746:
404:
2081:
In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language
90:, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.
741:{\displaystyle {\begin{aligned}\delta (q_{0},a,z)&=(q_{0},az)\\\delta (q_{0},a,a)&=(q_{0},aa)\\\delta (q_{0},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},\varepsilon ,z)&=(q_{f},\varepsilon )\end{aligned}}}
2797:
77:
Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.
376:
2620:, but there does not exist a context-free grammar generating this language. So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the
1962:
1243:
Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string. The process of producing this tree is called
1889:
2008:
1812:
1735:
409:
919:
836:
2614:
996:
1071:
2671:
3519:
2192:
157:
921:. This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset
1253:
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the
2247:
2300:
1522:
2415:
1613:
1559:
218:
2902:
2071:
2527:
According to
Hopcroft, Motwani, Ullman (2003), many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of
1454:
2513:
2139:
1152:
1481:
1379:
2666:
2045:
1334:
396:
1419:
2451:
2364:
2332:
2478:
1192:
1172:
1117:
3839:
3512:
230:
2621:
1892:
3439:
Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free
Languages and Push-Down Automata". In G. Rozenberg; A. Salomaa (eds.).
3726:
3505:
1906:
3651:
3449:
3741:
2982:
1265:
3666:
3488:
3412:
3834:
3819:
2937:
1269:
1221:
3695:
3173:
1817:
1967:
1964:. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference:
3144:
1634:
752:
1740:
1663:
3712:
3637:
3349:
Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal
Properties of Simple Phrase-Structure Grammars".
841:
758:
3809:
2542:
924:
3705:
2792:{\displaystyle \delta (\mathrm {state} _{1},\mathrm {read} ,\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )}
2617:
1280:
1013:
3881:
3630:
1623:
3782:
3777:
3163:"A Proof that if L = L1 β© L2 where L1 is CFL and L2 is Regular then L is Context Free Which Does Not use PDA's"
2147:
101:
3863:
Any language in each category is generated by a grammar and by an automaton in the category in the same line.
3793:
3731:
3656:
31:
3684:
3497:
2816:
2208:
1895:. As a consequence, context-free languages cannot be closed under complementation, as for any languages
1660:
The context-free languages are not closed under intersection. This can be seen by taking the languages
1292:
1207:
1096:
61:
2263:
1488:
3829:
3804:
3661:
3622:
2385:
2090:
54:
1574:
2629:
2528:
2086:
1534:
1199:
174:
2050:
3814:
3756:
3700:
2974:
2956:
1430:
1308:
225:
2483:
2100:
1250:. Known parsers have a time complexity that is cubic in the size of the string that is parsed.
1122:
3549:
3484:
3440:
3408:
1529:
1466:
1461:
1358:
87:
44:
2941:
2651:
2024:
1313:
381:
3798:
3751:
3718:
3564:
3460:
3134:
3011:
2966:
2917:
2625:
1092:
The context-free nature of the language makes it simple to parse with a pushdown automaton.
221:
1397:
3886:
3761:
3676:
3643:
3559:
3532:
3528:
2427:
2340:
2308:
159:, the language of all non-empty even-length strings, the entire first halves of which are
50:
3772:
3554:
3536:
3472:
3029:
3027:
3025:
2463:
1564:
1211:
1203:
1177:
1157:
1102:
86:
The set of all context-free languages is identical to the set of languages accepted by
3139:
3015:
2922:
3875:
3857:
3477:
3401:
2520:
2516:
1353:
1258:
1254:
1007:
64:, in particular, most arithmetic expressions are generated by context-free grammars.
17:
3162:
2999:
2978:
371:{\displaystyle M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},z,\{q_{f}\})}
3824:
3746:
3671:
2515: ? Efficient polynomial-time algorithms for the membership problem are the
1392:
3119:
2942:"Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication"
1303:
are context-free languages, the following languages are context-free as well:
1273:
2970:
1240:) CFG parsing, thus establishing some kind of lower bound for the latter.
1957:{\displaystyle A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}}
3351:
Zeitschrift fΓΌr
Phonetik, Sprachwissenschaft und Kommunikationsforschung
1246:
1087:
3380:
3002:(July 1965). "On the translation of languages from left to right".
2961:
2194: ? However, the intersection of a context-free language and a
3787:
2198:
language is context-free, hence the variant of the problem where
3344:
3342:
1903:, their intersection can be expressed by union and complement:
755:
CFLs. An example of an inherently ambiguous CFL is the union of
3501:
3856:
Each category of languages, except those marked by a , is a
3330:
John E. Hopcroft; Rajeev
Motwani; Jeffrey D. Ullman (2003).
751:
Unambiguous CFLs are a proper subset of all CFLs: there are
3403:
Introduction to
Automata Theory, Languages, and Computation
3332:
Introduction to
Automata Theory, Languages, and Computation
3120:"Note on the Boolean Properties of Context Free Languages"
2903:"General context-free recognition in less than cubic time"
2202:
is a regular grammar is decidable (see "Emptiness" below).
