462:
1612:
can be applied to a regular tree grammar to compute for each nonterminal the minimum weight of a derivable tree. Based on this information, it is straightforward to enumerate its language in increasing weight order. In particular, any nonterminal with infinite minimum weight produces the empty
1677:
1997:
1670:
1269:
Comon, Hubert; Dauchet, Max; Gilleron, Remi; LΓΆding, Christof; Jacquemard, Florent; Lugiez, Denis; Tison, Sophie; Tommasi, Marc (12 October 2007).
461:
1884:
1663:
1809:
1544:
Thatcher, J.W.; Wright, J.B. (1968). "Generalized Finite
Automata Theory with an Application to a Decision Problem of Second-Order Logic".
1899:
1824:
1317:
1467:
1992:
1977:
1853:
1199:
1870:
1795:
17:
1299:
1967:
1181:
1863:
1167:
2039:
1788:
1485:
Algorithms for
Regular Tree Grammar Network Search and Their Application to Mining Humanβviral Infection Patterns
1151:
1940:
1935:
1637:
Bogaert, B.; Tison, Sophie (1992). "Equality and
Disequality Constraints on Direct Subterms in Tree Automata".
2021:
Any language in each category is generated by a grammar and by an automaton in the category in the same line.
1609:
1951:
1889:
1814:
1591:
1358:
1249:
1228:
21:
1635:
Regular tree automata have been generalized to admit equality tests between sibling nodes in trees. See:
1894:
1842:
1655:
1425:
Gilleron, R.; Tison, S.; Tommasi, M. (1993). "Solving
Systems of Set Constraints using Tree Automata".
1987:
1962:
1819:
1780:
1455:
1363:
1254:
1596:
45:
33:
1972:
1914:
1858:
1561:
1445:
1376:
1323:
1174:
1146:
Rajeev Alur and
Parthasarathy Madhusudan related a subclass of regular binary tree languages to
1707:
1463:
1313:
1206:
1193:
1189:
1185:
1178:
854:, using the nonterminal set and the alphabet from above, but extending the production set by
1956:
1909:
1876:
1722:
1624:
1553:
1530:
1520:
1489:
1368:
1305:
941:
counterpart in
Standard ML, nor in any other functional language. It is a proper subset of
1919:
1834:
1801:
1717:
1690:
1686:
1133:
701:
697:
466:
214:
92:
41:
1459:
1930:
1712:
1694:
1584:
Algorithms on regular tree grammars are discussed from an efficiency-oriented view in:
1580:. Studies in Computer Science and Artificial Intelligence. Vol. 10. North-Holland.
1222:
1140:
29:
1525:
1508:
1301:
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04
550:(.,.) } is our ranked alphabet, arities indicated by dummy arguments (i.e. the symbol
2033:
2015:
1628:
449:
1565:
1605:
1395:
Emmelmann, Helmut (1991). "Code
Selection by Regularly Controlled Term Rewriting".
1327:
173:
37:
1380:
44:
can be seen as a special kind of regular tree grammar, describing a set of single-
1982:
1904:
1829:
1147:
740:
1588:
ACM Conference on
Functional Programming Languages and Computer Architecture
1372:
1343:
1309:
1292:
1645:
Allowing equality tests between deeper nodes leads to undecidability. See:
1139:
The regular tree languages are also the languages recognized by bottom-up
1270:
1586:
Aiken, A.; Murphy, B. (1991). "Implementing
Regular Tree Expressions".
1557:
1535:
1410:
Comon, Hubert (1990). "Equational
Formulas in Order-Sorted Algebras".
183:
according to their arities, where nonterminals are considered nullary.
1248:"Regular tree grammars as a formalism for scope underspecification".
700:; it is a tree of trees (main picture), whereas a derivation tree in
196:
implicitly defines a set of trees: any tree that can be derived from
1127:
Alternative characterizations and relation to other formal languages
1945:
1450:
473:
in linear (upper left table) and graphical (main picture) notation
460:
96:
1615:
Knuth, D.E. (1977). "A Generalization of Dijkstra's Algorithm".
