by\u016b. Let G be the dependency graph for w that has an edge\u016b \u2192v whenever there is an edge u \u2192 v in some D i , and no other edges. Because f is lexicalized, there is exactly one occurrence r in the output tuple w that does not correspond to any occurrence in some w i ; this is the occurrence of the anchor of f . Let D be the dependency tree for w that is obtained by adding to the graph G all edges of the form r \u2192r i , where r i is the root node of D i . By this construction, the occurrence r of the anchor becomes the root node of D, and the root nodes of the input dependency trees D i become its dependents. We then define",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Operations on Dependency Trees",
"sec_num": "3.2"
},
{
"text": "f (( w 1 , D 1 ), . . . , ( w m , D m )) = ( w, D) Figure 6",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Operations on Dependency Trees",
"sec_num": "3.2"
},
{
"text": "Operations on dependency trees.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Operations on Dependency Trees",
"sec_num": "3.2"
},
{
"text": "We consider a concrete application of an operation on dependency trees, illustrated in Figure 6 . In this example we have ",
"cite_spans": [],
"ref_spans": [
{
"start": 87,
"end": 95,
"text": "Figure 6",
"ref_id": null
}
],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "f = x 1 b, y x 2 w 1 = a, e w 2 = c d w = f ( w 1 , w 2 ) = a b,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "D 1 = ({(1, 1), (2, 1)}, {(1, 1) \u2192 (2, 1)}) D 2 = ({(1, 1), (1, 2)}, {(1, 1) \u2192 (1, 2)})",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "We show that f",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "(( w 1 , D 1 ), ( w 2 , D 2 )) = ( w, D), where D = (V, E) with V = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)} E = {(1, 1) \u2192 (2, 3), (1, 2) \u2192 (1, 1), (1, 2) \u2192 (2, 1), (2, 1) \u2192 (2, 2)}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "The correspondences between the occurrences u in the input tuples and the occurrences\u016b in the output tuple are as follows:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "for w 1 : (1, 1) = (1, 1) , (2, 1) = (2, 3) for w 2 : (1, 1) = (2, 1) , (1, 2) = (2, 2)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "By copying the edges from the input dependency trees, we obtain the intermediate dependency graph",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "G = (V, E ) for w, where E = {(1, 1) \u2192 (2, 3), (2, 1) \u2192 (2, 2)}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "The occurrence r of the anchor b of f in w is (1, 2); the nodes of G that correspond to the root nodes of D 1 and D 2 arer 1 = (1, 1) andr 2 = (2, 1). The dependency tree D is obtained by adding the edges r \u2192r 1 and r \u2192r 2 to G.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 3",
"sec_num": null
},
{
"text": "We now show how to extract lexicalized linear context-free rewriting systems from dependency treebanks. To this end, we adapt the standard technique for extracting context-free grammars from phrase structure treebanks (Charniak 1996) . Our technique was originally published by Kuhlmann and Satta (2009) . In recent work, Maier and Lichte (2011) have shown how to unify it with a similar technique for the extraction of range concatenation grammars from discontinuous constituent structures, due to Maier and S\u00f8gaard (2008) . To simplify our presentation we restrict our attention to treebanks containing simple dependency trees. A dependency tree and one of its construction trees.",
"cite_spans": [
{
"start": 218,
"end": 233,
"text": "(Charniak 1996)",
"ref_id": "BIBREF9"
},
{
"start": 278,
"end": 303,
"text": "Kuhlmann and Satta (2009)",
"ref_id": "BIBREF30"
},
{
"start": 322,
"end": 345,
"text": "Maier and Lichte (2011)",
"ref_id": "BIBREF34"
},
{
"start": 499,
"end": 523,
"text": "Maier and S\u00f8gaard (2008)",
"ref_id": "BIBREF35"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extraction of Dependency Grammars",
"sec_num": "4."
},
{
"text": "To extract a lexicalized LCFRS from a dependency treebank we proceed as follows. First, for each dependency tree (w, D) in the treebank, we compute a construction tree, a term t over yield functions that induces (w, D). Then we collect a set of production rules, one rule for each node of the construction trees. As an example, consider Figure 7 , which shows a dependency tree with one of its construction trees. (The analysis is taken from K\u00fcbler, McDonald, and Nivre [2009] .) From this construction tree we extract the following rules. The nonterminals (in bold) represent linear positions of nodes.",
"cite_spans": [
{
"start": 337,
"end": 345,
"text": "Figure 7",
"ref_id": null
},
{
"start": 442,
"end": 476,
"text": "K\u00fcbler, McDonald, and Nivre [2009]",
"ref_id": "BIBREF29"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extraction of Dependency Grammars",
"sec_num": "4."
},
{
"text": "1 \u2192 A 5 \u2192 on x (7) 2 \u2192 x hearing, y (1, 5) 6 \u2192 the 3 \u2192 x 1 is y 1 x 2 y 2 (2, 4) 7 \u2192 x issue (6) 4 \u2192 scheduled, x (8) 8 \u2192 today",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extraction of Dependency Grammars",
"sec_num": "4."
},
{
"text": "Rules like these can serve as the starting point for practical systems for data-driven, non-projective dependency parsing (Maier and Kallmeyer 2010) . Because the extraction of rules from construction trees is straightforward, the problem that we focus on in this section is how to obtain these trees in the first place. Our procedure for computing construction trees is based on the concept of \"blocks.\"",
"cite_spans": [
{
"start": 122,
"end": 148,
"text": "(Maier and Kallmeyer 2010)",
"ref_id": "BIBREF33"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extraction of Dependency Grammars",
"sec_num": "4."
},
{
"text": "Let D be a dependency tree. A segment of D is a contiguous, non-empty sequence of nodes of D, all of which belong to the same component of the string yield. Thus a segment contains its endpoints, as well as all nodes between the endpoints in the precedence order. For a node u of D, a block of u is a longest segment consisting of descendants of u. This means that the left endpoint of a block of u either is the first node in its component, or is preceded by a node that is not a descendant of u. A symmetric property holds for the right endpoint.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Blocks",
"sec_num": "4.1"
},
{
"text": "Consider the node 2 of the dependency tree in Figure 7 . The descendants of 2 fall into two blocks, marked by the dashed boxes: 1 2 and 5 6 7.",
"cite_spans": [],
"ref_spans": [
{
"start": 46,
"end": 54,
"text": "Figure 7",
"ref_id": "FIGREF4"
}
],
"eq_spans": [],
"section": "Example 4",
"sec_num": null
},
{
"text": "We use u and v as variables for blocks. Extending the precedence order on nodes, we say that a block u precedes a block v, denoted by u < v, if the right endpoint of u precedes the left endpoint of v.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 4",
"sec_num": null
},
{
"text": "To obtain a canonical construction tree t for a dependency tree (w, D) we label each node u of D with a yield function f as follows. Let w be the tuple consisting of the blocks of u, in the order of their precedence, and let w 1 , . . . , w m be the corresponding tuples for the children of u. We may view blocks as strings of nodes. Taking this view, we compute the (unique) yield function g with the property that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Computing Canonical Construction Trees",
"sec_num": "4.2"
},
{
"text": "g( w 1 , . . . , w m ) = w",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Computing Canonical Construction Trees",
"sec_num": "4.2"
},
{
"text": "The anchor of g is the node u, the rank of g corresponds to the number of children of u, the variables in the template of g represent the blocks of these children, and the components of the template represent the blocks of u. To obtain f , we take the template of g and replace the occurrence of u with the corresponding lexical item.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Computing Canonical Construction Trees",
"sec_num": "4.2"
},
{
"text": "Node 2 of the dependency tree shown in Figure 7 has two children, 1 and 5. We have w = 1 2, 5 6 7",
"cite_spans": [],
"ref_spans": [
{
"start": 39,
"end": 47,
"text": "Figure 7",
"ref_id": "FIGREF4"
}
],
"eq_spans": [],
"section": "Example 5",
"sec_num": null
},
{
"text": "w 1 = 1 w 2 = 5 6 7 g = x 2, y f = x hearing, y",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 5",
"sec_num": null
},
{
"text": "Note that in order to properly define f we need to assume some order on the children of u. The function g (and hence the construction tree t) is unique up to the specific choice of this order. In the following we assume that children are ordered from left to right based on the position of their leftmost descendants.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 5",
"sec_num": null
},
{
"text": "The algorithmically most interesting part of our extraction procedure is the computation of the yield function g. The template of g is uniquely determined by the left-to-right sequence of the endpoints of the blocks of u and its children. An efficient algorithm that can be used to compute these sequences is given in Table 1 .",
"cite_spans": [],
"ref_spans": [
{
"start": 318,
"end": 325,
"text": "Table 1",
"ref_id": "TABREF1"
}
],
"eq_spans": [],
"section": "Computing the Blocks of a Dependency Tree",
"sec_num": "4.3"
},
{
"text": "We start at a virtual root node \u22a5 (line 1) which serves as the parent of the real root node. For each node next in the precedence order of D, we follow the shortest path from the current node current to next. To determine this path, we compute the lowest common ancestor lca of the two nodes (lines 4-5), using a set of markings on the nodes. At the beginning of each iteration of the for loop in line 2, all ancestors of current (including the virtual root node \u22a5) are marked; therefore, we find lca by going upwards from next to the first node that is marked. To restore the loop invariant, we then unmark all nodes on the path from current to lca (lines 6-9). Each time we move down from a node to one of its children (line 12), we record the information that next is the left endpoint of a block of current. Symmetrically, each time we move up from a node to its parent (lines 8 and 17), we record the information that next \u2212 1 is the right endpoint of a block of current. The while loop in lines 15-18 takes us from the last node of the dependency tree back to the node \u22a5. ",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Description.",
"sec_num": "4.3.1"
},
{
"text": "= n 1 + n 2 + n 3 + n 4 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Description.",
"sec_num": "4.3.1"
},
{
"text": "Under the reasonable assumption that every line in Table 1 can be executed in constant time, the runtime of the algorithm clearly is in O(n).",
"cite_spans": [],
"ref_spans": [
{
"start": 51,
"end": 58,
"text": "Table 1",
"ref_id": "TABREF1"
}
],
"eq_spans": [],
"section": "Description.",
"sec_num": "4.3.1"
},
{
"text": "Because each iteration of loop 2 and loop 4 determines the right endpoint of a block, we have n 2 + n 4 = m. Similarly, as each iteration of loop 3 fixes the left endpoint of a block, we have n 3 = m. To determine n 1 , we note that every node that is pushed to the auxiliary stack in loop 1 is popped again in loop 3; therefore, n 1 = n 3 = m. Putting everything together, we have n = 3m, and we conclude that the runtime of the algorithm is in O(m).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Description.",
"sec_num": "4.3.1"
},
{
"text": "Note that this runtime is asymptotically optimal for the task we are considering.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Description.",
"sec_num": "4.3.1"
},
{
"text": "Our extraction technique produces a restricted type of lexicalized linear context-free rewriting system that we will refer to as \"canonical.\" In this section we provide a declarative characterization of these grammars, and show that every lexicalized LCFRS is equivalent to a canonical one.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Canonical Grammars",
"sec_num": "5."