1656:
Nonclosure under intersection, complement, and difference
1268:
which are defined as the set of languages accepted by a
1884:{\displaystyle A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}}
2003:{\displaystyle {\overline {L}}=\Sigma ^{*}\setminus L}
3170:
University of
Maryland Department of Computer Science
2674:
2654:
2545:
2486:
2466:
2430:
2388:
2343:
2311:
2266:
2211:
2150:
2103:
2053:
2027:
1970:
1909:
1820:
1814:, which are both context-free. Their intersection is
1743:
1666:
1577:
1537:
1491:
1469:
1433:
1400:
1361:
1316:
1264:
A special subclass of context-free languages are the
1180:
1160:
1125:
1105:
1016:
927:
844:
761:
407:
384:
233:
177:
104:
2253:
is a regular grammar is decidable, while that where
3381:"How to prove that a language is not context-free?"
2832:is given by the following production rules, taking
2021:is a regular language then both their intersection
1891:, which can be shown to be non-context-free by the
1807:{\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}}
1730:{\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}}
1232:) Boolean matrix multiplication to be reducible to
3476:
3448:. Vol. 1. Springer-Verlag. pp. 111β174.
3400:
2791:
2660:
2608:
2507:
2472:
2445:
2409:
2358:
2326:
2294:
2241:
2186:
2133:
2065:
2039:
2002:
1956:
1883:
1806:
1729:
1607:
1553:
1516:
1475:
1448:
1413:
1373:
1328:
1283:as an alternative approach to grammar and parser.
1186:
1166:
1146:
1111:
1065:
998:which is the intersection of these two languages.
990:
913:
830:
740:
390:
370:
212:
151:
3465:The Mathematical Theory of Context-Free Languages
914:{\displaystyle \{a^{n}b^{n}c^{m}d^{m}|n,m>0\}}
831:{\displaystyle \{a^{n}b^{m}c^{m}d^{n}|n,m>0\}}
60:Context-free languages have many applications in
27:Formal language generated by context-free grammar
2249: ? Again, the variant of the problem where
1210:, thus inheriting its complexity upper bound of
2609:{\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\}}
991:{\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\}}
3399:Hopcroft, John E.; Ullman, Jeffrey D. (1979).
3367:
3317:
3305:
3293:
3281:
3269:
3257:
3245:
3233:
3209:
3197:
3105:
3093:
3081:
3069:
3057:
3045:
3033:
2888:
2817:Matrix multiplication#Computational complexity
163:'s, and the entire second halves of which are
3513:
1174:is the language generated by a given grammar
8:
3060:, p. 131-132, Corollary of Theorem 6.2.
2603:
2546:
1878:
1833:
1801:
1750:
1724:
1673:
1602:
1578:
1295:under the following operations. That is, if
1066:{\displaystyle S\to SS~|~(S)~|~\varepsilon }
1008:language of all properly matched parentheses
985:
928:
908:
845:
825:
762:
362:
349:
318:
306:
300:
288:
282:
243:
146:
111:
3835:Counter-free (with aperiodic finite monoid)
2815:) was the then-best known upper bound. See
3545:
3520:
3506:
3498:
3336:Here: Sect.7.6, p.304, and Sect.9.7, p.411
3479:Introduction to the Theory of Computation
3138:
2960:
2921:
2772:
2763:
2746:
2725:
2708:
2699:
2682:
2673:
2653:
2589:
2583:
2573:
2563:
2553:
2544:
2485:
2465:
2456:Membership: Given a context-free grammar
2429:
2420:Finiteness: Given a context-free grammar
2387:
2342:
2310:
2286:
2265:
2210:
2149:
2102:
2052:
2026:
1988:
1971:
1969:
1938:
1925:
1922:
1908:
1860:
1850:
1840:
1819:
1777:
1767:
1757:
1742:
1700:
1690:
1680:
1665:
1576:
1542:
1536:
1496:
1490:
1468:
1432:
1405:
1399:
1360:
1315:
1179:
1159:
1124:
1104:
1052:
1032:
1015:
971:
965:
955:
945:
935:
926:
888:
882:
872:
862:
852:
843:
805:
799:
789:
779:
769:
760:
719:
684:
655:
620:
591:
556:
524:
489:
457:
422:
408:
406:
383:
356:
334:
276:
263:
250:
232:
196:
176:
128:
118:
103:
3036:, p. 131, Corollary of Theorem 6.1.
2828:A context-free grammar for the language
2622:pumping lemma for context-free languages
2378:Emptiness: Given a context-free grammar
2187:{\displaystyle L(A)\cap L(B)=\emptyset }
1893:pumping lemma for context-free languages
152:{\displaystyle L=\{a^{n}b^{n}:n\geq 1\}}
3221:
2910:Journal of Computer and System Sciences
2881:
2641:
2057:
1994:
1291:The class of context-free languages is
3727:Linear context-free rewriting language
2624:or a number of other methods, such as
2374:for arbitrary context-free languages:
3652:Linear context-free rewriting systems
3475:(1997). "2: Context-Free Languages".
