289:
568:
on the alphabet of 0 and 1 has the unambiguous context-free grammar S → 0S0 | 1S1 | ε. An arbitrary string of this language cannot be parsed without reading all its symbols first, which means that a pushdown automaton has to try alternative state transitions to accommodate for the different possible
791:
584:
While some context-free languages (the set of strings that can be generated by a grammar) have both ambiguous and unambiguous grammars, there exist context-free languages for which no unambiguous context-free grammar can exist. Such languages are called
103:…meaning that the nonterminal A can be derived to either itself again, or to the empty string. Thus the empty string has leftmost derivations of length 1, 2, 3, and indeed of any length, depending on how many times the rule A → A is used.
929:
1343:
609:
1098:
1017:
1424:
1308:
1177:
494:
is an unambiguous grammar for the language { 0+0, 0+1, 1+0, 1+1 }. While each of these four strings has only one leftmost derivation, it has two different derivations, for example
465:
mandatory. In other cases the grammar is left ambiguous, but the ambiguity is resolved by making the overall phrase grammar context-sensitive, such as by associating an
329:
Concretely, in many languages one may write conditionals in two valid forms: the if-then form, and the if-then-else form – in effect, making the else clause optional:
572:
Nevertheless, removing grammar ambiguity may produce a deterministic context-free grammar and thus allow for more efficient parsing. Compiler generators such as
1182:
More examples, and a general review of techniques for proving inherent ambiguity of context-free languages, are found given by
Bassino and Nicaud (2011).
802:
95:
The simplest example is the following ambiguous grammar (with start symbol A) for the trivial language that consists of only the empty string:
457:
This is resolved in various ways in different languages. Sometimes the grammar is modified so that it is unambiguous, such as by requiring an
1365:
1233:
295:
The language that it generates, however, is not inherently ambiguous; the following is a non-ambiguous grammar generating the same language:
326:
statement is optional, which results in nested conditionals having multiple ways of being recognized in terms of the context-free grammar.
55:
are always unambiguous, and are an important subclass of unambiguous grammars; there are non-deterministic unambiguous grammars, however.
1830:
538:
323:
288:
107:
52:
1649:
1753:
1726:
1433:
1349:
1317:
786:{\displaystyle \{x|x=a^{n}b^{m}a^{n^{\prime }}b^{m}{\text{ or }}x=a^{n}b^{m}a^{n}b^{m^{\prime }},{\text{ where }}n,n',m,m'\geq 1\}}
481:
The existence of multiple derivations of the same string does not suffice to indicate that the grammar is ambiguous; only multiple
1223:
542:
47:
admits an ambiguous grammar by introducing e.g. a duplicate rule. A language that only admits ambiguous grammars is called an
1703:
1100:
is inherently ambiguous. This set is context-free, since the union of two context-free languages is always context-free. But
1022:
941:
1808:
1496:
576:
include features for resolving some kinds of ambiguity, such as by using the precedence and associativity constraints.
527:
1833:- tool for analyzing context-free grammars with respect to language universality, ambiguity, and similar properties.
1107:
118:…meaning that the unique production can produce only the empty string, which is the unique string in the language.
1846:
1763:
Brabrand, Claus; Giegerich, Robert; Møller, Anders (March 2010). "Analyzing
Ambiguity of Context-Free Grammars".
67:
32:
1104:
give a proof that any context-free grammar for this union language cannot unambiguously parse strings of form
1334:
161:
tree (for the unambiguous grammar) or allowing both left- and right- association. This is elaborated below.
79:
1772:
600:
550:
75:
44:
219:→ A + A + A (First A is replaced by A+A. Replacement of the second A would yield a similar derivation)
498:
170:
59:
28:
564:
Unambiguous context-free grammars can be nondeterministic. For example, the language of even-length
1777:
1745:
1691:
1203:
596:
523:
36:
121:
In the same way, any grammar for a non-empty language can be made ambiguous by adding duplicates.