},
{
"text": "We are interested in a syntactic characterization of the yield functions that can occur in extracted grammars. We give such a characterization in terms of four properties, stated in the following. We use the following terminology and notation. Consider a yield function",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition of Canonical Grammars",
"sec_num": "5.1"
},
{
"text": "f : k 1 \u2022 \u2022 \u2022 k m \u2192 k , f = \u03b1 1 , . . . , \u03b1 k",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition of Canonical Grammars",
"sec_num": "5.1"
},
{
"text": "For variables x, y we write x < f y to state that x precedes y in the template of f , that is, in the string \u03b1 1 \u2022 \u2022 \u2022 \u03b1 k . Recall that, in the context of our extraction procedure, the components in the template of f represent the blocks of a node u, and the variables in the template represent the blocks of the children of u. For a variable x i,j we call i the argument index and j the component index of the variable.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition of Canonical Grammars",
"sec_num": "5.1"
},
{
"text": "For all 1 \u2264 i 1 , i 2 \u2264 m, if i 1 < i 2 then x i 1 ,1 < f x i 2 ,1 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 1",
"sec_num": null
},
{
"text": "This property is an artifact of our decision to order the children of a node from left to right based on the position of their leftmost descendants. A variable with argument index i represents a block of the ith child of u in that order. An example of a yield function that does not have Property 1 is x 2,1 x 1,1 , which defines a kind of \"reverse concatenation operation.\"",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 1",
"sec_num": null
},
{
"text": "Property 2 For all 1 \u2264 i \u2264 m and 1 \u2264 j 1 , j 2 \u2264 k i , if j 1 < j 2 then x i,j 1 < f x i,j 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 1",
"sec_num": null
},
{
"text": "This property reflects that, in our extraction procedure, the variable x i,j represents the jth block of the ith child of u, where the blocks of a node are ordered from left to right based on their precedence. An example of a yield function that violates the property is x 1,2 x 1,1 , which defines a kind of swapping operation. In the literature on LCFRSs and related formalisms, yield functions with Property 2 have been called monotone (Michaelis 2001; Kracht 2003) , ordered (Villemonte de la Clergerie 2002; Kallmeyer 2010), and non-permuting (Kanazawa 2009) .",
"cite_spans": [
{
"start": 439,
"end": 455,
"text": "(Michaelis 2001;",
"ref_id": "BIBREF38"
},
{
"start": 456,
"end": 468,
"text": "Kracht 2003)",
"ref_id": "BIBREF27"
},
{
"start": 548,
"end": 563,
"text": "(Kanazawa 2009)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Property 1",
"sec_num": null
},
{
"text": "No component \u03b1 h is the empty string.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 3",
"sec_num": null
},
{
"text": "This property, which is similar to \u03b5-freeness as known from context-free grammars, has been discussed for multiple context-free grammars (Seki et al. 1991, Property N3 in Lemma 2.2) and range concatenation grammars (Boullier 1998 , Section 5.1). For our extracted grammars it holds because each component \u03b1 h represents a block, and blocks are always non-empty.",
"cite_spans": [
{
"start": 215,
"end": 229,
"text": "(Boullier 1998",
"ref_id": "BIBREF4"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Property 3",
"sec_num": null
},
{
"text": "No component \u03b1 h contains a substring of the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 4",
"sec_num": null
},
{
"text": "x i,j 1 x i,j 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 4",
"sec_num": null
},
{
"text": "This property, which does not seem to have been discussed in the literature before, is a reflection of the facts that variables with the same argument index represent blocks of the same child node, and that these blocks are longest segments of descendants.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 4",
"sec_num": null
},
{
"text": "A yield function with Properties 1-4 is called canonical. An LCFRS is canonical if all of its yield functions are canonical.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Property 4",
"sec_num": null
},
{
"text": "A lexicalized LCFRS is canonical if and only if it can be extracted from a dependency treebank using the technique presented in Section 4.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 1",
"sec_num": null
},
{
"text": "We have already argued for the \"only if\" part of the claim. To prove the \"if\" part, it suffices to show that for every canonical, lexicalized yield function f , one can construct a dependency tree such that the construction tree extracted for this dependency tree contains f . This is an easy exercise.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "We conclude by noting that Properties 2-4 are also shared by the treebank grammars extracted from constituency treebanks using the technique by Maier and S\u00f8gaard (2008) .",
"cite_spans": [
{
"start": 144,
"end": 168,
"text": "Maier and S\u00f8gaard (2008)",
"ref_id": "BIBREF35"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "Two lexicalized LCFRSs are called strongly equivalent if they induce the same set of dependency trees. We show the following equivalence result:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Equivalence Between General and Canonical Grammars",
"sec_num": "5.2"
},
{
"text": "For every lexicalized LCFRS G one can construct a strongly equivalent lexicalized LCFRS G such that G is canonical.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 2",
"sec_num": null
},
{
"text": "Our proof of this lemma uses two normal-form results about multiple context-free grammars: Michaelis (2001, Section 2.4) provides a construction that transforms a multiple context-free grammar into a weakly equivalent multiple context-free grammar in which all rules satisfy Property 2, and Seki et al. (1991, Lemma 2 .2) present a corresponding construction for Property 3. Whereas both constructions are only quoted to preserve weak equivalence, we can verify that, in the special case where the input grammar is a lexicalized LCFRS, they also preserve the set of induced dependency trees. To complete the proof of Lemma 2, we show that every lexicalized LCFRS can be cast into normal forms that satisfy Property 1 and Property 4. It is not hard then to combine the four constructions into a single one that simultaneously establishes all properties of canonical yield functions.",
"cite_spans": [
{
"start": 303,
"end": 317,
"text": "(1991, Lemma 2",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "For every lexicalized LCFRS G one can construct a strongly equivalent lexicalized LCFRS G such that G only contains yield functions which satisfy Property 1.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 3",
"sec_num": null
},
{
"text": "The proof is very simple. Intuitively, Property 1 enforces a canonical naming of the arguments of yield functions. To establish it, we determine, for every yield function f , a permutation \u03c0 that renames the argument indices of the variables occurring in the template of f in such a way that the template meets Property 1. This renaming gives rise to a modified yield function f \u03c0 . We then replace every rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "A \u2192 f (A 1 , . . . , A m ) with the modified rule A \u2192 f \u03c0 (A \u03c0(1) , . . . , A \u03c0(m) ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "For every lexicalized LCFRS G one can construct a strongly equivalent lexicalized LCFRS G such that G only contains yield functions which satisfy Property 4.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 4",
"sec_num": null
},
{
"text": "The idea behind our construction of the grammar G is perhaps best illustrated by an example. Imagine that the grammar G generates the term t shown in Figure 8a . The yield function f 1 = x 1 c x 2 x 3 at the root node of that term violates Property 4, as its template contains the offending substring x 2 x 3 . We set up G in such a way that instead of t it generates the term t shown in Figure 8b in which f 1 is replaced with the yield function",
"cite_spans": [],
"ref_spans": [
{
"start": 150,
"end": 159,
"text": "Figure 8a",
"ref_id": null
},
{
"start": 388,
"end": 397,
"text": "Figure 8b",
"ref_id": null
}
],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The transformation implemented by the construction of the grammar G in Lemma 4.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "f 1 = x 1 c x 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "To obtain f 1 from f 1 we reduce the offending substring x 2 x 3 to the single variable x 2 . In order to ensure that t and t induce the same dependency tree (shown in Figure 8c ), we then adapt the function f 2 = x 1 b, y, x 2 at the first child of the root node: Dual to the reduction, we replace the two-component sequence y, x 2 in the template of f 2 with the single component y x 2 ; in this way we get f",
"cite_spans": [],
"ref_spans": [
{
"start": 168,
"end": 177,
"text": "Figure 8c",
"ref_id": null
}
],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "2 = x 1 b, y x 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "Because adaptation operations may introduce new offending substrings, we need a recursive algorithm to compute the rules of the grammar G . Such an algorithm is given in Table 2 . For every rule",
"cite_spans": [],
"ref_spans": [
{
"start": 170,
"end": 177,
"text": "Table 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "A \u2192 f (A 1 , . . . , A m ) of G we construct new rules (A, g) \u2192 f ((A 1 , g 1 ), . . . , (A m , g m ))",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "where g and the g i are yield functions encoding adaptation operations. As an example, the adaptation of the function f 2 in the term t may be encoded into the adaptor function x 1 , x 2 x 3 . The function f 2 can then be written as the composition of this function and f 2 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "f 2 = x 1 , x 2 x 3 \u2022 f 2 = x 1 , x 2 x 3 ( x 1 b, y, x 2 ) = x 1 b, y x 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "The yield function f and the adaptor functions g i are computed based on the template of the g-adapted yield function f , that is, the composed function g \u2022 f . In Table 2 we write this as f = reduce(f, g) and g i = adapt(f, g, i), respectively. Let us denote the template of the adapted function g \u2022 f by \u03c4. An i-block of \u03c4 is a maximal, non-empty substring of some component of \u03c4 that consists of variables with argument index i. To compute the template of g i we read the i-blocks of \u03c4 from left to right and rename the variables by changing their argument indices from i to 1. To compute the template of f we take the Table 2 Computing the production rules of an LCFRS in which all yield functions satisfy Property 4.",
"cite_spans": [],
"ref_spans": [
{
"start": 164,
"end": 171,
"text": "Table 2",
"ref_id": null
},
{
"start": 622,
"end": 629,
"text": "Table 2",
"ref_id": null
}
],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "Input: a linear context-free rewriting system G = (N, \u03a3, P, S) 1: P \u2190 \u2205; agenda \u2190 {(S, x )}; chart \u2190 \u2205 2: while agenda is not empty 3: remove some (A, g) from agenda 4:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "if (A, g) / \u2208 chart then 5:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "add (A, g) to chart 6:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "for each rule A \u2192 f (A 1 , . . . , A m ) \u2208 P do 7: f \u2190 reduce(f, g); g i \u2190 adapt(f, g, i) (1 \u2264 i \u2264 m) 8:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "for each i from 1 to m do 9:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "add (A i , g i ) to agenda 10:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "add (A, g) \u2192 f ((A 1 , g 1 ), . . . , (A m , g m ))",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "to P template \u03c4 and replace the jth i-block with the variable x i,j , for all argument indices i and component indices j.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "Our algorithm is controlled by an agenda and a chart, both containing pairs of the form (A, g) , where A is a nonterminal of G and g is an adaptor function. These pairs also constitute the nonterminals of the new grammar G . The fan-out of a nonterminal is the fan-out of g. The agenda is initialized with the pair (S, x ) where x is the identity function; this pair also represents the start symbol of G . To see that the algorithm terminates, one may observe that the fan-out of every nonterminal (A, g) added to the agenda is upper-bounded by the fan-out of A. Hence, there are only finitely many pairs (A, g) that may occur in the chart, and a finite number of iterations of the while-loop.",
"cite_spans": [],
"ref_spans": [
{
"start": 88,
"end": 94,
"text": "(A, g)",
"ref_id": null
}
],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "We conclude by noting that when constructing a canonical grammar, one needs to be careful about the order in which the individual constructions (for Properties 1-4) are combined. One order that works is Property 3 < Property 4 < Property 2 < Property 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 8",
"sec_num": null
},
{
"text": "Lexicalized linear context-free rewriting systems are able to account for arbitrarily nonprojective dependency trees. This expressiveness comes with a price: In this section we show that parsing with lexicalized LCFRSs is intractable, unless we are willing to restrict the class of grammars.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing and Recognition",
"sec_num": "6."
},
{
"text": "To ground our discussion of parsing complexity, we present a simple bottom-up parsing algorithm for LCFRSs, specified as a grammatical deduction system (Shieber, Schabes, and Pereira 1995) . Several similar algorithms have been described in the literature (Seki et al. 1991; Bertsch and Nederhof 2001; Kallmeyer 2010) . We assume that we are given a grammar G = (N, \u03a3, P, S) and a string w = a 1 \u2022 \u2022 \u2022 a n \u2208 V * to be parsed.",
"cite_spans": [
{
"start": 152,
"end": 188,
"text": "(Shieber, Schabes, and Pereira 1995)",
"ref_id": "BIBREF48"
},
{
"start": 256,
"end": 274,
"text": "(Seki et al. 1991;",
"ref_id": null
},
{
"start": 275,
"end": 301,
"text": "Bertsch and Nederhof 2001;",
"ref_id": "BIBREF1"
},
{
"start": 302,
"end": 317,
"text": "Kallmeyer 2010)",
"ref_id": "BIBREF24"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "Item form. The items of the deduction system take the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "[A, l 1 , r 1 , . . . , l k , r k ]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "where A \u2208 N with \u03d5(A) = k, and the remaining components are indices identifying the left and right endpoints of pairwise non-overlapping substrings of w. More formally, 0 \u2264 l h \u2264 r h \u2264 n, and for all h, h with h = h , either r h \u2264 l h or r h \u2264 l h . The intended interpretation of an item of this form is that A derives a term t \u2208 T(G) that yields the specified substrings of w, that is,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "A \u21d2 * G t and yield(t) = a l 1 +1 \u2022 \u2022 \u2022 a r 1 , . . . , a l k +1 \u2022 \u2022 \u2022 a r k",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "Goal item. The goal item is [S, 0, n] . By this item, there exists a term that can be derived from the start symbol S and yields the full string w .",
"cite_spans": [
{
"start": 28,
"end": 37,
"text": "[S, 0, n]",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "Inference rules. The inference rules of the deduction system are defined based on the rules in P. Each production rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "A \u2192 f (A 1 , . . . , A m ) with f : k 1 \u2022 \u2022 \u2022 k m \u2192 k , f = \u03b1 1 , . . . , \u03b1 k",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "is converted into a set of inference rules of the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "A 1 , l 1,1 , r 1,1 , . . . , l 1,k 1 , r 1,k 1 \u2022 \u2022 \u2022 A m , l m,1 , r m,1 , . . . , l m,k m , r m,k m A, l 1 , r 1 , . . . , l k , r k",