7:
1266:deterministic context-free languages
98:An example context-free language is
3483:. PWS Publishing. pp. 91β122.
3161:Beigel, Richard; Gasarch, William.
2535:Languages that are not context-free
3860:of the category directly above it.
2819:for bound improvements since then.
2782:
2779:
2776:
2773:
2759:
2756:
2753:
2750:
2747:
2732:
2729:
2726:
2718:
2715:
2712:
2709:
2695:
2692:
2689:
2686:
2683:
2404:
2283:
2242:{\displaystyle L(A)\subseteq L(B)}
2181:
1985:
25:
3467:. New York, NY, USA: McGraw-Hill.
3084:, p. 142-144, Exercise 6.4c.
2901:Valiant, Leslie G. (April 1975).
3455:from the original on 2011-05-16.
3407:(1st ed.). Addison-Wesley.
3150:from the original on 2018-11-26.
2988:from the original on 2003-04-27.
2337:Ambiguity: is every grammar for
2295:{\displaystyle L(A)=\Sigma ^{*}}
1517:{\displaystyle \varphi ^{-1}(L)}
1270:deterministic pushdown automaton
3260:, p. 203, Theorem 8.12(4).
3248:, p. 203, Theorem 8.12(2).
3200:, p. 203, Theorem 8.12(1).
3179:from the original on 2014-12-12
2410:{\displaystyle L(A)=\emptyset }
2017:is a context-free language and
1198:. Context-free recognition for
1095:Determining an instance of the
3320:, p. 137, Theorem 6.6(b).
3308:, p. 137, Theorem 6.6(a).
2786:
2742:
2736:
2678:
2590:
2502:
2496:
2440:
2434:
2398:
2392:
2353:
2347:
2321:
2315:
2276:
2270:
2236:
2230:
2221:
2215:
2175:
2169:
2160:
2154:
2128:
2122:
2113:
2107:
1608:{\displaystyle \{vu:uv\in L\}}
1511:
1505:
1443:
1437:
1141:
1135:
1053:
1046:
1040:
1033:
1020:
972:
889:
806:
731:
712:
702:
677:
667:
648:
638:
613:
603:
584:
574:
549:
539:
517:
507:
482:
472:
450:
440:
415:
365:
240:
197:
181:
1:
3140:10.1016/s0019-9958(60)90965-7
3108:, p. 142, Exercise 6.4a.
3096:, p. 142, Exercise 6.4b.
3048:, p. 142, Exercise 6.4d.
3016:10.1016/S0019-9958(65)90426-2
2923:10.1016/s0022-0000(75)80046-8
1554:{\displaystyle \varphi ^{-1}}
213:{\displaystyle S\to aSb~|~ab}
3442:Handbook of Formal Languages
3296:, p. 206, Theorem 8.16.
3284:, p. 205, Theorem 8.15.
3272:, p. 203, Theorem 8.11.
3212:, p. 202, Theorem 8.10.
2257:is regular is generally not.
2073:are context-free languages.
2066:{\displaystyle L\setminus D}
1976:
1949:
1943:
1930:
1010:is generated by the grammar
171:is generated by the grammar
3236:, p. 135, Theorem 6.5.
3118:Stephen Scheinberg (1960).
3072:, p. 132, Theorem 6.3.
2891:, p. 100, Theorem 4.7.
2370:The following problems are
2085:The following problems are
1449:{\displaystyle \varphi (L)}
1206:to be reducible to Boolean
3903:
3742:Deterministic context-free
3667:Deterministic context-free
3368:Hopcroft & Ullman 1979
3318:Hopcroft & Ullman 1979
3306:Hopcroft & Ullman 1979
3294:Hopcroft & Ullman 1979
3282:Hopcroft & Ullman 1979
3270:Hopcroft & Ullman 1979
3258:Hopcroft & Ullman 1979
3246:Hopcroft & Ullman 1979
3234:Hopcroft & Ullman 1979
3210:Hopcroft & Ullman 1979
3198:Hopcroft & Ullman 1979
3106:Hopcroft & Ullman 1979
3094:Hopcroft & Ullman 1979
3082:Hopcroft & Ullman 1979
3070:Hopcroft & Ullman 1979
3058:Hopcroft & Ullman 1979
3046:Hopcroft & Ullman 1979
3034:Hopcroft & Ullman 1979
2889:Hopcroft & Ullman 1979
2668:'s arguments and results:
2618:context-sensitive language
2082:with a different grammar.
1281:parsing expression grammar
1085:
3853:
3815:Nondeterministic pushdown
3543:
2508:{\displaystyle w\in L(G)}
2134:{\displaystyle L(A)=L(B)}
1147:{\displaystyle w\in L(G)}
1476:{\displaystyle \varphi }
1374:{\displaystyle L\cdot P}
224:. It is accepted by the
3426:. ACM Monograph Series.