66:
problem. If present, these ambiguities are generally resolved by adding precedence rules or other
1851:
1614:
1566:
1478:
537:
The efficiency of parsing a context-free grammar is determined by the automaton that accepts it.
1749:
1722:
1699:
1643:"Philippe Flajolet & Analytic Combinatorics: Inherent Ambiguity of Context-Free Languages"
1606:
1558:
1470:
1429:
1361:
1313:
1229:
554:
158:
70:
parsing rules, so the overall phrase grammar is unambiguous. Some parsing algorithms (such as
473:. In this latter case the grammar is unambiguous, but the context-free grammar is ambiguous.
1782:
1737:
1677:
1598:
1548:
1462:
1393:
1353:
1262:
932:
796:
519:
130:
20:
1642:
1738:
1419:
1415:
1303:
1299:
531:
1682:
1665:
1397:
1840:
1715:
1411:
1295:
558:
315:
309:
78:
parsers) can generate sets of parse trees (or "parse forests") from strings that are
71:
63:
1618:
1570:
1482:
1381:
1197:
1357:
1282:
1786:
1267:
1250:
180:
is ambiguous since there are two leftmost derivations for the string a + a + a:
924:{\displaystyle \{a^{n}b^{m}c^{m}|m,n\geq 1\}\cup \{a^{m}b^{m}c^{n}|m,n\geq 1\}}
1191:
565:
280:
40:
1610:
1562:
1474:
1450:
595:
The existence of inherently ambiguous context-free languages was proven with
1466:
546:
1553:
1536:
485:
derivations (or, equivalently, multiple parse trees) indicate ambiguity.
1586:
1602:
314:
A common example of ambiguity in computer programming languages is the
106:
This language also has an unambiguous grammar, consisting of a single
62:, the reference grammar is often ambiguous, due to issues such as the
1524:. Quarterly Progress Report, Research Laboratory of Electronics, MIT.
1384:(July 1965). "On the translation of languages from left to right".
799:
can be used to prove that certain context-free languages, such as
1497:"formal languages - Can regular expressions be made unambiguous?"
1335:"Analyzing Context-Free Grammars Using an Incremental SAT Solver"
279:
As another example, the grammar is ambiguous since there are two
573:
1344:
International
Colloquium on Automata, Languages and Programming
1824:
1225:
An
Introduction to the Theory of Formal Languages and Automata
1194:, a type of parser for ambiguous and nondeterministic grammars
728:
660:
358:
some ambiguous phrase structures can appear. The expression
1717:
Introduction to
Automata Theory, Languages, and Computation
1425:
Introduction to automata theory, languages, and computation
1312:(2nd ed.). Addison-Wesley. Theorem 9.20, pp. 405–406.
1309:
Introduction to automata theory, languages, and computation
1740:
Introduction to
Automata Theory, Languages and Computation
1736:
Hopcroft, John; Motwani, Rajeev; Ullman, Jeffrey (2001).
1641:
Fredérique
Bassino and Cyril Nicaud (December 16, 2011).
1333:
Axelsson, Roland; Heljanko, Keijo; Lange, Martin (2008).
557:
and can be parsed in polynomial time, for example by the
1449:
Book, R.; Even, S.; Greibach, S.; Ott, G. (Feb 1971).
1352:. Vol. 5126. Springer-Verlag. pp. 410–422.
1283:
An efficient augmented-context-free parsing algorithm
1110:
1093:{\displaystyle \{a^{n}b^{m}c^{m}d^{n}\mid n,m>0\}}
1025:
1012:{\displaystyle \{a^{n}b^{n}c^{m}d^{m}\mid n,m>0\}}
944:
805:
612:
592:
There are no inherently ambiguous regular languages.
526:
because it can be shown that it is equivalent to the
545:
and can be parsed in linear time, for example by an
48:
1714:
1428:(2nd ed.). Addison-Wesley. pp. 249–253.