"eq_num": "(3)"
}
],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "Each such rule is subject to the following constraints.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "Let 1 \u2264 h \u2264 k, v \u2208 V * , 1 \u2264 i \u2264 m,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "and 1 \u2264 j \u2264 k i . We write \u03b4(l, v) = r to assert that r = l + |v| and that v is the substring of w between indices l and r.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "If",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u03b1 h = v then \u03b4(l h , v) = r h (c1) If v x i,j is a prefix of \u03b1 h then \u03b4(l h , v) = l i,j",
"eq_num": "(c2)"
}
],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "If",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "x i,j v is a suffix of \u03b1 h then \u03b4(r i,j , v) = r h (c3) If x i,j v x i ,j is an infix of \u03b1 h then \u03b4(r i,j , v) = l i ,j",
"eq_num": "(c4)"
}
],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "These constraints ensure that the substrings corresponding to the premises of the inference rule can be combined into the substrings corresponding to the conclusion by means of the yield function f . Based on the deduction system, a tabular parser for LCFRSs can be implemented using standard dynamic programming techniques. This parser will compute a packed representation of the set of all derivation trees that the grammar G assigns to the string w. Such a packed representation is often called a shared forest (Lang 1994) . In combination with appropriate semirings, the shared forest is useful for many tasks in syntactic analysis and machine learning (Goodman 1999; Li and Eisner 2009).",
"cite_spans": [
{
"start": 514,
"end": 525,
"text": "(Lang 1994)",
"ref_id": "BIBREF31"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Algorithm",
"sec_num": "6.1"
},
{
"text": "We are interested in an upper bound on the runtime of the tabular parser that we have just presented. We can see that the parser runs in time O(|G||w| c ), where |G| denotes the size of some suitable representation of the grammar G, and c denotes the maximal number of instantiations of an inference rule (cf. McAllester 2002) . Let us write c( f ) for the specialization of c to inference rules for productions with yield function f . We refer to this value as the parsing complexity of f (cf. Gildea 2010). Then to show an upper bound on c it suffices to show an upper bound on the parsing complexities of the yield functions that the parser has to handle. An obvious such upper bound is",
"cite_spans": [
{
"start": 305,
"end": 326,
"text": "(cf. McAllester 2002)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Complexity",
"sec_num": "6.2"
},
{
"text": "c( f ) \u2264 2k + m i=1 2k i",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Complexity",
"sec_num": "6.2"
},
{
"text": "Here we imagine that we could choose each endpoint in Equation 3independently of all the others. By virtue of the constraints, however, some of the endpoints cannot be chosen freely; in particular, some of the substrings may be adjacent. In general, to show an upper bound c(f ) \u2264 b we specify a strategy for choosing b endpoints, and then argue that, given the constraints, these choices determine the remaining endpoints.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Parsing Complexity",
"sec_num": "6.2"
},
{
"text": "For a yield function f :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 5",
"sec_num": null
},
{
"text": "k 1 \u2022 \u2022 \u2022 k m \u2192 k we have c( f ) \u2264 k + m i=1 k i",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 5",
"sec_num": null
},
{
"text": "We adopt the following strategy for choosing endpoints: For 1 \u2264 i \u2264 k, choose the value of l h . Then, for 1 \u2264 i \u2264 m and 1 \u2264 j \u2264 k i , choose the value of r i,j . It is not hard to see that these choices suffice to determine all other endpoints. In particular, each left endpoint l i ,j will be shared either with the left endpoint l h of some component (by constraint c2), or with some right endpoint r i,j (by constraint c4).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The runtime of our parsing algorithm for LCFRSs is exponential in both the rank and the fan-out of the input grammar. One may wonder whether there are parsing algorithms that can be substantially faster. We now show that the answer to this question is likely to be negative even if we restrict ourselves to canonical lexicalized LCFRSs. To this end we study the universal recognition problem for this class of grammars.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Universal Recognition",
"sec_num": "6.3"
},
{
"text": "The universal recognition problem for a class of linear context-free rewriting systems is to decide, given a grammar G from the class in question and a string w, whether G yields w . A straightforward algorithm for solving this problem is to first compute the shared forest for G and w, and to return \"yes\" if and only if the shared forest is non-empty. Choosing appropriate data structures, the emptiness of shared forests can be decided in linear time and space with respect to the size of the forest. Therefore, the computational complexity of universal recognition is upper-bounded by the complexity of constructing the shared forest. Conversely, parsing cannot be faster than universal recognition.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Universal Recognition",
"sec_num": "6.3"
},
{
"text": "In the next three lemmas we prove that the universal recognition problem for canonical lexicalized LCFRSs is NP-complete unless we restrict ourselves to a class of grammars where both the fan-out and the rank of the yield functions are bounded by constants. Lemma 6, which shows that the universal recognition problem of lexicalized LCFRSs is in NP, distinguishes lexicalized LCFRSs from general LCFRSs, for which the universal recognition problem is known to be PSPACE-complete (Kaji et al. 1992) . The crucial difference between general and lexicalized LCFRSs is the fact that in the latter, the size of the generated terms is bounded by the length of the input string. Lemma 7 and Lemma 8, which establish two NP-hardness results for lexicalized LCFRSs, are stronger versions of the corresponding results for general LCFRSs presented by Satta (1992) , and are proved using similar reductions. They show that the hardness results hold under significant restrictions of the formalism: to lexicalized form and to canonical yield functions. Note that, whereas in Section 5.2 we have shown that every lexicalized LCFRS is equivalent to a canonical one, the normal form transformation increases the size of the original grammar by a factor that is at least exponential in the fan-out.",
"cite_spans": [
{
"start": 479,
"end": 497,
"text": "(Kaji et al. 1992)",
"ref_id": "BIBREF23"
},
{
"start": 840,
"end": 852,
"text": "Satta (1992)",
"ref_id": "BIBREF44"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Universal Recognition",
"sec_num": "6.3"
},
{
"text": "The universal recognition problem of lexicalized LCFRSs is in NP.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 6",
"sec_num": null
},
{
"text": "Let G be a lexicalized LCFRS, and let w be a string. To test whether G yields w , we guess a term t \u2208 T(G) and check whether t yields w . Let |t| denote the length of some string representation of t. Since the yield functions of G are lexicalized, |t| \u2264 |w||G|. Note that we have",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "|t| \u2264 |w||G| \u2264 |w| 2 + 2|w||G| + |G| 2 = (|w| + |G|) 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "Using a simple tabular algorithm, we can verify in time O(|w||G|) whether a candidate term t belongs to T(G). It is then straightforward to compute the string yield of t in time O(|w||G|). Thus we have a nondeterministic polynomial-time decider for the universal recognition problem.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "For the following two lemmas, recall the decision problem 3SAT, which is known to be NP-complete. An instance of 3SAT is a Boolean formula \u03c6 in conjunctive normal form where each clause contains exactly three literals, which may be either variables or negated variables. We write m for the number of distinct variables that occur in \u03c6, and n for the number of clauses. In the proofs the index i will always range over values from 1 to m, and the index j will range over values from 1 to n.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "In order to make the grammars in the following reductions more readable, we use yield functions with more than one lexical anchor. Our use of these yield functions is severely restricted, however, and each of our grammars can be transformed into a proper lexicalized LCFRS without affecting the correctness or polynomial size of the reductions.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The universal recognition problem for canonical lexicalized LCFRSs with unbounded fan-out and rank 1 is NP-hard.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 7",
"sec_num": null
},
{
"text": "To prove this claim, we provide a polynomial-time reduction of 3SAT. The basic idea is to use the derivations of the grammar to guess truth assignments for the variables, and to use the feature of unbounded fan-out to ensure that the truth assignment satisfies all clauses.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "Let \u03c6 be an instance of 3SAT. We construct a canonical lexicalized LCFRS G and a string w as follows. Let M denote the m \u00d7 n matrix with entries M i,j = (v i , c j ), that is, entries in the same row share the same variable, and entries in the same column share the same clause. We set up G in such a way that each of its derivations simulates a rowwise iteration over M. Before visiting a new row, the derivation chooses a truth value for the corresponding variable, and sticks to that choice until the end of the row. The string w takes the form",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "w = w 1 $ \u2022 \u2022 \u2022 $ w n where w j = c j,1 \u2022 \u2022 \u2022 c j,m c j,1 \u2022 \u2022 \u2022 c j,m",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "This string is built up during the iteration over M in a column-wise fashion, where each column corresponds to one component of a tuple with fan-out n. More specifically, for each entry (v i , c j ), the derivation generates one of two strings, denoted by \u03b3 i,j and\u03b3 i,j :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "\u03b3 i,j = c j,i \u2022 \u2022 \u2022 c j,m c j,1 \u2022 \u2022 \u2022 c j,i\u03b3i,j = c j,i",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The string \u03b3 i,j is generated only if v i can be used to satisfy c j under the hypothesized truth assignment. By this construction, every successful derivation of G represents a truth assignment that satisfies \u03c6. Conversely, using a satisfying truth assignment for \u03c6, we will be able to construct a derivation of G that yields w. To see how the traversal of the matrix M can be implemented by the grammar G, consider the grammar fragment in Figure 9 . Each of the rules specifies one possible step of the iteration for the pair (v i , c j ) under the truth assignment v i = true; rules with lefthand side F i,j (not shown here) specify possible steps under the assignment v i = false.",
"cite_spans": [],
"ref_spans": [
{
"start": 441,
"end": 449,
"text": "Figure 9",
"ref_id": "FIGREF5"
}
],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The universal recognition problem for canonical lexicalized LCFRSs with unbounded rank and fan-out 2 is NP-hard.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 8",
"sec_num": null
},
{
"text": "We provide another polynomial-time reduction of 3SAT to a grammar G and a string w, again based on the matrix M mentioned in the previous proof. Also as in the previous reduction, we set up the grammar G to simulate a row-wise iteration over M. The major difference this time is that the entries of M are not visited during one long rank 1 derivation, but during mn rather short fan-out 2 subderivations. The string w is",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "w = w ,1 \u2022 \u2022 \u2022 w ,m $ w ,1 \u2022 \u2022 \u2022 w ,n",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "where",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "w ,i = a i,1 \u2022 \u2022 \u2022 a i,n b i,1 \u2022 \u2022 \u2022 b i,n and w ,j = c 1,j \u2022 \u2022 \u2022 c m,j c 1,j \u2022 \u2022 \u2022 c m,j",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "During the traversal of M, for each entry (v i , c j ), we generate a tuple consisting of two substrings of w. The right component of the tuple consists of one the two strings \u03b3 i,j and\u03b3 i,j mentioned previously. As before, the string \u03b3 i,j is generated only if v i can be used to satisfy c j under the hypothesized truth assignment. The left component consists of one of two strings, denoted by \u03c3 i,j and\u03c3 i,j :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "\u03c3 i,1 = a i,1 \u2022 \u2022 \u2022 a i,n b i,1 \u03c3 i,j = b i,j (1 < j)\u03c3 i,n = a i,n b i,1 \u2022 \u2022 \u2022 b i,n\u03c3i,j = a i,j (j < n)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "These strings are generated to represent the truth assignments v i = true and v i = false, respectively. By this construction, each substring w ,i can be derived in exactly one of two ways, ensuring a consistent truth assignment for all subderivations that are linked to the same variable v i . The grammar G is defined as follows. There is one rather complex rule to rewrite the start symbol S; this rule sets up the general topology of w. Let I be the m \u00d7 n matrix with entries I i,j = (j \u2212 1)m + i. Define x 1 to be the sequence of variables of the form x h,1 , where the argument index i is taken from a row-wise reading of the matrix I; in this case, the argument indices in x will simply go up from 1 to mn. Now define x 2 to be the sequence of variables of the form x h,2 , where h is taken from a column-wise reading of the matrix I. Then S can be expanded with the rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "S \u2192 x 1 $ x 2 (V 1,1 , . . . , V 1,n , . . . , V m,1 , . . . , V m,n )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "Note that there is one nonterminal V i,j for each variable-clause pair (v i , c j ). These nonterminals can be rewritten using the following rules:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "V i,1 \u2192 \u03c3 i,1 , x (T i,1 ) V i,j \u2192 \u03c3 i,j , x (T i,j ) V i,n \u2192 \u03c3 i,n , x (F i,n ) V i,j \u2192 \u03c3 i,j , x (F i,j )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "The remaining rules rewrite the nonterminals T i,j and F i,j :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "T i,j \u2192 \u03b3 i,j (if v i occurs in c j ) T i,j \u2192 \u03b3 i,j F i,j \u2192 \u03b3 i,j (ifv i occurs in c j ) F i,j \u2192 \u03b3 i,j",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "It is not hard to see that both G and w can be constructed in polynomial time.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Proof",
"sec_num": null
},
{
"text": "To obtain efficient parsing, we would like to have grammars with as low a fan-out as possible. Therefore it is interesting to know how low we can go without losing too much coverage. In lexicalized LCFRSs extracted from dependency treebanks, the fan-out of a grammar has a structural correspondence in the maximal number of blocks per subtree, a measure known as \"block-degree.\" In this section we formally define block-degree, and evaluate grammar coverage under different bounds on this measure.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Block-Degree",
"sec_num": "7."