3127:Information and Control
3004:Information and Control
2661:{\displaystyle \delta }
2040:{\displaystyle L\cap D}
1329:{\displaystyle L\cup P}
1272:and can be parsed by a
398:is defined as follows:
391:{\displaystyle \delta }
220:. This language is not
3820:Deterministic pushdown
3696:Recursively enumerable
3422:Salomaa, Arto (1973).
2793:
2662:
2610:
2509:
2474:
2447:
2411:
2360:
2328:
2296:
2243:
2188:
2135:
2089:for arbitrarily given
2067:
2041:
2004:
1958:
1885:
1808:
1731:
1648:by a regular language
1618:the prefix closure of
1609:
1555:
1518:
1477:
1450:
1415:
1375:
1330:
1202:grammars was shown by
1188:
1168:
1148:
1113:
1099:; i.e. given a string
1067:
992:
915:
832:
742:
392:
372:
214:
153:
32:formal language theory
2971:10.1145/505241.505242
2836:as the start symbol:
2794:
2663:
2611:
2531:, Perles, and Shamir
2510:
2475:
2448:
2412:
2361:
2329:
2297:
2244:
2189:
2136:
2091:context-free grammars
2068:
2047:and their difference
2042:
2005:
1959:
1886:
1809:
1732:
1610:
1556:
1519:
1478:
1451:
1416:
1414:{\displaystyle L^{*}}
1376:
1331:
1208:matrix multiplication
1189:
1169:
1149:
1114:
1068:
993:
916:
833:
743:
393:
373:
215:
154:
62:programming languages
36:context-free language
18:Context free language
3805:Tree stack automaton
3224:, p. 59, Theorem 6.7
2807:In Valiant's paper,
2672:
2652:
2543:
2484:
2464:
2446:{\displaystyle L(A)}
2428:
2386:
2359:{\displaystyle L(A)}
2341:
2327:{\displaystyle L(A)}
2309:
2264:
2209:
2148:
2101:
2051:
2025:
1968:
1907:
1818:
1741:
1664:
1575:
1535:
1530:inverse homomorphism
1489:
1467:
1431:
1398:
1359:
1314:
1178:
1158:
1123:
1119:, determine whether
1103:
1082:Context-free parsing
1014:
925:
842:
759:
753:inherently ambiguous
405:
382:
231:
175:
102:
73:Context-free grammar
55:context-free grammar
3713:range concatenation
3638:range concatenation
2334:a regular language?
1200:Chomsky normal form
1194:; is also known as
2864:. The grammar for
2789:
2658:
2606:
2521:Earley's Algorithm
2505:
2470:
2443:
2407:
2356:
2324:
2292:
2239:
2184:
2131:
2063:
2037:
2000:
1954:
1881:
1804:
1727:
1605:
1551:
1514:
1473:
1446:
1411:
1371:
1326:
1287:Closure properties
1259:Earley's Algorithm
1184:
1164:
1144:
1109:
1097:membership problem
1063:
988:
911:
828:
738:
736:
388:
368:
226:pushdown automaton
210:
149:
3869:
3868:
3848:
3847:
3810:Embedded pushdown
3706:Context-sensitive
3631:Context-sensitive
3565:Abstract machines
3550:Chomsky hierarchy
3461:Ginsburg, Seymour
3334:. Addison Wesley.
2473:{\displaystyle w}
2260:Universality: is
2144:Disjointness: is
1979:
1952:
1946:
1933:
1204:Leslie G. Valiant
1187:{\displaystyle G}
1167:{\displaystyle L}
1112:{\displaystyle w}
1059:
1051:
1039:
1031:
203:
195:
88:pushdown automata
42:), also called a
16:(Redirected from
3894:
3882:Formal languages
3864:
3861:
3825:Visibly pushdown
3799:Thread automaton
3747:Visibly pushdown
3715:
3672:Visibly pushdown
3640:
3627:(no common name)
3546:
3533:formal languages
3522:
3515:
3508:
3499:
3494:
3482:
3468:
3456:
3454:
3447:
3427:
3424:Formal Languages
3418:
3406:
3385:
3384:
3377:
3371:
3365:
3359:
3358:
3346:
3337:
3335:
3327:
3321:
3315:
3309:
3303:
3297:
3291:
3285:
3279:
3273:
3267:
3261:
3255:
3249:
3243:
3237:
3231:
3225:
3219:
3213:
3207:
3201:
3195:
3189:
3188:
3186:
3184:
3178:
3167:
3158:
3152:
3151:
3149:
3142:
3124:
3115:
3109:
3103:
3097:
3091:
3085:
3079:
3073:
3067:
3061:
3055:
3049:
3043:
3037:
3031:
3020:
3019:
2996:
2990:
2989:
2987:
2964:
2946:
2940:(January 2002).