1285:." Computational linguistics 13.1-2 (1987): 31-46.
1171:
1092:
1011:
923:
785:
534:for detecting ambiguity of context-free grammars.
1587:"A helpful result for proving inherent ambiguity"
1255:Electronic Notes in Theoretical Computer Science
522:of whether an arbitrary grammar is ambiguous is
477:An unambiguous grammar with multiple derivations
1666:"Inherent ambiguity of minimal linear grammars"
1200:, another type of parser for ambiguous grammars
1713:Hopcroft, John E.; Ullman, Jeffrey D. (1979).
1101:
530:. At least, there are tools implementing some
510:Only the former derivation is a leftmost one.
1172:{\displaystyle a^{n}b^{n}c^{n}d^{n},(n>0)}
8:
1251:"SPPF-Style Parsing From Earley Recognizers"
1087:
1026:
1006:
945:
918:
865:
859:
806:
780:
613:
355:Statement | ... Condition → ...
133:of unary strings of a given character, say
1776:
1744:(2nd ed.). Addison Wesley. pp.
1681:
1552:
1266:
1145:
1135:
1125:
1115:
1109:
1063:
1053:
1043:
1033:
1024:
982:
972:
962:
952:
943:
898:
892:
882:
872:
839:
833:
823:
813:
804:
738:
727:
722:
712:
702:
692:
677:
671:
659:
654:
644:
634:
619:
611:
1800:
1214:
1696:Introduction to Formal Language Theory
1451:"Ambiguity in Graphs and Expressions"
149:…but also has the ambiguous grammar:
7:
539:Deterministic context-free grammars
53:Deterministic context-free grammars
1249:Scott, Elizabeth (April 1, 2008).
569:lengths of a semi-parsed string.
549:. They are a strict subset of the
332:In a grammar containing the rules
14:
1664:Gross, Maurice (September 1964).
1350:Lecture Notes in Computer Science
1721:(1st ed.). Addison-Wesley.
1655:from the original on 2022-09-25.
931:, are inherently ambiguous. See
488:For example, the simple grammar
318:problem. In many languages, the
287:
283:for the string a + a − a:
157:These correspond to producing a
141:), has the unambiguous grammar:
1827:- a grammar ambiguity analyzer.
1765:Science of Computer Programming
1535:Parikh, Rohit J. (1966-10-01).
543:deterministic pushdown automata
1520:Parikh, Rohit (January 1961).
1455:IEEE Transactions on Computers
1346:(ICALP'08), Reykjavik, Iceland
1166:
1154:
899:
840:
620:
580:Inherently ambiguous languages
514:Recognizing ambiguous grammars
16:Type of a context-free grammar
1:
1683:10.1016/S0019-9958(64)90422-X
1398:10.1016/S0019-9958(65)90426-2
1228:. John Benjamins Publishing.
446:is associated with the first
49:inherently ambiguous language
1358:10.1007/978-3-540-70583-3_34
1222:Willem J. M. Levelt (2008).
1102:Hopcroft & Ullman (1979)
603:in an MIT research report.
35:that can have more than one
1807:The following example uses
1787:10.1016/j.scico.2009.11.002
1591:Mathematical Systems Theory
1585:Ogden, William (Sep 1968).
1537:"On Context-Free Languages"
1522:Language-generating devices
1268:10.1016/j.entcs.2008.03.044
528:Post correspondence problem
182:
1868:
507:S ⇒ A + A ⇒ A + 0 ⇒ 0 + 0
307:
793:is inherently ambiguous.
442:depending on whether the
268:
250:
232:
214:
195:
31:for which there exists a
1771:(3). Elsevier: 176–191.