},
{
"text": "Recall the concept of \"blocks\" that was defined in Section 4.2. The block-degree of a node u of a dependency tree D is the number of distinct blocks of u. The block-degree of D is the maximal block-degree of its nodes. 2 Figure 10 shows two non-projective dependency trees. For D 1 , consider the node 2. The descendants of 2 fall into two blocks, marked by the dashed boxes. Because this is the maximal number of blocks per node in D 1 , the block-degree of D 1 is 2. Similarly, we can verify that the block-degree of the dependency tree D 2 is 3.",
"cite_spans": [],
"ref_spans": [
{
"start": 221,
"end": 230,
"text": "Figure 10",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Definition of Block-Degree",
"sec_num": "7.1"
},
{
"text": "Block-degree.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 10",
"sec_num": null
},
{
"text": "A dependency tree is projective if its block-degree is 1. In a projective dependency tree, each subtree corresponds to a substring of the underlying tuple of strings. In a nonprojective dependency tree, a subtree may span over several, discontinuous substrings.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 10",
"sec_num": null
},
{
"text": "Using a straightforward extension of the algorithm in Table 1 , the block-degrees of all nodes of a dependency tree D can be computed in time O(m), where m is the total number of blocks. To compute the block-degree of D, we simply take the maximum over the degrees of each node. We can also adapt this procedure to test whether D is projective, by aborting the computation as soon as we discover that some node has more than one block. The runtime of this test is linear in the number of nodes of D.",
"cite_spans": [],
"ref_spans": [
{
"start": 54,
"end": 61,
"text": "Table 1",
"ref_id": "TABREF1"
}
],
"eq_spans": [],
"section": "Computing the Block-Degrees",
"sec_num": "7.2"
},
{
"text": "In a lexicalized LCFRS extracted from a dependency treebank, there is a one-to-one correspondence between the blocks of a node u and the components of the template of the yield function f extracted for u. In particular, the fan-out of f is exactly the block-degree of u. As a consequence, any bound on the block-degree of the trees in the treebank translates into a bound on the fan-out of the extracted grammar. This has consequences for the generative capacity of the grammars: As Seki et al. (1991) show, the class of LCFRSs with fan-out k > 1 can generate string languages that cannot be generated by the class of LCFRSs with fan-out k \u2212 1.",
"cite_spans": [
{
"start": 483,
"end": 501,
"text": "Seki et al. (1991)",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Block-Degree in Extracted Grammars",
"sec_num": "7.3"
},
{
"text": "It may be worth emphasizing that the one-to-one correspondence between blocks and tuple components is a consequence of two characteristic properties of extracted grammars (Properties 3 and 4), and does not hold for non-canonical lexicalized LCFRSs.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Block-Degree in Extracted Grammars",
"sec_num": "7.3"
},
{
"text": "The following term induces a two-node dependency tree with block-degree 1, but contains yield functions with fan-out 2: a x 1 x 2 ( b, \u03b5 ). Note that the yield functions in this term violate both Property 3 and Property 4.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 7",
"sec_num": null
},
{
"text": "In order to assess the consequences of different bounds on the fan-out, we now evaluate the block-degree of dependency trees in real-world data. Specifically, we look into five dependency treebanks used in the 2006 CoNLL shared task on dependency parsing (Buchholz and Marsi 2006) : the Prague Arabic Dependency Treebank (Haji\u010d et al. 2004) , the Prague Dependency Treebank of Czech (B\u00f6hmov\u00e1 et al. 2003) , the Danish Dependency Treebank (Kromann 2003) , the Slovene Dependency Treebank (D\u017eeroski et al. 2006) , and the Metu-Sabanc\u0131 treebank of Turkish (Oflazer et al. 2003) . The full data used in the CoNLL shared task also included treebanks that were produced by conversion of corpora originally annotated with structures other than dependencies, which is a potential source of \"noise\" that one has to take into account when interpreting any findings. Here, we consider only genuine dependency treebanks. More specifically, our statistics concern the training sections of the treebanks that were set off for the task. For similar results on other data sets, see Kuhlmann and Nivre (2006) , Havelka (2007) , and Maier and Lichte (2011) .",
"cite_spans": [
{
"start": 255,
"end": 280,
"text": "(Buchholz and Marsi 2006)",
"ref_id": "BIBREF7"
},
{
"start": 321,
"end": 340,
"text": "(Haji\u010d et al. 2004)",
"ref_id": "BIBREF16"
},
{
"start": 383,
"end": 404,
"text": "(B\u00f6hmov\u00e1 et al. 2003)",
"ref_id": "BIBREF3"
},
{
"start": 438,
"end": 452,
"text": "(Kromann 2003)",
"ref_id": "BIBREF28"
},
{
"start": 487,
"end": 509,
"text": "(D\u017eeroski et al. 2006)",
"ref_id": null
},
{
"start": 553,
"end": 574,
"text": "(Oflazer et al. 2003)",
"ref_id": "BIBREF40"
},
{
"start": 1066,
"end": 1091,
"text": "Kuhlmann and Nivre (2006)",
"ref_id": "BIBREF30"
},
{
"start": 1094,
"end": 1108,
"text": "Havelka (2007)",
"ref_id": "BIBREF17"
},
{
"start": 1115,
"end": 1138,
"text": "Maier and Lichte (2011)",
"ref_id": "BIBREF34"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Coverage on Dependency Treebanks",
"sec_num": "7.4"
},
{
"text": "Our results are given in Table 3 . For each treebank, we list the number of rules extracted from that treebank, as well as the number of corresponding dependency trees. We then list the number of rules that we lose if we restrict ourselves to rules with fanout = 1, or rules with fan-out \u2264 2, as well as the number of dependency trees that we lose because their construction trees contain at least one such rule. We count rule tokens, meaning that two otherwise identical rules are counted twice if they were extracted from different trees, or from different nodes in the same tree.",
"cite_spans": [],
"ref_spans": [
{
"start": 25,
"end": 32,
"text": "Table 3",
"ref_id": "TABREF2"
}
],
"eq_spans": [],
"section": "Coverage on Dependency Treebanks",
"sec_num": "7.4"
},
{
"text": "By putting the bound at fan-out 1, we lose between 0.74% (Arabic) and 1.75% (Slovene) of the rules, and between 11.16% (Arabic) and 23.15% (Czech) of the trees in the treebanks. This loss is quite substantial. If we instead put the bound at fan-out \u2264 2, then rule loss is reduced by between 94.16% (Turkish) and 99.76% (Arabic), and tree loss is reduced by between 94.31% (Turkish) and 99.39% (Arabic). This outcome is surprising. For example, Holan et al. (1998) argue that it is impossible to give a theoretical upper bound for the block-degree of reasonable dependency analyses of Czech. Here we find that, if we are ready to accept a loss of as little as 0.02% of the rules extracted from the Prague Dependency Treebank, and up to 0.5% of the trees, then such an upper bound can be set at a block-degree as low as 2.",
"cite_spans": [
{
"start": 444,
"end": 463,
"text": "Holan et al. (1998)",
"ref_id": "BIBREF19"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Coverage on Dependency Treebanks",
"sec_num": "7.4"
},
{
"text": "The parsing of LCFRSs is exponential both in the fan-out and in the rank of the grammars. In this section we study \"well-nestedness,\" another restriction on the nonprojectivity of dependency trees, and show how enforcing this constraint allows us to restrict our attention to the class of LCFRSs with rank 2. Example 8 Figure 11 shows three non-projective dependency trees. Both D 1 and D 2 are well-nested: D 1 does not contain any overlapping sets of descendants at all. In D 2 , although 1 and 2 overlap, it is also the case that 1 \u2287 2 . In contrast, D 3 is ill-nested, as",
"cite_spans": [],
"ref_spans": [
{
"start": 319,
"end": 328,
"text": "Figure 11",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Well-Nestedness",
"sec_num": "8."
},
{
"text": "\u2208 u and v l , v r \u2208 v such that u l < v l < u r < v r or v l < u l < v r < u",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness",
"sec_num": "8."
},
{
"text": "2 3 but 2 \u2229 3 = \u2205",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness",
"sec_num": "8."
},
{
"text": "The following lemma characterizes well-nestedness in terms of blocks.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness",
"sec_num": "8."
},
{
"text": "A dependency tree is ill-nested if and only if it contains two sibling nodes u, v and blocks",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 9",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "u 1 , u 2 of u and v 1 , v 2 of v such that u 1 < v 1 < u 2 < v 2",
"eq_num": "(4)"
}
],
"section": "Lemma 9",
"sec_num": null
},
{
"text": "Proof Let D be a dependency tree. Suppose that D contains a configuration of the form (4). This configuration witnesses that the sets u and v overlap. Because u, v are siblings, u \u2229 v = \u2205. Therefore we conclude that D is ill-nested. Conversely now, suppose that D is ill-nested. In this case, there exist two nodes u and v such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 9",
"sec_num": null
},
{
"text": "u v and u \u2229 v = \u2205 ( * )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 9",
"sec_num": null
},
{
"text": "Well-nestedness and ill-nestedness.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 11",
"sec_num": null
},
{
"text": "Here, we may assume u and v to be siblings: otherwise, we may replace either u or v with its parent node, and property ( * ) will continue to hold. Because u v , there exist descendants u l , u r \u2208 u and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 11",
"sec_num": null
},
{
"text": "v l , v r \u2208 v such that u l < v l < u r < v r or v l < u l < v r < u r",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 11",
"sec_num": null
},
{
"text": "Without loss of generality, assume that we have the first case. The nodes u l and u r belong to different blocks of u, say u 1 and u 2 ; and the nodes v l and v r belong to different blocks of v, say v 1 and v 2 . Then it is not hard to verify Equation (4).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 11",
"sec_num": null
},
{
"text": "Note that projective dependency trees are always well-nested; in these structures, every node has exactly one block, so configuration (4) is impossible. For every k > 1, there are both well-nested and ill-nested dependency trees with block-degree k.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 11",
"sec_num": null
},
{
"text": "Based on Lemma 9, testing whether a dependency tree D is well-nested can be done in time linear in the number of blocks in D using a simple subsequence test as follows. We run the algorithm given in Table 1 , maintaining a stack s [u] for every node u. The first time we make a down step to u, we push u to the stack for the parent of u; every other time, we pop the stack for the parent until we either find u as the topmost element, or the stack becomes empty. In the latter case, we terminate the computation and report that D is ill-nested; if the computation can be completed without any stack ever becoming empty, we report that D is well-nested.",
"cite_spans": [
{
"start": 231,
"end": 234,
"text": "[u]",
"ref_id": null
}
],
"ref_spans": [
{
"start": 199,
"end": 206,
"text": "Table 1",
"ref_id": "TABREF1"
}
],
"eq_spans": [],
"section": "Testing for Well-Nestedness",
"sec_num": "8.2"
},
{
"text": "To show that the algorithm is sound, suppose that some stack s [p] becomes empty when making a down step to some child v of p. In this case, the node v must have been popped from s [p] when making a down step to some other child u of p, and that child must have already been on the stack before the first down step to v. This witnesses the existence of a configuration of the form in Equation (4).",
"cite_spans": [
{
"start": 63,
"end": 66,
"text": "[p]",
"ref_id": null
},
{
"start": 181,
"end": 184,
"text": "[p]",
"ref_id": null
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Testing for Well-Nestedness",
"sec_num": "8.2"
},
{
"text": "Just like block-degree, well-nestedness can be characterized in terms of yield functions. Recall the notation x < f y from Section 5.1. A yield function",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness in Extracted Grammars",
"sec_num": "8.3"
},
{
"text": "f : k 1 \u2022 \u2022 \u2022 k m \u2192 k , f = \u03b1 1 , . . . , \u03b1 k",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness in Extracted Grammars",
"sec_num": "8.3"
},
{
"text": "is ill-nested if there are argument indices 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness in Extracted Grammars",
"sec_num": "8.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "\u2264 i 1 , i 2 \u2264 m with i 1 = i 2 and component indices 1 \u2264 j 1 , j 1 \u2264 k i 1 , 1 \u2264 j 2 , j 2 \u2264 k i 2 such that x i 1 ,j 1 < f x i 2 ,j 2 < f x i 1 ,j 1 < f x i 2 ,j 2",
"eq_num": "(5)"
}
],
"section": "Well-Nestedness in Extracted Grammars",
"sec_num": "8.3"
},
{
"text": "Otherwise, we say that f is well-nested. As an immediate consequence of Lemma 9, a restriction to well-nested dependency trees translates into a restriction to well-nested yield functions in the extracted grammars. This puts them into the class of what Kanazawa (2009) calls \"well-nested multiple context-free grammars.\" 3 These grammars have a number of interesting properties that set them apart from general LCFRSs; in particular, they have a standard pumping lemma (Kanazawa 2009) . The yield languages generated by well-nested multiple context-free grammars form a proper subhierarchy within the languages generated by general LCFRSs (Kanazawa and Salvati 2010) . Perhaps the most prominent subclass of well-nested LCFRSs is the class of tree-adjoining grammars (Joshi and Schabes 1997) . Similar to the situation with block-degree, the correspondence between structural well-nestedness and syntactic well-nestedness is tight only for canonical grammars. For non-canonical grammars, syntactic well-nestedness alone does not imply structural well-nestedness, nor the other way around.",
"cite_spans": [
{
"start": 253,
"end": 268,
"text": "Kanazawa (2009)",
"ref_id": null
},
{
"start": 469,
"end": 484,
"text": "(Kanazawa 2009)",
"ref_id": null
},
{
"start": 639,
"end": 666,
"text": "(Kanazawa and Salvati 2010)",
"ref_id": "BIBREF25"
},
{
"start": 767,
"end": 791,
"text": "(Joshi and Schabes 1997)",