1427:
10th Annual Symposium on Theoretical Aspects of Computer Science
1209:
about a finite algebra (which is always a regular tree language)
922:) is the set of all finite lists of boolean values that contain
430:
denotes the set of all trees composed from symbols of Ξ£, while β
1659:
1649:
Automates d'Arbres avec Tests d'ΓgalitΓ©s entre Cousins Germains
1440:
Burghardt, Jochen (2002). "Axiomatization of Finite Algebras".
446:
A language generated by some regular tree grammar is called a
1093:
are also regular tree languages, and it is decidable whether
2014:
Each category of languages, except those marked by a , is a
739:
corresponds to the algebraic data type declarations (in the
714:
is the set of all finite lists of boolean values, that is,
1503:
Regular tree grammars were already described in 1968 by:
176:, i.e. the set of all trees composed from symbols in
1444:. LNAI. Vol. 2479. Springer. pp. 222β234.
1049:
both are regular tree languages, then the tree sets
809:) corresponds to a Standard-ML value of type BList.
95:(i.e., an alphabet whose symbols have an associated
1641:. LNCS. Vol. 577. Springer. pp. 161β172.
1429:. LNCS. Vol. 665. Springer. pp. 505β514.
1399:. Workshops in Computing. Springer. pp. 3β29.
1163:Applications of regular tree grammars include:
1132:Regular tree grammars are a generalization of
1671:
1397:Code Generation - Concepts, Tools, Techniques
361:means a tree with exactly one hole in it; if
8:
1143:and nondeterministic top-down tree automata.
963:), too, as the following derivation shows:
1993:Counter-free (with aperiodic finite monoid)
1271:"Tree Automata Techniques and Applications"
1225:β a generalization of regular tree grammars
952:). The above example term happens to be in
861:, consisting of the following productions:
125:is a finite set of productions of the form
1703:
1678:
1664:
1656:
1576:Nivat, Maurice; Podelski, Andreas (1992).
1595:
1534:
1524:
1449:
1362:
1253:
704:is a tree of strings (upper left table).
1604:Given a mapping from trees to weights,
1240:
628:An example derivation from the grammar
577:consists of the following productions:
369:denotes the result of filling the tree
1885:Linear context-free rewriting language
1810:Linear context-free rewriting systems
1509:"The Minimalization of Tree Automata"
1205:The set of all truths expressible in
7:
1574:A book devoted to tree grammars is:
437:denotes successive applications of β
212:. This set of trees is known as the
1442:Advances in Artificial Intelligence
1344:"Adding nesting structure to words"
2018:of the category directly above it.
1610:Dijkstra's shortest-path algorithm
696:The image shows the corresponding
109:is the starting nonterminal, with
14:
1342:Alur, R.; Madhusudan, P. (2009).
1291:Alur, R.; Madhusudan, P. (2004).
567:is our starting nonterminal, and
88:is a finite set of nonterminals,
707:The tree language generated by
380:The tree language generated by
222:. More formally, the relation β
1617:Information Processing Letters
1:
1526:10.1016/s0019-9958(68)90917-0
527:} is our set of nonterminals,
1629:10.1016/0020-0190(77)90002-3
1293:"Visibly pushdown languages"
18:theoretical computer science
1578:Tree Automata and Languages
1546:Mathematical Systems Theory
1483:Ziv-Ukelson, Smoly (2016).
1170:in compiler code generation
2056:
1900:Deterministic context-free
1825:Deterministic context-free
1152:visibly pushdown languages
2011:
1973:Nondeterministic pushdown
1701:
812:For another example, let
307:), if there is a context
239:) is defined as follows:
745:
265:derived in a single step
60:is defined by the tuple
32:that describes a set of
1513:Information and Control
1507:Brainerd, W.S. (1968).