1342:Proceedings of the 35th
553:, which are accepted by
382:can be parsed as either
343:Statement |
260:
242:
224:
206:
186:
165:Addition and subtraction
137:(the regular expression
1670:Information and Control
1467:10.1109/t-c.1971.223204
1386:Information and Control
532:semi-decision procedure
80:syntactically ambiguous
1173:
1094:
1013:
925:
787:
501:A + A ⇒ 0 + A ⇒ 0 + 0
1554:10.1145/321356.321364
1174:
1095:
1014:
926:
788:
551:context-free grammars
299:A → A + a | A − a | a
176:A → A + A | A − A | a
60:programming languages
45:context-free language
1108:
1023:
942:
803:
610:
587:inherently ambiguous
491:S → A + A A → 0 | 1
461:statement or making
171:context free grammar
29:context-free grammar
1204:Syntactic ambiguity
37:leftmost derivation
1698:. Addison-Wesley.
1631:p.99-103, Sect.4.7
1603:10.1007/bf01694004
1541:Journal of the ACM
1169:
1090:
1009:
921:
783:
43:. Every non-empty
1692:Michael, Harrison
1367:978-3-540-70582-6
1281:Tomita, Masaru. "
1235:978-90-272-3250-2
741:
740: where
680:
555:pushdown automata
469:with the nearest
277:
276:
159:right-associative
68:context-sensitive
25:ambiguous grammar
1859:
1847:Formal languages
1825:dk.brics.grammar
1812:
1805:
1790:
1780:
1759:
1743:
1732:
1720:
1709:
1687:
1685:
1657:
1656:
1654:
1647:
1638:
1632:
1629:
1623:
1622:
1582:
1576:
1575:Here: Theorem 3.
1574:
1556:
1532:
1526:
1525:
1517:
1511:
1510:
1508:
1507:
1493:
1487:
1486:
1446:
1440:
1439:
1408:
1402:
1401:
1378:
1372:
1371:
1339:
1330:
1324:
1323:
1292:
1286:
1279:
1273:
1272:
1270:
1246:
1240:
1239:
1219:
1178:
1176:
1175:
1170:
1150:
1149:
1140:
1139:
1130:
1129:
1120:
1119:
1099:
1097:
1096:
1091:
1068:
1067:
1058:
1057:
1048:
1047:
1038:
1037:
1018:
1016:
1015:
1010:
987:
986:
977:
976:
967:
966:
957:
956:
930:
928:
927:
922:
902:
897:
896:
887:
886:
877:
876:
843:
838:
837:
828:
827:
818:
817:
792:
790:
789:
784:
773:
756:
742:
739:
734:
733:
732:
731:
717:
716:
707:
706:
697:
696:
681:
678:
676:
675:
666:
665:
664:
663:
649:
648:
639:
638:
623:
597:Parikh's theorem
541:are accepted by
520:decision problem
472:
468:
464:
460:
453:
449:
445:
321:
291:
183:
140:
136:
131:regular language
91:Trivial language
21:computer science
1867:
1866:
1862:
1861:
1860:
1858:
1857:
1856:
1837:
1836:
1821:
1816:
1815:
1806:
1802:
1797:
1762:
1756:
1735:
1729:
1712:
1706:
1690:
1663:
1660:
1652:
1645:
1640:
1639:
1635:
1630:
1626:
1584:
1583:
1579:
1534:
1533:
1529:
1519:
1518:
1514:
1505:
1503:
1495:
1494:
1490:
1448:
1447:
1443:
1436:
1420:Ullman, Jeffrey
1416:Motwani, Rajeev
1410:
1409:
1405:
1380:
1379:
1375:
1368:
1337:
1332:
1331:
1327:
1320:
1304:Ullman, Jeffrey
1300:Motwani, Rajeev
1294:
1293:
1289:
1280:
1276:
1248:
1247:
1243:
1236:
1221:
1220:
1216:
1212:
1188:
1141:
1131:
1121:
1111:
1106:
1105:
1059:
1049:
1039:
1029:
1021:
1020:
978:
968:
958:
948:
940:
939:
888:
878:
868:
829:
819:
809:
801:
800:
766:
749:
723:
718:
708:
698:
688:
667:
655:
650:
640:
630:
608:
607:
582:
516:
508:
502:
492:
479:
470:
466:
462:
458:
451:
447:
443:
440:
410:
380:
356:
319:
312:
306:
167:
153:A → aA | Aa | ε
138:
134:
127:
108:production rule
93:
88:
17:
12:
11:
5:
1865:
1863:
1855:
1854:
1849:
1839:
1838:
1835:
1834:
1828:
1820:
1819:External links
1817:
1814:
1813:
1799:
1798:
1796:
1793:
1792:
1791:
1778:10.1.1.86.3118
1760:
1754:
1733:
1727:
1710:
1704:
1688:
1676:(3): 366–368.