"ref_id": "BIBREF22"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Well-Nestedness in Extracted Grammars",
"sec_num": "8.3"
},
{
"text": "To estimate the coverage of well-nested grammars, we extend the evaluation presented in Section 7.4. Table 4 shows how many rules and trees in the five dependency treebanks we lose if we restrict ourselves to well-nested yield functions with fan-out \u2264 2. The losses reported in Table 3 are repeated here for comparison. Although the coverage of well-nested rules is significantly smaller than the coverage of rules without this requirement, rule loss is still reduced by between 92.65% (Turkish) and 99.51% (Arabic) when compared to the fan-out = 1 baseline.",
"cite_spans": [],
"ref_spans": [
{
"start": 101,
"end": 108,
"text": "Table 4",
"ref_id": null
},
{
"start": 278,
"end": 285,
"text": "Table 3",
"ref_id": "TABREF2"
}
],
"eq_spans": [],
"section": "Coverage on Dependency Treebanks",
"sec_num": "8.4"
},
{
"text": "Our main interest in well-nestedness comes from the following:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Binarization of Well-Nested Grammars",
"sec_num": "8.5"
},
{
"text": "The universal recognition problem for well-nested lexicalized LCFRS with fan-out k and unbounded rank can be decided in time",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Lemma 10",
"sec_num": null
},
{
"text": "To prove this lemma, we will provide an algorithm for the binarization of wellnested lexicalized LCFRSs. In the context of LCFRSs, a binarization is a procedure for transforming a grammar into an equivalent one with rank at most 2. Binarization, either explicit at the level or the grammar or implicit at the level of some parsing algorithm, is essential for achieving efficient recognition algorithms, in particular the usual cubic-time algorithms for context-free grammars. Note that our binarization only Table 4 Loss in coverage under the restriction to yield functions with fan-out = 1, fan-out \u2264 2, and to well-nested yield functions with fan-out \u2264 2 (last column). preserves weak equivalence; in effect, it reduces the universal recognition problem for well-nested lexicalized LCFRSs to the corresponding problem for well-nested LCFRSs with rank 2. Many interesting semiring computations on the original grammar can be simulated on the binarized grammar, however. A direct parsing algorithm for wellnested dependency trees has been presented by G\u00f3mez-Rodr\u00edguez, Carroll, and Weir (2011) . The binarization that we present here is a special case of the binarization proposed by G\u00f3mez-Rodr\u00edguez, Kuhlmann, and Satta (2010) . They show that every well-nested LCFRS can be transformed (at the cost of a linear size increase) into a weakly equivalent one in which all yield functions are either constants (that is, have rank 0) or binary functions of one of two types:",
"cite_spans": [
{
"start": 1052,
"end": 1093,
"text": "G\u00f3mez-Rodr\u00edguez, Carroll, and Weir (2011)",
"ref_id": "BIBREF14"
},
{
"start": 1201,
"end": 1227,
"text": "Kuhlmann, and Satta (2010)",
"ref_id": "BIBREF15"
}
],
"ref_spans": [
{
"start": 508,
"end": 515,
"text": "Table 4",
"ref_id": null
}
],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "x 1 , . . . , x k 1 y 1 , . . . , y k 2 : k 1 k 2 \u2192 (k 1 + k 2 \u2212 1) (concatenation) (6) x 1 , . . . , x j y 1 , . . . , y k 2 x j+1 , . . . , x k 1 : k 1 k 2 \u2192 (k 1 + k 2 \u2212 2) (wrapping)",
"eq_num": "(7)"
}
],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "A concatenation function takes a k 1 -tuple and a k 2 -tuple and returns the (k 1 + k 2 \u2212 1)tuple that is obtained by concatenating the two arguments. The simplest concatenation function is the standard concatenation operation x y . We will write conc : k 1 k 2 to refer to a concatenation function of the type given in Equation (6). By counting endpoints, we see that the parsing complexity of concatenation functions is c(conc :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "k 1 k 2 ) \u2264 2k 1 + 2k 2 \u2212 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "A wrapping function takes a k 1 -tuple (for some k 1 \u2265 2) and a k 2 -tuple and returns the (k 1 + k 2 \u2212 2)-tuple that is obtained by \"wrapping\" the first argument around the second argument, filling some gap in the former. The simplest function of this type is x 1 y x 2 , which wraps a 2-tuple around a 1-tuple. We write wrap : k 1 k 2 j to refer to a wrapping function of the type given in Equation (7). The parsing complexity is c(wrap :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "k 1 k 2 j) \u2264 2k 1 + 2k 2 \u2212 2 (for all choices of j)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "The constants of the binarized grammar have the form \u03b5 , \u03b5, \u03b5 , and a , where a is the anchor of some yield function of the original grammar.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "8.5.1 Parsing Complexity. Before presenting the actual binarization, we determine the parsing complexity of the binarized grammar. Because the binarization preserves the fan-out of the original grammar, and because in a grammar with fan-out k, for concatenation functions conc : k 1 k 2 we have k 1 + k 2 \u2212 1 \u2264 k and for wrapping functions wrap : k 1 k 2 j we have k 1 + k 2 \u2212 2 \u2264 k, we can rewrite the general parsing complexities as c(conc :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "k 1 k 2 ) \u2264 2k 1 + 2k 2 \u2212 1 = 2(k 1 + k 2 \u2212 1) + 1 \u2264 2k + 1 c(wrap : k 1 k 2 j) \u2264 2k 1 + 2k 2 \u2212 2 = 2(k 1 + k 2 \u2212 2) + 2 \u2264 2k + 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "Thus the maximal parsing complexity in the binarized grammar is 2k + 2; this is achieved by wrapping operations. This gives the bound stated in Lemma 10.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "that if for some argument index 1 \u2264 i \u2264 m, either f 1 or f 2 contains any variable with argument index i, then it contains all such variables. The two sequences B 1 , . . . , B m 1 and C 1 , . . . , C m 2 are obtained from A 1 , . . . , A m by collecting the nonterminal A i if the variables with argument index i belong to the template of f 1 and f 2 , respectively. The nonterminals B and C are fresh nonterminals. We do not create rules for f 1 and f 2 if they are identity functions.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "O |G| \u2022 |w| 2k+2",
"sec_num": null
},
{
"text": "We illustrate the binarization by showing how to transform the rule",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "A \u2192 x 1 a x 2 y 1 , y 2 , y 3 x 3 (A 1 , A 2 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "The template x 1 a x 2 y 1 , y 2 , y 3 x 3 is complex and matches Case 3 in Figure 12 , because its first component starts with the variable x 1 and its last component ends with the variable x 3 . We therefore split the template into two smaller parts x 1 a x 2 , x 3 and y 1 , y 2 , y 3 . The function y 1 , y 2 , y 3 is an identity. We therefore create two rules:",
"cite_spans": [],
"ref_spans": [
{
"start": 76,
"end": 85,
"text": "Figure 12",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "A \u2192 f 1 (X, A 2 ) , f 1 = wrap : 2 3 1 = x 1 y 1 , y 2 , y 3 x 2 X \u2192 x 1 a x 2 , x 3 (A 1 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "Note that the index j for the wrapping function was chosen to be j = 2 because there were more component boundaries between x 2 and x 3 than between x 1 and x 2 . The template x 1 a x 2 , x 3 requires further decomposition according to Case 3. This time, the two smaller parts are the identity function x 1 , x 2 , x 3 and the constant a . We therefore create the following rules:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "X \u2192 f 2 (A 1 , Y) , f 2 = wrap : 3 1 1 = x 1 y x 2 , x 3 Y \u2192 a",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "At this point, the transformation ends.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Example 9",
"sec_num": null
},
{
"text": "We need to show that the fan-out of the binarized grammar does not exceed the fan-out of the original grammar. We reason as follows. Starting from some initial yield function f 0 : k 1 \u2022 \u2022 \u2022 k m \u2192 k, each step of the binarization decomposes some yield function f into two new yield functions f 1 , f 2 . Let us denote the fan-outs of the three functions by h, h 1 , h 2 , respectively. We have h = h 1 + h 2 \u2212 1 in Case 1 and Case 2 (8)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Correctness.",
"sec_num": "8.5.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "h = h 1 + h 2 \u2212 2 in Case 3",
"eq_num": "(9)"
}
],
"section": "Correctness.",
"sec_num": "8.5.3"
},
{
"text": "From Equation (8) it is clear that in Case 1 and Case 2, both h 1 and h 2 are upperbounded by h. In Case 3 we have h 1 \u2265 2, which together with Equation (9) implies that h 2 \u2264 h. However, h 1 is upper-bounded by h only if h 2 \u2265 2; if h 2 = 1, then h 1 may be greater than h. As an example, consider the decomposition of x 1 a x 2 (fan-out 1) into the wrapping function x 1 , x 2 (fan-out 2) and the constant a (fan-out 1). But because in Case 3 the index j is chosen to maximize the number of component boundaries between the variables x i,j and x i,j+1 , the assumption h 2 = 1 implies that each of the h 1 components of f 1 contains at least one variable with argument index i-if there were a component without such a variable, then the two variables that surrounded that component would have given rise to a different choice of j. Hence we deduce that h 1 \u2264 k i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Correctness.",
"sec_num": "8.5.3"
},
{
"text": "In this article, we have presented a formalism for non-projective dependency grammar based on linear context-free rewriting systems, along with a technique for extracting grammars from dependency treebanks. We have shown that parsing with the full class of these grammars is intractable. Therefore, we have investigated two constraints on the non-projectivity of dependency trees, block-degree and well-nestedness. Jointly, these two constraints define a class of \"mildly\" non-projective dependency grammars that can be parsed in polynomial time.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "9."
},
{
"text": "Our results in Sections 7 and 8 allow us to relate the formal power of an LCFRS to the structural properties of the dependency structures that it induces. Although we have used this relation to identify a class of dependency grammars that can be parsed in polynomial time, it also provides us with a new perspective on the question about the descriptive adequacy of a grammar formalism. This question has traditionally been discussed on the basis of strong and weak generative capacity (Bresnan et al. 1982; Huybregts 1984; Shieber 1985) . A notion of generative capacity based on dependency trees makes a useful addition to this discussion, in particular when comparing formalisms for which no common concept of strong generative capacity exists. As an example for a result in this direction, see Koller and Kuhlmann (2009) .",
"cite_spans": [
{
"start": 486,
"end": 507,
"text": "(Bresnan et al. 1982;",
"ref_id": "BIBREF6"
},
{
"start": 508,
"end": 523,
"text": "Huybregts 1984;",
"ref_id": "BIBREF21"
},
{
"start": 524,
"end": 537,
"text": "Shieber 1985)",
"ref_id": "BIBREF47"
},
{
"start": 798,
"end": 824,
"text": "Koller and Kuhlmann (2009)",
"ref_id": "BIBREF26"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "9."
},
{
"text": "We have defined the dependency trees that an LCFRS induces by means of a compositional mapping on the derivations. While we would claim that compositionality is a generally desirable property, the particular notion of induction is up for discussion. In particular, our interpretation of derivations may not always be in line with how the grammar producing these derivations is actually used. One formalism for which such a mismatch between derivation trees and dependency trees has been pointed out is treeadjoining grammar (Rambow, Vijay-Shanker, and Weir 1995; Candito and Kahane 1998) . Resolving this mismatch provides an interesting line of future work.",
"cite_spans": [
{
"start": 524,
"end": 562,
"text": "(Rambow, Vijay-Shanker, and Weir 1995;",
"ref_id": "BIBREF42"
},
{
"start": 563,
"end": 587,
"text": "Candito and Kahane 1998)",
"ref_id": "BIBREF8"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "9."
},
{
"text": "One aspect that we have not discussed here is the linguistic adequacy of blockdegree and well-nestedness. Each of our dependency grammars is restricted to a finite block-degree. As a consequence of this restriction, our dependency grammars are not expressive enough to capture linguistic phenomena that require unlimited degrees of non-projectivity, such as the \"scrambling\" in German subordinate clauses (Becker, Rambow, and Niv 1992) . The question whether it is reasonable to assume a bound on the block-degree of dependency trees, perhaps for some performance-based reason, is open. Likewise, it is not clear whether well-nestedness is a \"natural\" constraint on dependency analyses (Chen-Main and Joshi 2010; Maier and Lichte 2011).",
"cite_spans": [
{
"start": 405,
"end": 435,
"text": "(Becker, Rambow, and Niv 1992)",
"ref_id": "BIBREF0"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "9."
},
{
"text": "Although most of the results that we have presented in this article are of a theoretical nature, some of them have found their way into practical systems. In particular, the extraction technique from Section 4 is used by the data-driven dependency parser of Maier and Kallmeyer (2010) .",
"cite_spans": [
{
"start": 258,
"end": 284,
"text": "Maier and Kallmeyer (2010)",
"ref_id": "BIBREF33"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "9."
},
{
"text": "We draw the nodes of a dependency tree as circles, and the edges as arrows pointing towards the dependent (away from the root node). FollowingHays (1964), we use dotted lines to help us keep track of the positions of the nodes in the linear order, and to associate nodes with lexical items.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "We note that, instead of counting the blocks of each node, one may also count the gaps between these blocks and define the \"gap-degree\" of a dependency tree(Holan et al. 1998).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "Kanazawa (2009) calls a multiple context-free grammar well-nested if each of its rules is non-deleting, non-permuting (our Property 2), and well-nested according to (5).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "In order for these parts to make well-defined templates, we will in general need to rename the variables. We leave this renaming implicit here.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