1373:10.1145/1516512.1516518
1310:10.1145/1007352.1007390
1202:about mathematical sets
926:at least once. The set
743:programming language):
56:A regular tree grammar
1978:Deterministic pushdown
1854:Recursively enumerable
1229:Tree-adjoining grammar
474:
22:formal language theory
1608:'s generalization of
1168:Instruction selection
1134:regular word grammars
464:
1963:Tree stack automaton
1647:Tommasi, M. (1991).
1590:. pp. 427β447.
1304:. pp. 202β211.
172:) is the associated
42:regular word grammar
26:regular tree grammar
1871:range concatenation
1796:range concatenation
1460:2014arXiv1403.7347B
1031:Language properties
725:) happens to equal
365:is such a context,
200:using the rule set
188:Derivation of trees
1558:10.1007/BF01691346
1487:. J. of Comp. Bio.
1351:Journal of the ACM
1332:Sect.4, Theorem 5,
1175:decision procedure
475:
2027:
2026:
2006:
2005:
1968:Embedded pushdown
1864:Context-sensitive
1789:Context-sensitive
1723:Abstract machines
1708:Chomsky hierarchy
1207:first-order logic
1184:of formulas over
1179:first-order logic
373:into the hole of
311:and a production
2047:
2040:Formal languages
2022:
2019:
1983:Visibly pushdown
1957:Thread automaton
1905:Visibly pushdown
1873:
1830:Visibly pushdown
1798:
1785:(no common name)
1704:
1691:formal languages
1680:
1673:
1666:
1657:
1652:
1642:
1632:
1601:
1599:
1581:
1569:
1540:
1538:
1528:
1491:
1488:
1480:
1474:
1473:
1453:
1437:
1431:
1430:
1422:
1416:
1415:
1407:
1401:
1400:
1392:
1386:
1384:
1366:
1348:
1339:
1333:
1331:
1297:
1288:
1282:
1281:
1279:
1277:
1266:
1260:
1259:
1257:
1245:
1192:(β) as the only
1108:
1078:
853:
798:Every member of
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
419:
384:is the language
325:
287:
262:
182:
160:
142:
118:
99:) disjoint from
2055:
2054:
2050:
2049:
2048:
2046:
2045:
2044:
2030:
2029:
2028:
2023:
2020:
2013:
2007:
2002:
1924:
1868:
1847:
1793:
1774:
1697:
1695:formal grammars
1687:Automata theory
1684:
1646:
1639:Proc. 9th STACS
1636:
1614:
1613:language. See:
1585:
1575:
1543:
1506:
1500:
1498:Further reading
1495:
1494:
1482:
1481:
1477:
1470:
1439:
1438:
1434:
1424:
1423:
1419:
1409:
1408:
1404:
1394:
1393:
1389:
1364:10.1.1.145.9971
1346:
1341:
1340:
1336:
1320:
1295:
1290:
1289:
1285:
1275:
1273:
1268:
1267:
1263:
1255:10.1.1.164.5484
1247:
1246:
1242:
1237:
1219:
1161:
1129:
1122:
1115:
1107:
1100:
1094:
1092:
1085:
1077:
1070:
1063:
1056:
1050:
1048:
1041:
1033:
986:
971:
962:
951:
936:
921:
906:
891:
870:
860:
851:
844:
837:
830:
826:
819:
813:
808:
796:
795:
792:
789:
786:
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780:
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774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
738:
731:
724:
713:
698:derivation tree
634:
576:
562:
533:
518:
508:
501:
494:
490:
483:
472:
467:derivation tree
459:
442:
436:
429:
414:
403:
385:
348:
335:
312:
306:
300:
294:
281:
274:
268:
256:
249:
243:
234:
227:
204:is said to be
190:
177:
167:
154:
144:
134:
110:
93:ranked alphabet
54:
12:
11:
5:
2053:
2051:
2043:
2042:
2032:
2031:
2025:
2024:
2012:
2009:
2008:
2004:
2003:
2001:
2000:
1998:Acyclic finite
1995:
1990:
1985:
1980:
1975:
1970:
1965:
1959:
1954:
1949:
1948:Turing Machine
1943:
1941:Linear-bounded
1938:
1933:
1931:Turing machine
1927:
1925:
1923:
1922:
1917:
1912:
1907:
1902:
1897:
1892:
1890:Tree-adjoining
1887:
1882:
1879:
1874:
1866:
1861:
1856:
1850:
1848:
1846:
1845:
1840:
1837:
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1827:
1822:
1817:
1815:Tree-adjoining
1812:
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1654:
1653:
1643:
1633:
1602:
1597:10.1.1.39.3766
1582:
1572:
1571:
1570:
1541:
1519:(5): 484β491.