1659:
1658:
1633:
1624:
1597:(3): 191–194.
1577:
1547:(4): 570–581.
1527:
1512:
1488:
1461:(2): 149–153.
1441:
1434:
1412:Hopcroft, John
1403:
1392:(6): 607–639.
1373:
1366:
1325:
1318:
1296:Hopcroft, John
1287:
1274:
1241:
1234:
1213:
1211:
1208:
1207:
1206:
1201:
1195:
1187:
1184:
1168:
1165:
1162:
1159:
1156:
1153:
1148:
1144:
1138:
1134:
1128:
1124:
1118:
1114:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1066:
1062:
1056:
1052:
1046:
1042:
1036:
1032:
1028:
1008:
1005:
1002:
999:
996:
993:
990:
985:
981:
975:
971:
965:
961:
955:
951:
947:
920:
917:
914:
911:
908:
905:
901:
895:
891:
885:
881:
875:
871:
867:
864:
861:
858:
855:
852:
849:
846:
842:
836:
832:
826:
822:
816:
812:
808:
782:
779:
776:
772:
769:
765:
762:
759:
755:
752:
748:
745:
737:
730:
726:
721:
715:
711:
705:
701:
695:
691:
687:
684:
679: or
674:
670:
662:
658:
653:
647:
643:
637:
633:
629:
626:
622:
618:
615:
581:
578:
515:
512:
506:
496:
490:
478:
475:
414:
384:
360:
334:
324:If–then(–else)
308:Main article:
305:
302:
301:
300:
293:
292:
275:
274:
271:
269:
266:
263:
261:
257:
256:
253:
251:
248:
245:
243:
239:
238:
235:
233:
230:
227:
225:
221:
220:
217:
215:
212:
209:
207:
203:
202:
199:
196:
193:
190:
187:
178:
177:
166:
163:
155:
154:
147:
146:
126:
123:
116:
115:
101:
100:
92:
89:
87:
84:
15:
13:
10:
9:
6:
4:
3:
2:
1864:
1853:
1850:
1848:
1845:
1844:
1842:
1832:
1829:
1826:
1823:
1822:
1818:
1810:
1804:
1801:
1794:
1788:
1784:
1779:
1774:
1770:
1766:
1761:
1757:
1755:9780201441246
1751:
1747:
1742:
1741:
1734:
1730:
1728:9780201029888
1724:
1719:
1718:
1711:
1707:
1701:
1697:
1693:
1689:
1684:
1679:
1675:
1671:
1667:
1662:
1661:
1651:
1644:
1637:
1634:
1628:
1625:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1592:
1588:
1581:
1578:
1572:
1568:
1564:
1560:
1555:
1550:
1546:
1542:
1538:
1531:
1528:
1523:
1516:
1513:
1502:
1498:
1492:
1489:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1445:
1442:
1437:
1435:0-201-44124-1
1431:
1427:
1426:
1421:
1417:
1413:
1407:
1404:
1399:
1395:
1391:
1387:
1383:
1377:
1374:
1369:
1363:
1359:
1355:
1351:
1347:
1345:
1336:
1329:
1326:
1321:
1319:0-201-44124-1
1315:
1311:
1310:
1305:
1301:
1297:
1291:
1288:
1284:
1278:
1275:
1269:
1264:
1260:
1256:
1252:
1245:
1242:
1237:
1231:
1227:
1226:
1218:
1215:
1209:
1205:
1202:
1199:
1196:
1193:
1190:
1189:
1185:
1183:
1180:
1163:
1160:
1157:
1151:
1146:
1142:
1136:
1132:
1126:
1122:
1116:
1112:
1103:
1084:
1081:
1078:
1075:
1072:
1069:
1064:
1060:
1054:
1050:
1044:
1040:
1034:
1030:
1003:
1000:
997:
994:
991:
988:
983:
979:
973:
969:
963:
959:
953:
949:
938:The union of
936:
935:for a proof.