}
],
"back_matter": [
{
"text": "The author gratefully acknowledges financial support from The German Research Foundation (Sonderforschungsbereich 378, project MI 2) and The Swedish Research Council (diary no. 2008-296).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Acknowledgments",
"sec_num": null
},
{
"text": "Binarization of well-nested LCFRSs (complex cases).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Figure 12",
"sec_num": null
},
{
"text": "We now turn to the actual binarization. Consider a rulewhere f is not already a concatenation function, wrapping function, or constant. We decompose this rule into up to three rulesas follows. We match the template of f against one of three cases, shown schematically in Figure 12 . In each case we select a concatenation or wrapping function f (shown in the right half of the figure), and split up the template of f into two parts defining yield functions f 1 and f 2 , respectively. In Figure 12 , f 1 is drawn shaded, and f 2 is drawn nonshaded. 4 The split of f partitions the variables that occur in the template, in the sense",
"cite_spans": [
{
"start": 549,
"end": 550,
"text": "4",
"ref_id": null
}
],
"ref_spans": [
{
"start": 271,
"end": 280,
"text": "Figure 12",
"ref_id": null
},
{
"start": 488,
"end": 497,
"text": "Figure 12",
"ref_id": null
}
],
"eq_spans": [],
"section": "Binarization.",
"sec_num": "8.5.2"
}
],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"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": "Becker, Tilman, Owen Rambow, and Michael Niv. 1992. The derivational generative power of formal systems, or: Scrambling is beyond LCFRS. IRCS Report 92-38, University of Pennsylvania, Philadelphia, PA.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "On the complexity of some extensions of RCG parsing",
"authors": [
{
"first": "Eberhard",
"middle": [],
"last": "Bertsch",
"suffix": ""
},
{
"first": "Mark",
"middle": [
"-"
],
"last": "",
"suffix": ""
}
],
"year": 2001,
"venue": "Proceedings of the Seventh International Workshop on Parsing Technologies (IWPT)",
"volume": "",
"issue": "",
"pages": "66--77",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Bertsch, Eberhard and Mark-Jan Nederhof. 2001. On the complexity of some extensions of RCG parsing. In Proceedings of the Seventh International Workshop on Parsing Technologies (IWPT), pages 66-77, Beijing.",
"links": null
},
"BIBREF2": {
"ref_id": "b2",
"title": "Well-nested drawings as models of syntactic structure",
"authors": [
{
"first": "Manuel",
"middle": [],
"last": "Bodirsky",
"suffix": ""
},
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
},
{
"first": "Mathias",
"middle": [],
"last": "M\u00f6hl",
"suffix": ""
}
],
"year": 2005,
"venue": "Proceedings of the 10th Conference on Formal Grammar (FG) and Ninth Meeting on Mathematics of Language (MOL)",
"volume": "",
"issue": "",
"pages": "195--203",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Bodirsky, Manuel, Marco Kuhlmann, and Mathias M\u00f6hl. 2005. Well-nested drawings as models of syntactic structure. In Proceedings of the 10th Conference on Formal Grammar (FG) and Ninth Meeting on Mathematics of Language (MOL), pages 195-203, Edinburgh.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "The Prague Dependency Treebank: A three-level annotation scenario",
"authors": [
{
"first": "Alena",
"middle": [],
"last": "B\u00f6hmov\u00e1",
"suffix": ""
},
{
"first": "Jan",
"middle": [],
"last": "Haji\u010d",
"suffix": ""
},
{
"first": "Eva",
"middle": [],
"last": "Haji\u010dov\u00e1",
"suffix": ""
},
{
"first": "Barbora",
"middle": [],
"last": "Hladk\u00e1",
"suffix": ""
}
],
"year": 2003,
"venue": "",
"volume": "",
"issue": "",
"pages": "103--127",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "B\u00f6hmov\u00e1, Alena, Jan Haji\u010d, Eva Haji\u010dov\u00e1, and Barbora Hladk\u00e1. 2003. The Prague Dependency Treebank: A three-level annotation scenario. In Abeill\u00e9, Anne, editor. Treebanks: Building and Using Parsed Corpora. Kluwer Academic Publishers, Dordrecht, chapter 7, pages 103-127.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Proposal for a natural language processing syntactic backbone",
"authors": [
{
"first": "Pierre",
"middle": [],
"last": "Boullier",
"suffix": ""
}
],
"year": 1998,
"venue": "INRIA Rocquencourt",
"volume": "3342",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Boullier, Pierre. 1998. Proposal for a natural language processing syntactic backbone. Rapport de recherche 3342, INRIA Rocquencourt, Paris, France.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "of Text, Speech and Language Technology",
"authors": [
{
"first": "Pierre",
"middle": [],
"last": "Boullier",
"suffix": ""
}
],
"year": 2004,
"venue": "New Developments in Parsing Technology",
"volume": "23",
"issue": "",
"pages": "269--289",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Boullier, Pierre. 2004. Range Concatenation Grammars. In Harry C. Bunt, John Carroll, and Giorgio Satta, editors, New Developments in Parsing Technology, volume 23 of Text, Speech and Language Technology. Kluwer Academic Publishers, Dordrecht, pages 269-289.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "Cross-serial dependencies in Dutch",
"authors": [
{
"first": "Joan",
"middle": [],
"last": "Bresnan",
"suffix": ""
},
{
"first": "Ronald",
"middle": [
"M"
],
"last": "Kaplan",
"suffix": ""
},
{
"first": "Stanley",
"middle": [],
"last": "Peters",
"suffix": ""
},
{
"first": "Annie",
"middle": [],
"last": "Zaenen",
"suffix": ""
}
],
"year": 1982,
"venue": "Linguistic Inquiry",
"volume": "13",
"issue": "4",
"pages": "613--635",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Bresnan, Joan, Ronald M. Kaplan, Stanley Peters, and Annie Zaenen. 1982. Cross-serial dependencies in Dutch. Linguistic Inquiry, 13(4):613-635.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "CoNLL-X shared task on multilingual dependency parsing",
"authors": [
{
"first": "Sabine",
"middle": [],
"last": "Buchholz",
"suffix": ""
},
{
"first": "Erwin",
"middle": [],
"last": "Marsi",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the Tenth Conference on Computational Natural Language Learning (CoNLL)",
"volume": "",
"issue": "",
"pages": "149--164",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Buchholz, Sabine and Erwin Marsi. 2006. CoNLL-X shared task on multilingual dependency parsing. In Proceedings of the Tenth Conference on Computational Natural Language Learning (CoNLL), pages 149-164, New York, NY.",
"links": null
},
"BIBREF8": {
"ref_id": "b8",
"title": "Can the TAG derivation tree represent a semantic graph? An answer in the light of Meaning-Text Theory",
"authors": [
{
"first": "Marie-H\u00e9l\u00e8ne",
"middle": [],
"last": "Candito",
"suffix": ""
},
{
"first": "Sylvain",
"middle": [],
"last": "Kahane",
"suffix": ""
}
],
"year": 1998,
"venue": "Proceedings of the Fourth Workshop on Tree Adjoining Grammars and Related Formalisms (TAG+)",
"volume": "",
"issue": "",
"pages": "21--24",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Candito, Marie-H\u00e9l\u00e8ne and Sylvain Kahane. 1998. Can the TAG derivation tree represent a semantic graph? An answer in the light of Meaning-Text Theory. In Proceedings of the Fourth Workshop on Tree Adjoining Grammars and Related Formalisms (TAG+), pages 21-24, Philadelphia, PA.",
"links": null
},
"BIBREF9": {
"ref_id": "b9",
"title": "Tree-bank grammars",
"authors": [
{
"first": "Eugene",
"middle": [],
"last": "Charniak",
"suffix": ""
}
],
"year": 1996,
"venue": "Proceedings of the 13th National Conference on Artificial Intelligence (AAAI) and Eighth Innovative Applications of Artificial Intelligence Conference (IAAI)",
"volume": "2",
"issue": "",
"pages": "1031--1036",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Charniak, Eugene. 1996. Tree-bank grammars. In Proceedings of the 13th National Conference on Artificial Intelligence (AAAI) and Eighth Innovative Applications of Artificial Intelligence Conference (IAAI), volume 2, pages 1031-1036, Portland, OR.",
"links": null
},
"BIBREF10": {
"ref_id": "b10",
"title": "Unavoidable ill-nestedness in natural language and the adequacy of tree local-MCTAG induced dependency structures",
"authors": [
{
"first": "Joan",
"middle": [],
"last": "Chen-Main",
"suffix": ""
},
{
"first": "K",
"middle": [],
"last": "Aravind",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Joshi",
"suffix": ""
}
],
"year": 2010,
"venue": "Proceedings of the Tenth International Conference on Tree Adjoining Grammars and Related Formalisms (TAG+)",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {
"DOI": [
"10.1093/logcom/exs012"
]
},
"num": null,
"urls": [],
"raw_text": "Chen-Main, Joan and Aravind K. Joshi. 2010. Unavoidable ill-nestedness in natural language and the adequacy of tree local-MCTAG induced dependency structures. In Proceedings of the Tenth International Conference on Tree Adjoining Grammars and Related Formalisms (TAG+), New Haven, CT. Available at http://dx. doi.org/10.1093/logcom/exs012.",
"links": null
},
"BIBREF11": {
"ref_id": "b11",
"title": "Optimal head-driven parsing complexity for linear context-free rewriting systems",
"authors": [
{
"first": "Pierluigi",
"middle": [],
"last": "Crescenzi",
"suffix": ""
},
{
"first": "Daniel",
"middle": [],
"last": "Gildea",
"suffix": ""
},
{
"first": "Andrea",
"middle": [],
"last": "Marino",
"suffix": ""
},
{
"first": "Gianluca",
"middle": [],
"last": "Rossi",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "1388--1391",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Crescenzi, Pierluigi, Daniel Gildea, Andrea Marino, Gianluca Rossi, and Giorgio Satta. 2011. Optimal head-driven parsing complexity for linear context-free rewriting systems. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics (ACL), pages 450-459, Portland, OR. D\u017eeroski, Sa\u0161o, Toma\u017e Erjavec, Nina Ledinek, Petr Pajas, Zdenek\u017dabokrtsky, and Andreja\u017dele. 2006. Towards a Slovene dependency treebank. In Fifth International Conference on Language Resources and Evaluations (LREC), pages 1388-1391, Genoa.",
"links": null
},
"BIBREF12": {
"ref_id": "b12",
"title": "Dependency systems and phrase-structure systems",
"authors": [
{
"first": "Haim",
"middle": [],
"last": "Gaifman",
"suffix": ""
}
],
"year": 1965,
"venue": "Information and Control",
"volume": "8",
"issue": "3",
"pages": "304--337",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Gaifman, Haim. 1965. Dependency systems and phrase-structure systems. Information and Control, 8(3):304-337.",
"links": null
},
"BIBREF13": {
"ref_id": "b13",
"title": "Optimal parsing strategies for linear context-free rewriting systems",
"authors": [
{
"first": "Daniel",
"middle": [],
"last": "Gildea",
"suffix": ""
}
],
"year": 2010,
"venue": "Proceedings of Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL)",
"volume": "",
"issue": "",
"pages": "769--776",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Gildea, Daniel. 2010. Optimal parsing strategies for linear context-free rewriting systems. In Proceedings of Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL), pages 769-776, Los Angeles, CA.",
"links": null
},
"BIBREF14": {
"ref_id": "b14",
"title": "Dependency parsing schemata and mildly non-projective dependency parsing",
"authors": [
{
"first": "Carlos",
"middle": [],
"last": "G\u00f3mez-Rodr\u00edguez",
"suffix": ""
},
{
"first": "John",
"middle": [],
"last": "Carroll",
"suffix": ""
},
{
"first": "David",
"middle": [
"J"
],
"last": "Weir",
"suffix": ""
}
],
"year": 2011,
"venue": "Computational Linguistics",
"volume": "37",
"issue": "3",
"pages": "541--586",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "G\u00f3mez-Rodr\u00edguez, Carlos, John Carroll, and David J. Weir. 2011. Dependency parsing schemata and mildly non-projective dependency parsing. Computational Linguistics, 37(3):541-586.",
"links": null
},
"BIBREF15": {
"ref_id": "b15",
"title": "An optimal-time binarization algorithm for linear context-free rewriting systems with fan-out two",
"authors": [
{
"first": "Carlos",
"middle": [],
"last": "G\u00f3mez-Rodr\u00edguez",
"suffix": ""
},
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
},
{
"first": "Carlos",
"middle": [],
"last": "G\u00f3mez-Rodr\u00edguez",
"suffix": ""
},
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
},
{
"first": "David",
"middle": [
"J"
],
"last": "Weir",
"suffix": ""
},
{
"first": ";",
"middle": [],
"last": "",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 1999,
"venue": "Proceedings of the Joint Conference of the 47th Annual Meeting of the Association for Computational Linguistics (ACL) and the Fourth International Joint Conference on Natural Language Processing of the Asian Federation of Natural Language Processing (IJCNLP)",
"volume": "25",
"issue": "",
"pages": "573--605",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "G\u00f3mez-Rodr\u00edguez, Carlos, Marco Kuhlmann, and Giorgio Satta. 2010. Efficient parsing of well-nested linear context-free rewriting systems. In Proceedings of Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL), pages 276-284, New Haven, CT. G\u00f3mez-Rodr\u00edguez, Carlos, Marco Kuhlmann, Giorgio Satta, and David J. Weir. 2009. Optimal reduction of rule length in linear context-free rewriting systems. In Proceedings of Human Language Technologies: The 2009 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL), pages 539-547, Boulder, CO. G\u00f3mez-Rodr\u00edguez, Carlos and Giorgio Satta. 2009. An optimal-time binarization algorithm for linear context-free rewriting systems with fan-out two. In Proceedings of the Joint Conference of the 47th Annual Meeting of the Association for Computational Linguistics (ACL) and the Fourth International Joint Conference on Natural Language Processing of the Asian Federation of Natural Language Processing (IJCNLP), pages 985-993, Singapore. Goodman, Joshua. 1999. Semiring parsing. Computational Linguistics, 25(4):573-605.",
"links": null
},
"BIBREF16": {
"ref_id": "b16",
"title": "Prague Arabic Dependency Treebank: Development in data and tools",
"authors": [
{
"first": "Jan",
"middle": [],
"last": "Haji\u010d",
"suffix": ""
},
{
"first": "Otakar",
"middle": [],
"last": "Smr\u017e",
"suffix": ""
},
{
"first": "Petr",
"middle": [],
"last": "Zem\u00e1nek",
"suffix": ""
},
{