1499:
1496:
1493:
1492:
1475:
1468:
1432:
1417:
1402:
1387:
1334:
1319:978-1581138528
1318:
1283:
1261:
1239:
1238:
1236:
1233:
1232:
1231:
1226:
1223:Set constraint
1218:
1215:
1214:
1213:
1210:
1203:
1196:
1190:set membership
1171:
1160:
1157:
1156:
1155:
1144:
1137:
1128:
1125:
1120:
1113:
1109:, and whether
1105:
1098:
1090:
1083:
1075:
1068:
1061:
1054:
1046:
1039:
1032:
1029:
984:
969:
960:
949:
934:
919:
909:
908:
904:
889:
884:
868:
858:
849:
842:
835:
828:
824:
817:
806:
746:
736:
732:. The grammar
729:
722:
711:
632:
626:
625:
624:
623:
605:
596:
587:
574:
568:
560:
555:
531:
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409:
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279:
272:
254:
247:
232:
223:
189:
186:
185:
184:
165:
152:
120:
104:
89:
53:
50:
34:directed trees
30:formal grammar
13:
10:
9:
6:
4:
3:
2:
2052:
2041:
2038:
2037:
2035:
2017:
2016:proper subset
2010:
1999:
1996:
1994:
1991:
1989:
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1872:
1867:
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1857:
1855:
1852:
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1849:
1844:
1843:Non-recursive
1841:
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1514:
1510:
1505:
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1501:
1497:
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1469:3-540-44185-9
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1256:
1251:
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1234:
1230:
1227:
1224:
1221:
1220:
1216:
1212:Graph-search
1211:
1208:
1204:
1201:
1197:
1195:
1191:
1187:
1183:
1180:
1176:
1172:
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1166:
1165:
1164:
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1149:
1145:
1142:
1141:tree automata
1138:
1135:
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1130:
1126:
1124:
1119:
1112:
1104:
1097:
1089:
1082:
1074:
1067:
1060:
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1045:
1038:
1030:
1028:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
983:
979:
975:
968:
964:
959:
955:
948:
944:
940:
933:
929:
925:
918:
914:
911:The language
903:
899:
895:
888:
885:
882:
878:
874:
867:
864:
863:
862:
857:
848:
841:
834:
823:
816:
810:
805:
801:
744:
742:
735:
728:
721:
717:
710:
705:
703:
702:word grammars
699:
694:
692:
688:
684:
680:
676:
672:
668:
664:
660:
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648:
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586:
582:
579:
578:
573:
569:
566:
559:
556:
554:has arity 2),
553:
549:
545:
541:
537:
529:
526:
522:
515:
512:
511:
510:
505:
498:
487:
480:
468:
463:
456:
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452:
451:
450:tree language
444:
441:
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396:
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51:
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43:
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35:
31:
27:
23:
19:
1952:Nested stack
1895:Context-free
1820:Context-free
1781:Unrestricted
1648:
1638:
1620:
1616:
1606:Donald Knuth
1587:
1577:
1552:(1): 57β81.
1549:
1545:
1516:
1512:
1484:
1478:
1441:
1435:
1426:
1420:
1411:
1405:
1396:
1390:
1354:
1350:
1337:
1300:
1286:
1274:. Retrieved
1264:
1243:
1162:
1159:Applications
1148:nested words
1117:
1110:
1102:
1095:
1087:
1080:
1072:
1065:
1058:
1051:
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