934:
915:
912:
909:
906:
903:
893:
889:
883:
879:
873:
869:
862:
856:
853:
850:
847:
844:
834:
830:
824:
820:
814:
810:
798:
797:Ogden's lemma
794:
777:
774:
770:
767:
763:
760:
757:
753:
750:
746:
743:
735:
724:
719:
713:
709:
703:
699:
693:
689:
685:
682:
672:
668:
656:
651:
645:
641:
635:
631:
627:
624:
616:
606:The language
604:
602:
598:
593:
590:
588:
579:
577:
575:
570:
567:
562:
560:
559:CYK algorithm
556:
552:
548:
544:
540:
535:
533:
529:
525:
521:
513:
511:
505:
500:
495:
489:
486:
484:
476:
474:
455:
439:
435:
431:
427:
424:
421:
417:
413:
408:
405:
401:
397:
394:
391:
387:
383:
378:
374:
370:
367:
363:
359:
354:
350:
346:
342:
338:
333:
330:
327:
325:
317:
316:dangling else
311:
310:Dangling else
304:Dangling else
303:
298:
297:
296:
290:
286:
285:
284:
282:
272:
270:
267:
264:
262:
259:
258:
254:
252:
249:
246:
244:
241:
240:
236:
234:
231:
228:
226:
223:
222:
218:
216:
213:
210:
208:
205:
204:
200:
197:
194:
191:
188:
185:
184:
181:
175:
174:
173:
172:
164:
162:
160:
152:
151:
150:
144:
143:
142:
132:
124:
122:
119:
113:
112:
111:
109:
104:
98:
97:
96:
90:
85:
83:
81:
77:
73:
69:
65:
64:dangling else
61:
58:For computer
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1803:
1768:
1764:
1739:
1716:
1695:
1673:
1669:
1636:
1627:
1594:
1590:
1580:
1544:
1540:
1530:
1521:
1515:
1504:. Retrieved
1501:MathOverflow
1500:
1491:
1458:
1454:
1444:
1423:
1406:
1389:
1385:
1382:Knuth, D. E.
1376:
1341:
1328:
1307:
1290:
1277:
1261:(2): 53–67.
1258:
1254:
1244:
1224:
1217:
1198:Chart parser
1181:
937:
795:
605:
601:Rohit Parikh
594:
591:
586:
583:
571:
563:
536:
517:
509:
503:
493:
487:
482:
480:
456:
441:
437:
433:
429:
425:
422:
419:
415:
411:
406:
403:
399:
395:
392:
389:
385:
381:
376:
372:
368:
365:
361:
357:
352:
348:
344:
340:
336:
335:Statement →
331:
328:
313:
294:
278:
273:→ a + a + a
265:→ a + a + a
255:→ a + a + A
247:→ a + a + A
237:→ a + A + A
229:→ a + A + A
179:
168:
156:
148:
128:
125:Unary string
120:
117:
105:
102:
94:
57:
24:
18:
1831:CFGAnalyzer
599:in 1961 by
566:palindromes
524:undecidable
281:parse trees
1841:Categories
1705:0201029553
1506:2023-02-23
1210:References
1192:GLR parser
450:or second
351:Statement
347:Condition
339:Condition
145:A → aA | ε
41:parse tree
1852:Ambiguity
1773:CiteSeerX
1611:0025-5661
1563:0004-5411
1475:0018-9340
1070:∣
989:∣
933:this page
913:≥
863:∪
854:≥
775:≥
729:′
661:′
547:LR parser
99:A → A | ε
1694:(1978).