"first": "Jan\u0161naidauf",
"middle": [],
"last": "",
"suffix": ""
},
{
"first": "Emanuel",
"middle": [],
"last": "Be\u0161ka",
"suffix": ""
}
],
"year": 2004,
"venue": "Proceedings of the International Conference on Arabic Language Resources and Tools",
"volume": "",
"issue": "",
"pages": "110--117",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Haji\u010d, Jan, Otakar Smr\u017e, Petr Zem\u00e1nek, Jan\u0160naidauf, and Emanuel Be\u0161ka. 2004. Prague Arabic Dependency Treebank: Development in data and tools. In Proceedings of the International Conference on Arabic Language Resources and Tools, pages 110-117, Cairo.",
"links": null
},
"BIBREF17": {
"ref_id": "b17",
"title": "Beyond projectivity: Multilingual evaluation of constraints and measures on non-projective structures",
"authors": [
{
"first": "Ji\u0159\u00ed",
"middle": [],
"last": "Havelka",
"suffix": ""
}
],
"year": 2007,
"venue": "Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "608--615",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Havelka, Ji\u0159\u00ed. 2007. Beyond projectivity: Multilingual evaluation of constraints and measures on non-projective structures. In Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics (ACL), pages 608-615, Prague.",
"links": null
},
"BIBREF18": {
"ref_id": "b18",
"title": "Dependency theory: A formalism and some observations",
"authors": [
{
"first": "David",
"middle": [
"G"
],
"last": "Hays",
"suffix": ""
}
],
"year": 1964,
"venue": "Language",
"volume": "40",
"issue": "4",
"pages": "511--525",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Hays, David G. 1964. Dependency theory: A formalism and some observations. Language, 40(4):511-525.",
"links": null
},
"BIBREF19": {
"ref_id": "b19",
"title": "Two useful measures of word order complexity",
"authors": [
{
"first": "Tom\u00e1\u0161",
"middle": [],
"last": "Holan",
"suffix": ""
},
{
"first": "Vladislav",
"middle": [],
"last": "Kubo\u0148",
"suffix": ""
},
{
"first": "Karel",
"middle": [],
"last": "Oliva",
"suffix": ""
},
{
"first": "Martin",
"middle": [],
"last": "Pl\u00e1tek",
"suffix": ""
}
],
"year": 1998,
"venue": "Proceedings of the Workshop on Processing of Dependency-Based Grammars",
"volume": "",
"issue": "",
"pages": "21--29",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Holan, Tom\u00e1\u0161, Vladislav Kubo\u0148, Karel Oliva, and Martin Pl\u00e1tek. 1998. Two useful measures of word order complexity. In Proceedings of the Workshop on Processing of Dependency-Based Grammars, pages 21-29, Montr\u00e9al.",
"links": null
},
"BIBREF20": {
"ref_id": "b20",
"title": "Language Networks. The New Word Grammar",
"authors": [
{
"first": "Richard",
"middle": [],
"last": "Hudson",
"suffix": ""
}
],
"year": 2007,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Hudson, Richard. 2007. Language Networks. The New Word Grammar. Oxford University Press, Oxford.",
"links": null
},
"BIBREF21": {
"ref_id": "b21",
"title": "The weak inadequacy of context-free phrase structure grammars",
"authors": [
{
"first": "Riny",
"middle": [],
"last": "Huybregts",
"suffix": ""
}
],
"year": 1984,
"venue": "Foris",
"volume": "",
"issue": "",
"pages": "81--99",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Huybregts, Riny. 1984. The weak inadequacy of context-free phrase structure grammars. In Ger de Haan, Mieke Trommelen, and Wim Zonneveld, editors, Van periferie naar kern. Foris, Dordrecht, pages 81-99.",
"links": null
},
"BIBREF22": {
"ref_id": "b22",
"title": "Tree-Adjoining Grammars",
"authors": [
{
"first": "Aravind",
"middle": [
"K"
],
"last": "Joshi",
"suffix": ""
},
{
"first": "Yves",
"middle": [],
"last": "Schabes",
"suffix": ""
}
],
"year": 1997,
"venue": "Grzegorz Rozenberg and Arto Salomaa",
"volume": "3",
"issue": "",
"pages": "69--123",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joshi, Aravind K. and Yves Schabes. 1997. Tree-Adjoining Grammars. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages, volume 3. Springer, Berlin, pages 69-123.",
"links": null
},
"BIBREF23": {
"ref_id": "b23",
"title": "The universal recognition problems for multiple context-free grammars and for linear context-free rewriting systems",
"authors": [
{
"first": "Yuichi",
"middle": [],
"last": "Kaji",
"suffix": ""
},
{
"first": "Ryuichi",
"middle": [],
"last": "Nakanishi",
"suffix": ""
},
{
"first": "Hiroyuki",
"middle": [],
"last": "Seki",
"suffix": ""
},
{
"first": "Tadao",
"middle": [],
"last": "Kasami",
"suffix": ""
}
],
"year": 1992,
"venue": "IEICE Transactions on Information and Systems",
"volume": "",
"issue": "1",
"pages": "78--88",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kaji, Yuichi, Ryuichi Nakanishi, Hiroyuki Seki, and Tadao Kasami. 1992. The universal recognition problems for multiple context-free grammars and for linear context-free rewriting systems. IEICE Transactions on Information and Systems, E75-D(1):78-88.",
"links": null
},
"BIBREF24": {
"ref_id": "b24",
"title": "The pumping lemma for well-nested multiple context-free languages",
"authors": [
{
"first": "Laura",
"middle": [],
"last": "Kallmeyer",
"suffix": ""
}
],
"year": 2009,
"venue": "Developments in Language Theory. Proceedings of the 13th International Conference",
"volume": "5583",
"issue": "",
"pages": "312--325",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kallmeyer, Laura. 2010. Parsing Beyond Context-Free Grammars. Springer, Berlin. Kanazawa, Makoto. 2009. The pumping lemma for well-nested multiple context-free languages. In Developments in Language Theory. Proceedings of the 13th International Conference, DLT 2009, volume 5583 of Lecture Notes in Computer Science, pages 312-325, Stuttgart.",
"links": null
},
"BIBREF25": {
"ref_id": "b25",
"title": "The copying power of well-nested multiple context-free grammars",
"authors": [
{
"first": "Makoto",
"middle": [],
"last": "Kanazawa",
"suffix": ""
},
{
"first": "Sylvain",
"middle": [],
"last": "Salvati",
"suffix": ""
}
],
"year": 2010,
"venue": "Language and Automata Theory and Applications. Proceedings of the 4th International Conference",
"volume": "6031",
"issue": "",
"pages": "344--355",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kanazawa, Makoto and Sylvain Salvati. 2010. The copying power of well-nested multiple context-free grammars. In Adrian-Horia Dediu, Henning Fernau, and Carlos Mart\u00edn-Vide, editors, Language and Automata Theory and Applications. Proceedings of the 4th International Conference, LATA 2010, volume 6031 of Lecture Notes in Computer Science, pages 344-355, Trier.",
"links": null
},
"BIBREF26": {
"ref_id": "b26",
"title": "Dependency trees and the strong generative capacity of CCG",
"authors": [
{
"first": "Alexander",
"middle": [],
"last": "Koller",
"suffix": ""
},
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
}
],
"year": 2009,
"venue": "Proceedings of the 12th Conference of the European Chapter of the Association for Computational Linguistics (EACL)",
"volume": "",
"issue": "",
"pages": "460--468",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Koller, Alexander and Marco Kuhlmann. 2009. Dependency trees and the strong generative capacity of CCG. In Proceedings of the 12th Conference of the European Chapter of the Association for Computational Linguistics (EACL), pages 460-468, Athens.",
"links": null
},
"BIBREF27": {
"ref_id": "b27",
"title": "The Mathematics of Language",
"authors": [
{
"first": "Marcus",
"middle": [],
"last": "Kracht",
"suffix": ""
}
],
"year": 2003,
"venue": "Studies in Generative Grammar. Mouton de Gruyter",
"volume": "63",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kracht, Marcus. 2003. The Mathematics of Language, volume 63 of Studies in Generative Grammar. Mouton de Gruyter, Paris.",
"links": null
},
"BIBREF28": {
"ref_id": "b28",
"title": "The Danish Dependency Treebank and the underlying linguistic theory",
"authors": [
{
"first": "Matthias",
"middle": [],
"last": "Kromann",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Trautner",
"suffix": ""
}
],
"year": 2003,
"venue": "Proceedings of the Second Workshop on Treebanks and Linguistic Theories (TLT)",
"volume": "",
"issue": "",
"pages": "217--220",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kromann, Matthias Trautner. 2003. The Danish Dependency Treebank and the underlying linguistic theory. In Proceedings of the Second Workshop on Treebanks and Linguistic Theories (TLT), pages 217-220, V\u00e4xj\u00f6.",
"links": null
},
"BIBREF29": {
"ref_id": "b29",
"title": "Dependency Parsing. Synthesis Lectures on Human Language Technologies",
"authors": [
{
"first": "Sandra",
"middle": [],
"last": "K\u00fcbler",
"suffix": ""
},
{
"first": "Ryan",
"middle": [],
"last": "Mcdonald",
"suffix": ""
},
{
"first": "Joakim",
"middle": [],
"last": "Nivre",
"suffix": ""
}
],
"year": 2009,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "K\u00fcbler, Sandra, Ryan McDonald, and Joakim Nivre. 2009. Dependency Parsing. Synthesis Lectures on Human Language Technologies. Morgan and Claypool.",
"links": null
},
"BIBREF30": {
"ref_id": "b30",
"title": "Treebank grammar techniques for non-projective dependency parsing",
"authors": [
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
},
{
"first": "Joakim",
"middle": [],
"last": "Nivre",
"suffix": ""
},
{
"first": ";",
"middle": [],
"last": "Sydney",
"suffix": ""
},
{
"first": "Marco",
"middle": [],
"last": "Kuhlmann",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 2006,
"venue": "Proceedings of the 21st International Conference on Computational Linguistics (COLING) and 44th Annual Meeting of the Association for Computational Linguistics (ACL) Main Conference Poster Sessions",
"volume": "",
"issue": "",
"pages": "478--486",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kuhlmann, Marco and Joakim Nivre. 2006. Mildly non-projective dependency structures. In Proceedings of the 21st International Conference on Computational Linguistics (COLING) and 44th Annual Meeting of the Association for Computational Linguistics (ACL) Main Conference Poster Sessions, pages 507-514, Sydney. Kuhlmann, Marco and Giorgio Satta. 2009. Treebank grammar techniques for non-projective dependency parsing. In Proceedings of the 12th Conference of the European Chapter of the Association for Computational Linguistics (EACL), pages 478-486, Athens.",
"links": null
},
"BIBREF31": {
"ref_id": "b31",
"title": "Recognition can be harder than parsing",
"authors": [
{
"first": "Bernard",
"middle": [],
"last": "Lang",
"suffix": ""
}
],
"year": 1994,
"venue": "Computational Intelligence",
"volume": "10",
"issue": "4",
"pages": "486--494",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Lang, Bernard. 1994. Recognition can be harder than parsing. Computational Intelligence, 10(4):486-494.",
"links": null
},
"BIBREF32": {
"ref_id": "b32",
"title": "First-and second-order expectation semirings with applications to minimum-risk training on translation forests",
"authors": [
{
"first": "Zhifei",
"middle": [],
"last": "Li",
"suffix": ""
},
{
"first": "Jason",
"middle": [],
"last": "Eisner",
"suffix": ""
}
],
"year": 2009,
"venue": "Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing (EMNLP)",
"volume": "",
"issue": "",
"pages": "40--51",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Li, Zhifei and Jason Eisner. 2009. First-and second-order expectation semirings with applications to minimum-risk training on translation forests. In Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 40-51, Singapore.",
"links": null
},
"BIBREF33": {
"ref_id": "b33",
"title": "Discontinuity and non-projectivity: Using mildly context-sensitive formalisms for data-driven parsing",
"authors": [
{
"first": "Wolfgang",
"middle": [],
"last": "Maier",
"suffix": ""
},
{
"first": "Laura",
"middle": [],
"last": "Kallmeyer",
"suffix": ""
}
],
"year": 2010,
"venue": "Proceedings of the Tenth International Conference on Tree Adjoining Grammars and Related Formalisms (TAG+)",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Maier, Wolfgang and Laura Kallmeyer. 2010. Discontinuity and non-projectivity: Using mildly context-sensitive formalisms for data-driven parsing. In Proceedings of the Tenth International Conference on Tree Adjoining Grammars and Related Formalisms (TAG+), New Haven, CT.",
"links": null
},
"BIBREF34": {
"ref_id": "b34",
"title": "Characterizing discontinuity in constituent treebanks",
"authors": [
{
"first": "Wolfgang",
"middle": [],
"last": "Maier",
"suffix": ""
},
{
"first": "Timm",
"middle": [],
"last": "Lichte",
"suffix": ""
}
],
"year": 2009,
"venue": "Formal Grammar. Proceedings of the 14th International Conference",
"volume": "5591",
"issue": "",
"pages": "167--182",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Maier, Wolfgang and Timm Lichte. 2011. Characterizing discontinuity in constituent treebanks. In Philippe de Groote, Markus Egg, and Laura Kallmeyer, editors, Formal Grammar. Proceedings of the 14th International Conference, FG 2009, Revised Selected Papers, volume 5591 of Lecture Notes in Computer Science, pages 167-182, Bordeaux.",
"links": null
},
"BIBREF35": {
"ref_id": "b35",
"title": "Treebanks and mild context-sensitivity",
"authors": [
{
"first": "Wolfgang",
"middle": [],
"last": "Maier",
"suffix": ""
},
{
"first": "Anders",
"middle": [],
"last": "S\u00f8gaard",
"suffix": ""
}
],
"year": 2008,
"venue": "Proceedings of the 13th Conference on Formal Grammar (FG)",
"volume": "",
"issue": "",
"pages": "61--76",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Maier, Wolfgang and Anders S\u00f8gaard. 2008. Treebanks and mild context-sensitivity. In Proceedings of the 13th Conference on Formal Grammar (FG), pages 61-76, Hamburg.",
"links": null
},
"BIBREF36": {
"ref_id": "b36",
"title": "On the complexity analysis of static analyses",
"authors": [
{
"first": "David",
"middle": [],
"last": "Mcallester",
"suffix": ""
}
],
"year": 2002,
"venue": "Journal of the Association for Computing Machinery",
"volume": "49",
"issue": "4",
"pages": "512--537",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "McAllester, David. 2002. On the complexity analysis of static analyses. Journal of the Association for Computing Machinery, 49(4):512-537.",
"links": null
},
"BIBREF37": {
"ref_id": "b37",
"title": "Derivational minimalism is mildly context-sensitive",
"authors": [
{
"first": "Igor",
"middle": [],
"last": "Mel'\u010duk",
"suffix": ""
}
],
"year": 1988,
"venue": "Logical Aspects of Computational Linguistics, Third International Conference",
"volume": "2014",
"issue": "",