1650:Archived
1619:13197551
1571:12263468
1483:20676251
1422:(2001).
1306:(2001).
1186:See also
771:′
754:′
483:leftmost
211:→ a + A
201:→ A + A
192:→ A + A
86:Examples
1811:syntax
1809:Pascal
1775:
1752:
1725:
1702:
1617:
1609:
1569:
1561:
1481:
1473:
1432:
1364:
1316:
1232:
412:or as
322:in an
72:Earley
33:string
1795:Notes
1653:(PDF)
1646:(PDF)
1615:S2CID
1567:S2CID
1479:S2CID
1338:(PDF)
1019:with
459:endif
423:begin
393:begin
114:A → ε
27:is a
23:, an
1750:ISBN
1723:ISBN
1700:ISBN
1607:ISSN
1559:ISSN
1471:ISSN
1459:C-20
1430:ISBN
1362:ISBN
1314:ISBN
1230:ISBN
1161:>
1082:>
1001:>
574:YACC
518:The
504:and
467:else
463:else
444:else
434:else
430:then
420:then
407:else
400:then
390:then
377:else
373:then
366:then
353:else
349:then
341:then
320:else
169:The
129:The
1783:doi
1746:217
1678:doi
1599:doi
1549:doi
1463:doi
1394:doi
1354:doi
1263:doi
1259:203
561:.
438:end
436:s2
409:s2
404:end
379:s2
135:'a'
76:GLR
74:or
39:or
19:In
1843::
1781:.
1769:75
1767:.
1748:.
1672:.
1668:.
1648:.
1613:.
1605:.
1593:.
1589:.
1565:.
1557:.
1545:13
1543:.
1539:.
1499:.
1477:.
1469:.
1457:.
1453:.
1418:;
1414:;
1388:.
1360:.
1348:.
1340:.
1302:;
1298:;
1257:.
1253:.
1179:.
589:.
497:S
471:if
454:.
452:if
448:if
432:s
428:b
426:if
418:a
416:if
402:s
398:b
396:if
388:a
386:if
375:s
371:b
369:if
364:a
362:if
345:if
337:if
139:a*
110::
82:.
51:.
1789:.
1785::
1758:.
1731:.
1708:.
1686:.
1680::
1674:7
1621:.
1601::
1595:2
1573:.
1551::
1509:.
1485:.
1465::
1438:.
1400:.
1396::
1390:8
1370:.
1356::
1322:.
1271:.
1265::
1238:.
1167:)
1164:0
1158:n
1155:(
1152:,
1147:n
1143:d
1137:n
1133:c
1127:n
1123:b
1117:n
1113:a
1088:}
1085:0
1079:m
1076:,
1073:n
1065:n
1061:d
1055:m
1051:c
1045:m
1041:b
1035:n
1031:a
1027:{
1007:}
1004:0
998:m
995:,
992:n
984:m
980:d
974:m
970:c
964:n
960:b
954:n
950:a
946:{
919:}
916:1
910:n
907:,
904:m
900:|
894:n
890:c
884:m
880:b
874:m
870:a
866:{
860:}
857:1
851:n
848:,
845:m
841:|
835:m
831:c
825:m
821:b
815:n
811:a
807:{
781:}
778:1
768:m
764:,
761:m
758:,
751:n
747:,
744:n
736:,
725:m
720:b
714:n
710:a
704:m
700:b
694:n
690:a
686:=
683:x
673:m
669:b
657:n
652:a
646:m
642:b
636:n
632:a
628:=
625:x
621:|
617:x
614:{
499:⇒
198:A
189:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