"pages": "179--198",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Mel'\u010duk, Igor. 1988. Dependency Syntax: Theory and Practice. State University of New York Press, Albany, NY. Michaelis, Jens. 1998. Derivational minimalism is mildly context-sensitive. In Logical Aspects of Computational Linguistics, Third International Conference, LACL 1998, Selected Papers, volume 2014 of Lecture Notes in Computer Science, pages 179-198, Grenoble.",
"links": null
},
"BIBREF38": {
"ref_id": "b38",
"title": "On Formal Properties of Minimalist Grammars",
"authors": [
{
"first": "Jens",
"middle": [],
"last": "Michaelis",
"suffix": ""
}
],
"year": 2001,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Michaelis, Jens. 2001. On Formal Properties of Minimalist Grammars. Ph.D. thesis, Universit\u00e4t Potsdam, Potsdam, Germany.",
"links": null
},
"BIBREF39": {
"ref_id": "b39",
"title": "The CoNLL 2007 shared task on dependency parsing",
"authors": [
{
"first": "Joakim",
"middle": [],
"last": "Nivre",
"suffix": ""
},
{
"first": "Johan",
"middle": [],
"last": "Hall",
"suffix": ""
},
{
"first": "Sandra",
"middle": [],
"last": "K\u00fcbler",
"suffix": ""
},
{
"first": "Ryan",
"middle": [],
"last": "Mcdonald",
"suffix": ""
},
{
"first": "Jens",
"middle": [],
"last": "Nilsson",
"suffix": ""
},
{
"first": "Sebastian",
"middle": [],
"last": "Riedel",
"suffix": ""
},
{
"first": "Deniz",
"middle": [],
"last": "Yuret",
"suffix": ""
}
],
"year": 2007,
"venue": "Proceedings of the Joint Conference on Empirical Methods in Natural Language Processing (EMNLP) and Computational Natural Language Learning (CoNLL)",
"volume": "",
"issue": "",
"pages": "915--932",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Nivre, Joakim, Johan Hall, Sandra K\u00fcbler, Ryan McDonald, Jens Nilsson, Sebastian Riedel, and Deniz Yuret. 2007. The CoNLL 2007 shared task on dependency parsing. In Proceedings of the Joint Conference on Empirical Methods in Natural Language Processing (EMNLP) and Computational Natural Language Learning (CoNLL), pages 915-932, Prague.",
"links": null
},
"BIBREF40": {
"ref_id": "b40",
"title": "Treebanks: Building and Using Parsed Corpora",
"authors": [
{
"first": "Kemal",
"middle": [],
"last": "Oflazer",
"suffix": ""
},
{
"first": "Bilge",
"middle": [],
"last": "Say",
"suffix": ""
},
{
"first": "G\u00f6khan",
"middle": [],
"last": "Dilek Zeynep Hakkani-T\u00fcr",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "T\u00fcr",
"suffix": ""
}
],
"year": 2003,
"venue": "",
"volume": "",
"issue": "",
"pages": "261--277",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Oflazer, Kemal, Bilge Say, Dilek Zeynep Hakkani-T\u00fcr, and G\u00f6khan T\u00fcr. 2003. Building a Turkish treebank. In Abeill\u00e9, Anne, editor. Treebanks: Building and Using Parsed Corpora. Kluwer Academic Publishers, Dordrecht, chapter 15, pages 261-277.",
"links": null
},
"BIBREF41": {
"ref_id": "b41",
"title": "A formal look at dependency grammars and phrase-structure grammars, with special consideration of word-order phenomena",
"authors": [
{
"first": "Owen",
"middle": [],
"last": "Rambow",
"suffix": ""
},
{
"first": "Aravind",
"middle": [
"K"
],
"last": "Joshi",
"suffix": ""
}
],
"year": 1997,
"venue": "Recent Trends in Meaning-Text Theory",
"volume": "39",
"issue": "",
"pages": "167--190",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Rambow, Owen and Aravind K. Joshi. 1997. A formal look at dependency grammars and phrase-structure grammars, with special consideration of word-order phenomena. In Leo Wanner, editor, Recent Trends in Meaning-Text Theory, volume 39 of Studies in Language, Companion Series. John Benjamins, Amsterdam, pages 167-190.",
"links": null
},
"BIBREF42": {
"ref_id": "b42",
"title": "D-Tree grammars",
"authors": [
{
"first": "Owen",
"middle": [],
"last": "Rambow",
"suffix": ""
},
{
"first": "K",
"middle": [],
"last": "Vijay-Shanker",
"suffix": ""
},
{
"first": "David",
"middle": [
"J"
],
"last": "Weir",
"suffix": ""
}
],
"year": 1995,
"venue": "Proceedings of the 33rd Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "151--158",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Rambow, Owen, K. Vijay-Shanker, and David J. Weir. 1995. D-Tree grammars. In Proceedings of the 33rd Annual Meeting of the Association for Computational Linguistics (ACL), pages 151-158, Cambridge, MA.",
"links": null
},
"BIBREF43": {
"ref_id": "b43",
"title": "Optimal rank reduction for linear context-free rewriting systems with fan-out two",
"authors": [
{
"first": "Beno\u00eet",
"middle": [],
"last": "Sagot",
"suffix": ""
},
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 2010,
"venue": "Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "525--533",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Sagot, Beno\u00eet and Giorgio Satta. 2010. Optimal rank reduction for linear context-free rewriting systems with fan-out two. In Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics (ACL), pages 525-533, Uppsala.",
"links": null
},
"BIBREF44": {
"ref_id": "b44",
"title": "Recognition of linear context-free rewriting systems",
"authors": [
{
"first": "Giorgio",
"middle": [],
"last": "Satta",
"suffix": ""
}
],
"year": 1992,
"venue": "Proceedings of the 30th Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "89--95",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Satta, Giorgio. 1992. Recognition of linear context-free rewriting systems. In Proceedings of the 30th Annual Meeting of the Association for Computational Linguistics (ACL), pages 89-95, Newark, DE.",
"links": null
},
"BIBREF45": {
"ref_id": "b45",
"title": "Parsing strategies with 'lexicalized' grammars: Application to tree adjoining grammars",
"authors": [
{
"first": "Yves",
"middle": [],
"last": "Schabes",
"suffix": ""
},
{
"first": "P",
"middle": [
"A"
],
"last": "Philadelphia",
"suffix": ""
},
{
"first": "Yves",
"middle": [],
"last": "Schabes",
"suffix": ""
},
{
"first": "Anne",
"middle": [],
"last": "Abeill\u00e9",
"suffix": ""
},
{
"first": "Aravind",
"middle": [
"K"
],
"last": "Joshi",
"suffix": ""
}
],
"year": 1988,
"venue": "Proceedings of the Twelfth International Conference on Computational Linguistics (COLING)",
"volume": "88",
"issue": "",
"pages": "191--229",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Schabes, Yves. 1990. Mathematical and Computational Aspects of Lexicalized Grammars. Ph.D. thesis, University of Pennsylvania, Philadelphia, PA. Schabes, Yves, Anne Abeill\u00e9, and Aravind K. Joshi. 1988. Parsing strategies with 'lexicalized' grammars: Application to tree adjoining grammars. In Proceedings of the Twelfth International Conference on Computational Linguistics (COLING), pages 578-583, Budapest. Seki, Hiroyuki, Takashi Matsumura, Mamoru Fujii, and Tadao Kasami. 1991. On Multiple Context-Free Grammars. Theoretical Computer Science, 88(2):191-229.",
"links": null
},
"BIBREF46": {
"ref_id": "b46",
"title": "The Meaning of the Sentence in Its Semantic and Pragmatic Aspects",
"authors": [
{
"first": "Petr",
"middle": [],
"last": "Sgall",
"suffix": ""
},
{
"first": "Eva",
"middle": [],
"last": "Haji\u010dov\u00e1",
"suffix": ""
},
{
"first": "Jarmila",
"middle": [],
"last": "Panevov\u00e1",
"suffix": ""
}
],
"year": 1986,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Sgall, Petr, Eva Haji\u010dov\u00e1, and Jarmila Panevov\u00e1. 1986. The Meaning of the Sentence in Its Semantic and Pragmatic Aspects. Springer, Berlin.",
"links": null
},
"BIBREF47": {
"ref_id": "b47",
"title": "Evidence against the context-freeness of natural language",
"authors": [
{
"first": "Stuart",
"middle": [
"M"
],
"last": "Shieber",
"suffix": ""
}
],
"year": 1985,
"venue": "Linguistics and Philosophy",
"volume": "8",
"issue": "3",
"pages": "333--343",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Shieber, Stuart M. 1985. Evidence against the context-freeness of natural language. Linguistics and Philosophy, 8(3):333-343.",
"links": null
},
"BIBREF48": {
"ref_id": "b48",
"title": "Principles and implementation of deductive parsing",
"authors": [
{
"first": "Stuart",
"middle": [
"M"
],
"last": "Shieber",
"suffix": ""
},
{
"first": "Yves",
"middle": [],
"last": "Schabes",
"suffix": ""
},
{
"first": "Fernando",
"middle": [],
"last": "Pereira",
"suffix": ""
}
],
"year": 1995,
"venue": "Journal of Logic Programming",
"volume": "24",
"issue": "1-2",
"pages": "3--36",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Shieber, Stuart M., Yves Schabes, and Fernando Pereira. 1995. Principles and implementation of deductive parsing. Journal of Logic Programming, 24(1-2):3-36.",
"links": null
},
"BIBREF49": {
"ref_id": "b49",
"title": "Combinatory categorial grammar",
"authors": [
{
"first": "Mark",
"middle": [],
"last": "Steedman",
"suffix": ""
},
{
"first": "Jason",
"middle": [],
"last": "Baldridge",
"suffix": ""
}
],
"year": 2011,
"venue": "Non-Transformational Syntax: Formal and Explicit Models of Grammar",
"volume": "",
"issue": "",
"pages": "181--224",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Steedman, Mark and Jason Baldridge. 2011. Combinatory categorial grammar. In Robert D. Borsley and Kersti B\u00f6rjars, editors, Non-Transformational Syntax: Formal and Explicit Models of Grammar. Wiley-Oxford, Blackwell, chapter 5, pages 181-224.",
"links": null
},
"BIBREF50": {
"ref_id": "b50",
"title": "\u00c9l\u00e9ments de syntaxe structurale",
"authors": [
{
"first": "Lucien",
"middle": [],
"last": "Tesni\u00e8re",
"suffix": ""
}
],
"year": 1959,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Tesni\u00e8re, Lucien. 1959.\u00c9l\u00e9ments de syntaxe structurale. Klinksieck, Paris.",
"links": null
},
"BIBREF51": {
"ref_id": "b51",
"title": "Characterizing structural descriptions produced by various grammatical formalisms",
"authors": [
{
"first": "K",
"middle": [],
"last": "Vijay-Shanker",
"suffix": ""
},
{
"first": "J",
"middle": [],
"last": "David",
"suffix": ""
},
{
"first": "Aravind",
"middle": [
"K"
],
"last": "Weir",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Joshi",
"suffix": ""
}
],
"year": 1987,
"venue": "Proceedings of the 25th Annual Meeting of the Association for Computational Linguistics (ACL)",
"volume": "",
"issue": "",
"pages": "104--111",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Vijay-Shanker, K., David J. Weir, and Aravind K. Joshi. 1987. Characterizing structural descriptions produced by various grammatical formalisms. In Proceedings of the 25th Annual Meeting of the Association for Computational Linguistics (ACL), pages 104-111, Stanford, CA.",
"links": null
},
"BIBREF52": {
"ref_id": "b52",
"title": "Parsing mildly context-sensitive languages with thread automata",
"authors": [
{
"first": "\u00c9ric",
"middle": [],
"last": "Villemonte De La Clergerie",
"suffix": ""
}
],
"year": 2002,
"venue": "Proceedings of the 19th International Conference on Computational Linguistics (COLING)",
"volume": "",
"issue": "",
"pages": "1--7",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Villemonte de la Clergerie,\u00c9ric. 2002. Parsing mildly context-sensitive languages with thread automata. In Proceedings of the 19th International Conference on Computational Linguistics (COLING), pages 1-7, Taipei.",
"links": null
},
"BIBREF53": {
"ref_id": "b53",
"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": "Weir, David J. 1988. Characterizing Mildly Context-Sensitive Grammar Formalisms. Ph.D. thesis, University of Pennsylvania, Philadelphia, PA.",
"links": null
}
},
"ref_entries": {
"FIGREF0": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "Nested dependencies and cross-serial dependencies."
},
"FIGREF1": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "Lexicalized linear context-free rewriting systems."
},
"FIGREF2": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "Lexicalized linear context-free rewriting systems induce dependency trees."
},
"FIGREF3": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "Dependency trees."
},
"FIGREF4": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "Figure 7 A dependency tree and one of its construction trees."
},
"FIGREF5": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "fragment of the grammar used in the proof of Lemma 7."
},
"FIGREF6": {
"num": null,
"uris": null,
"type_str": "figure",
"text": "r A dependency tree D is called well-nested if for all pairs of nodes u, v of D u v implies that u \u2229 v = \u2205 In other words, u and v may overlap only if u is an ancestor of v, or v is an ancestor of u. If this implication does not hold, then D is called ill-nested."
},
"TABREF1": {
"type_str": "table",
"text": "Computing the blocks of a simple dependency tree.Input: a string w and a simple dependency tree D for w 1: current \u2190 \u22a5; mark current 2: for each node next of D from 1 to |w| do",
"content": "<table><tr><td>3:</td><td>lca \u2190 next; stack \u2190 []</td><td/></tr><tr><td>4:</td><td>while lca is not marked do</td><td>loop 1</td></tr><tr><td>5:</td><td>push lca to stack; lca \u2190 the parent of lca</td><td/></tr><tr><td>6:</td><td>while current = lca do</td><td>loop 2</td></tr><tr><td>7:</td><td>next \u2212 1 is the right endpoint of a block of current</td><td/></tr><tr><td>8:</td><td>move up from current to the parent of current</td><td/></tr><tr><td>9:</td><td>unmark current; current \u2190 the parent of current</td><td/></tr><tr><td>10:</td><td>while stack is not empty do</td><td>loop 3</td></tr><tr><td>11:</td><td>current \u2190 pop stack; mark current</td><td/></tr><tr><td>12:</td><td>move down from the parent of current to current</td><td/></tr><tr><td>13:</td><td>next is the left endpoint of a block of current</td><td/></tr><tr><td>14:</td><td/><td/></tr></table>",
"html": null,
"num": null
},
"TABREF2": {
"type_str": "table",
"text": "Loss in coverage under the restriction to yield functions with fan-out = 1 and fan-out \u2264 2.Let D be a dependency tree, and let u and v be nodes of D. The descendants of u and v overlap, denoted by u v , if there exist nodes u l , u r",
"content": "<table><tr><td/><td/><td/><td colspan=\"2\">fan-out = 1</td><td colspan=\"2\">fan-out \u2264 2</td></tr><tr><td/><td>rules</td><td>trees</td><td>rules</td><td>trees</td><td colspan=\"2\">rules trees</td></tr><tr><td>Arabic</td><td>5,839</td><td>1,460</td><td>411</td><td>163</td><td>1</td><td>1</td></tr><tr><td>Czech</td><td colspan=\"4\">1,322,111 72,703 22,283 16,831</td><td>328</td><td>312</td></tr><tr><td>Danish</td><td>99,576</td><td>5,190</td><td>1,229</td><td>811</td><td>11</td><td>9</td></tr><tr><td>Slovene</td><td>30,284</td><td>1,534</td><td>530</td><td>340</td><td>14</td><td>11</td></tr><tr><td>Turkish</td><td>62,507</td><td>4,997</td><td>924</td><td>580</td><td>54</td><td>33</td></tr></table>",
"html": null,
"num": null
}
}
}
} |