2934:
2928:
2927:
2925:
2907:
2898:
2892:
2886:
2869:
2826:
2820:
2805:
2799:
2798:
2796:
2795:
2790:
2785:
2768:
2767:
2762:
2735:
2721:
2704:
2703:
2698:
2667:
2665:
2664:
2659:
2646:
2630:Parikh's theorem
2615:
2613:
2612:
2607:
2593:
2588:
2587:
2578:
2577:
2568:
2567:
2558:
2557:
2514:
2512:
2511:
2506:
2479:
2477:
2476:
2471:
2452:
2450:
2449:
2444:
2416:
2414:
2413:
2408:
2365:
2363:
2362:
2357:
2333:
2331:
2330:
2325:
2301:
2299:
2298:
2293:
2291:
2290:
2248:
2246:
2245:
2240:
2205:Containment: is
2193:
2191:
2190:
2185:
2140:
2138:
2137:
2132:
2097:Equivalence: is
2072:
2070:
2069:
2064:
2046:
2044:
2043:
2038:
2009:
2007:
2006:
2001:
1993:
1992:
1980:
1972:
1963:
1961:
1960:
1955:
1953:
1948:
1947:
1939:
1934:
1926:
1923:
1890:
1888:
1887:
1882:
1865:
1864:
1855:
1854:
1845:
1844:
1813:
1811:
1810:
1805:
1782:
1781:
1772:
1771:
1762:
1761:
1736:
1734:
1733:
1728:
1705:
1704:
1695:
1694:
1685:
1684:
1626:of strings from
1622:(the set of all
1614:
1612:
1611:
1606:
1560:
1558:
1557:
1552:
1550:
1549:
1523:
1521:
1520:
1515:
1504:
1503:
1482:
1480:
1479:
1474:
1455:
1453:
1452:
1447:
1420:
1418:
1417:
1412:
1410:
1409:
1380:
1378:
1377:
1372:
1346:the reversal of
1335:
1333:
1332:
1327:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1153:
1151:
1150:
1145:
1118:
1116:
1115:
1110:
1072:
1070:
1069:
1064:
1057:
1056:
1049:
1037:
1036:
1029:
997:
995:
994:
989:
975:
970:
969:
960:
959:
950:
949:
940:
939:
920:
918:
917:
912:
892:
887:
886:
877:
876:
867:
866:
857:
856:
837:
835:
834:
829:
809:
804:
803:
794:
793:
784:
783:
774:
773:
747:
745:
744:
739:
737:
724:
723:
689:
688:
660:
659:
625:
624:
596:
595:
561:
560:
529:
528:
494:
493:
462:
461:
427:
426:
397:
395:
394:
389:
377:
375:
374:
369:
361:
360:
339:
338:
281:
280:
268:
267:
255:
254:
219:
217:
216:
211:
201:
200:
193:
170:
166:
162:
158:
156:
155:
150:
133:
132:
123:
122:
21:
3902:
3901:
3897:
3896:
3895:
3893:
3892:
3891:
3872:
3871:
3870:
3865:
3862:
3855:
3849:
3844:
3766:
3710:
3689:
3635:
3616:
3539:
3537:formal grammars
3529:Automata theory
3526:
3491:
3473:Sipser, Michael
3471:
3459:
3452:
3445:
3438:
3435:
3433:Further reading
3430:
3421:
3415:
3398:
3394:
3389:
3388:
3379:
3378:
3374:
3366:
3362:
3348:
3347:
3340:
3329:
3328:
3324:
3316:
3312:
3304:
3300:
3292:
3288:
3280:
3276:
3268:
3264:
3256:
3252:
3244:
3240:
3232:
3228:
3220:
3216:
3208:
3204:
3196:
3192:
3182:
3180:
3176:
3165:
3160:
3159:
3155:
3147:
3122:
3117:
3116:
3112:
3104:
3100:
3092:
3088:
3080:
3076:
3068:
3064:
3056:
3052:
3044:
3040:
3032:
3023:
2998:
2997:
2993:
2985:
2944:
2936:
2935:
2931:
2905:
2900:
2899:
2895:
2887:
2883:
2878:
2873:
2872:
2827:
2823:
2806:
2802:
2745:
2681:
2670:
2669:
2650:
2649:
2647:
2643:
2638:
2579:
2569:
2559:
2549:
2541:
2540:
2537:
2482:
2481:
2462:
2461:
2426:
2425:
2384:
2383:
2339:
2338:
2307:
2306:
2305:Regularity: is
2282:
2262:
2261:
2207:
2206:
2146:
2145:
2099:
2098:
2079:
2049:
2048:
2023:
2022:
1984:
1966:
1965:
1924:
1905:
1904:
1856:
1846:
1836:
1816:
1815:
1773:
1763:
1753:
1739:
1738:
1696:
1686:
1676:
1662:
1661:
1658:
1573:
1572:
1538:
1533:
1532:
1492:
1487:
1486:
1465:
1464:
1429:
1428:
1401:
1396:
1395:
1357:
1356:
1312:
1311:
1289:
1220:). Conversely,
1176:
1175:
1156:
1155:
1121:
1120:
1101:
1100:
1090:
1084:
1079:
1012:
1011:
1004:
961:
951:
941:
931:
923:
922:
878:
868:
858:
848:
840:
839:
795:
785:
775:
765:
757:
756:
735:
734:
715:
705:
680:
671:
670:
651:
641:
616:
607:
606:
587:
577:
552:
543:
542:
520:
510:
485:
476:
475:
453:
443:
418:
403:
402:
380:
379:
352:
330:
272:
259:
246:
229:
228:
173:
172:
168:
164:
160:
124:
114:
100:
99:
96:
84:
75:
70:
53:generated by a
47:type-2 language
28:
23:
22:
15:
12:
11:
5:
3900:
3898:
3890:
3889:
3884:
3874:
3873:
3867:
3866:
3854:
3851:
3850:
3846:
3845:
3843:
3842:
3840:Acyclic finite
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3801:
3796:
3791:
3790:Turing Machine
3785:
3783:Linear-bounded
3780:
3775:
3773:Turing machine
3769:
3767:
3765:
3764:
3759:
3754:
3749:
3744:
3739:
3734:
3732:Tree-adjoining
3729:
3724:
3721:
3716:
3708:
3703:
3698:
3692:
3690:
3688:
3687:
3682:
3679:
3674:
3669:
3664:
3659:
3657:Tree-adjoining
3654:
3649:
3646:
3641:
3633:
3628:
3625:
3619:
3617:
3615:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3571:
3568:
3567:
3562:
3557:
3552:
3544:
3541:
3540:
3527:
3525:
3524:
3517:
3510:
3502:
3496:
3495:
3489:
3469:
3457:
3434:
3431:
3429:
3428:
3419:
3413:
3395:
3393:
3390:
3387:
3386:
3372:
3360:
3338:
3322:
3310:
3298:
3286:
3274:
3262:
3250:
3238:
3226:
3222:Salomaa (1973)
3214:
3202:
3190:
3153:
3133:(4): 372β375.
3110:
3098:
3086:
3074:
3062:
3050:
3038:
3021:
3010:(6): 607β639.
2991:
2929:
2916:(2): 308β315.
2893:
2880:
2879:
2877:
2874:
2871:
2870:
2821:
2800:
2788:
2784:
2781:
2778:
2775:
2771:
2766:
2761:
2758:
2755:
2752:
2749:
2744:
2741:
2738:
2734:
2731:
2728:
2724:
2720:
2717:
2714:
2711:
2707:
2702:
2697:
2694:
2691:
2688:
2685:
2680:
2677:
2657:
2640:
2639:
2637:
2634:
2605:
2602:
2599:
2596:
2592:
2586:
2582:
2576:
2572:
2566:
2562:
2556:
2552:
2548:
2536:
2533:
2525:
2524:
2504:
2501:
2498:
2495:
2492:
2489:
2469:
2460:, and a word
2454:
2442:
2439:
2436:
2433:
2418:
2406:
2403:
2400:
2397:
2394:
2391:
2368:
2367:
2355:
2352:
2349:
2346:
2335:
2323:
2320:
2317:
2314:
2303:
2289:
2285:
2281:
2278:
2275:
2272:
2269:
2258:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2203:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2142:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2078:
2075:
2062:
2059:
2056:
2036:
2033:
2030:
1999:
1996:
1991:
1987:
1983:
1978:
1975:
1951:
1945:
1942:
1937:
1932:
1929:
1921:
1918:
1915:
1912:
1880:
1877:
1874:
1871:
1868:
1863:
1859:
1853:
1849:
1843:
1839:
1835:
1832:
1829:
1826:
1823:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1780:
1776:
1770:
1766:
1760:
1756:
1752:
1749:
1746:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1703:
1699:
1693:
1689:
1683:
1679:
1675:
1672:
1669:
1657:
1654:
1653:
1652:
1631:
1616:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1571:(the language
1565:circular shift
1561:
1548:
1545:
1541:
1513:
1510:
1507:
1502:
1499:
1495:
1483:
1472:
1445:
1442:
1439:
1436:
1425:
1408:
1404:
1389:
1370:
1367:
1364:
1350:
1344:
1325:
1322:
1319:
1288:
1285:
1183:
1163:
1143:
1140:
1137:
1134:
1131:
1128:
1108:
1086:Main article:
1083:
1080:
1078:
1075:
1062:
1055:
1048:
1045:
1042:
1035:
1028:
1025:
1022:
1019:
1003:
1000:
987:
984:
981:
978:
974:
968:
964:
958:
954:
948:
944:
938:
934:
930:
910:
907:
904:
901:
898:
895:
891:
885:
881:
875:
871:
865:
861:
855:
851:
847:
827:
824:
821:
818:
815:
812:
808:
802:
798:
792:
788:
782:
778:
772:
768:
764:
749:
748:
733:
730:
727:
722:
718:
714:
711:
708:
706:
704:
701:
698:
695:
692:
687:
683:
679:
676:
673:
672:
669:
666:
663:
658:
654:
650:
647:
644:
642:
640:
637:
634:
631:
628:
623:
619:
615:
612:
609:
608:
605:
602:
599:
594:
590:
586:
583:
580:
578:
576:
573:
570:
567:
564:
559:
555:
551:
548:
545:
544:
541:
538:
535:
532:
527:
523:
519:
516:
513:
511:
509:
506:
503:
500:
497:
492:
488:
484:
481:
478:
477:
474:
471:
468:
465:
460:
456:
452:
449:
446:
444:
442:
439:
436:
433:
430:
425:
421:
417:
414:
411:
410:
387:
367:
364:
359:
355:
351:
348:
345:
342:
337:
333:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
279:
275:
271:
266:
262:
258:
253:
249:
245:
242:
239:
236:
209:
206:
199:
192:
189:
186:
183:
180:
148:
145:
142:
139:
136:
131:
127:
121:
117:
113:
110:
107:
95:
92:
83:
80:
74:
71:
69:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3899:
3888:
3885:
3883:
3880:
3879:
3877:
3859:
3858:proper subset
3852:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3802:
3800:
3797:
3795:
3792:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3770:
3768:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3728:
3725:
3722:
3720:
3717:
3714:
3709:
3707:
3704:
3702:
3699:
3697:
3694:
3693:
3691:
3686:
3685:Non-recursive
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3647:
3645:
3642:
3639:
3634:
3632:
3629:
3626:
3624:
3621:
3620:
3618:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3572:
3570:
3569:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3547:
3542:
3538:
3534:
3530:
3523:
3518:
3516:
3511:
3509:
3504:
3503:
3500:
3492:
3490:0-534-94728-X
3486:
3481:
3480:
3474:
3470:
3466:
3462:
3458:
3451:
3444:
3443:
3437:
3436:
3432:
3425:
3420:
3416:
3414:9780201029888
3410:
3405:
3404:
3397:
3396:
3391:
3382:
3376:
3373:
3369:
3364:
3361:
3357:(2): 143β172.
3356:
3352:
3345:
3343:
3339:
3333:
3326:
3323:
3319:
3314:
3311:
3307:
3302:
3299:
3295:
3290:
3287:
3283:
3278:
3275:
3271:
3266:
3263:
3259:
3254:
3251:
3247:
3242:
3239:
3235:
3230:
3227:
3223:
3218:
3215:
3211:
3206:
3203:
3199:
3194:
3191:
3175:
3171:
3164:
3157:
3154:
3146:
3141:
3136:
3132:
3128:
3121:
3114:
3111:
3107:
3102:
3099:
3095:
3090:
3087:
3083:
3078:
3075:
3071:
3066:
3063:
3059:
3054:
3051:
3047:
3042:
3039:
3035:
3030:
3028:
3026:
3022:
3017:
3013:
3009:
3005:
3001:
2995:
2992:
2984:
2980:
2976:
2972:
2968:
2963:
2958:
2954:
2950:
2943:
2939:
2933:
2930:
2924:
2919:
2915:
2911:
2904:
2897:
2894:
2890:
2885:
2882:
2875:
2868:is analogous.
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2825:
2822:
2818:
2814:
2810:
2804:
2801:
2769:
2764:
2739:
2722:
2705:
2700:
2675:
2655:
2645:
2642:
2635:
2633:
2631:
2627:
2626:Ogden's lemma
2623:
2619:
2600:
2597:
2594:
2584:
2580:
2574:
2570:
2564:
2560:
2554:
2550:
2534:
2532:
2530:
2522:
2518:
2517:CYK algorithm
2499:
2493:
2490:
2487:
2467:
2459:
2455:
2437:
2431:
2423:
2419:
2401:
2395:
2389:
2381:
2377:
2376:
2375:
2373:
2350:
2344:
2336:
2318:
2312:
2304:
2287:
2279:
2273:
2267:
2259:
2256:
2252:
2233:
2227:
2224:
2218:
2212:
2204:
2201:
2197:
2178:
2172:
2166:
2163:
2157:
2151:
2143:
2125:
2119:
2116:
2110:
2104:
2096:
2095:
2094:
2092:
2088:
2083:
2076:
2074:
2060:
2054:
2034:
2031:
2028:
2020:
2016:
2011:
1997:
1989:
1981:
1973:
1940:
1935:
1927:
1919:
1916:
1913:
1910:
1902:
1898:
1894:
1875:
1872:
1869:
1866:
1861:
1857:
1851:
1847:
1841:
1837:
1830:
1827:
1824:
1821:
1798:
1795:
1792:
1789:
1786:
1783:
1778:
1774:
1768:
1764:
1758:
1754:
1747:
1744:
1721:
1718:
1715:
1712:
1709:
1706:
1701:
1697:
1691:
1687:
1681:
1677:
1670:
1667:
1655:
1651:
1647:
1643:
1639:
1636:
1632:
1629:
1625:
1621:
1617:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1570:
1566:
1562:
1546:
1543:
1539:
1531:
1527:
1508:
1500:
1497:
1493:
1484:
1470:
1463:
1459:
1440:
1434:
1426:
1424:
1406:
1402:
1394:
1390:
1388:
1384:
1368:
1365:
1362:
1355:
1354:concatenation
1351:
1349:
1345:
1343:
1339:
1323:
1320:
1317:
1310:
1306:
1305:
1304:
1302:
1298:
1294:
1286:
1284:
1282:
1277:
1275:
1271:
1267:
1262:
1260:
1256:
1255:CYK algorithm
1251:
1249:
1248:
1241:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1214:
1209:
1205:
1201:
1197:
1181:
1161:
1138:
1132:
1129:
1126:
1106:
1098:
1093:
1089:
1081:
1076:
1074:
1060:
1043:
1026:
1023:
1017:
1009:
1002:Dyck language
1001:
999:
982:
979:
976:
966:
962:
956:
952:
946:
942:
936:
932:
905:
902:
899:
896:
893:
883:
879:
873:
869:
863:
859:
853:
849:
822:
819:
816:
813:
810:
800:
796:
790:
786:
780:
776:
770:
766:
754:
728:
725:
720:
716:
709:
707:
699:
696:
693:
690:
685:
681:
674:
664:
661:
656:
652:
645:
643:
635:
632:
629:
626:
621:
617:
610:
600:
597:
592:
588:
581:
579:
571:
568:
565:
562:
557:
553:
546:
536:
533:
530:
525:
521:
514:
512:
504:
501:
498:
495:
490:
486:
479:
469:
466:
463:
458:
454:
447:
445:
437:
434:
431:
428:
423:
419:
412:
401:
400:
399:
385:
357:
353:
346:
343:
340:
335:
331:
327:
324:
321:
315:
312:
309:
303:
297:
294:
291:
285:
277:
273:
269:
264:
260:
256:
251:
247:
237:
234:
227:
223:
207:
204:
190:
187:
184:
178:
143:
140:
137:
134:
129:
125:
119:
115:
108:
105:
93:
91:
89:
81:
79:
72:
67:
65:
63:
58:
56:
52:
48:
46:
41:
37:
33:
19:
3794:Nested stack
3737:Context-free
3736:
3662:Context-free
3623:Unrestricted
3478:
3464:
3441:
3423:
3402:
3375:
3363:
3354:
3350:
3331:
3325:
3313:
3301:
3289:
3277:
3265:
3253:
3241:
3229:
3217:
3205:
3193:
3181:. Retrieved
3169:
3156:
3130:
3126:
3113:
3101:
3089:
3077:
3065:
3053:
3041:
3007:
3003:
3000:Knuth, D. E.
2994:
2952:
2948:
2938:Lee, Lillian
2932:
2913:
2909:
2896:
2884:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2824:
2812:
2808:
2803:
2644:
2538:
2526:
2457:
2421:
2379:
2371:
2369:
2254:
2250:
2199:
2195:
2084:
2080:
2077:Decidability
2018:
2014:
2013:However, if
2012:
1900:
1896:
1659:
1649:
1645:
1641:
1637:
1627:
1619:
1568:
1525:
1462:homomorphism
1457:
1422:
1386:
1382:
1347:
1341:
1337:
1300:
1296:
1290:
1278:
1274:LR(k) parser
1263:
1252:
1245:
1242:
1237:
1233:
1229:
1225:
1217:
1212:
1195:
1094:
1091:
1005:
750:
97:
85:
76:
59:
43:
39:
35:
29:
3803:restricted
3392:Works cited
2955:(1): 1β15.
2648:meaning of
2087:undecidable
1393:Kleene star
1222:Lillian Lee
1196:recognition
3876:Categories
2962:cs/0112018
2876:References
2529:Bar-Hillel
2366:ambiguous?
1485:the image
1427:the image
1224:has shown
1077:Properties
68:Background
3757:Star-free
3711:Positive
3701:Decidable
3636:Positive
3560:Languages
2676:δ
2656:δ
2491:∈
2405:∅
2372:decidable
2288:∗
2284:Σ
2225:⊆
2182:∅
2164:∩
2093:A and B:
2058:∖
2032:∩
1995:∖
1990:∗
1986:Σ
1977:¯
1950:¯
1944:¯
1936:∪
1931:¯
1914:∩
1873:≥
1867:∣
1825:∩
1796:≥
1784:∣
1719:≥
1707:∣
1597:∈
1544:−
1540:φ
1528:under an
1498:−
1494:φ
1471:φ
1435:φ
1407:∗
1366:⋅
1321:∪
1279:See also
1130:∈
1061:ε
1021:→
729:ε
694:ε
675:δ
665:ε
611:δ
601:ε
547:δ
480:δ
413:δ
386:δ
325:δ
182:→
141:≥
3555:Grammars
3463:(1966).
3450:Archived
3174:Archived
3145:Archived
2983:Archived
2539:The set
1635:quotient
1624:prefixes
1460:under a
94:Examples
82:Automata
51:language
3778:Decider
3752:Regular
3719:Indexed
3677:Regular
3644:Indexed
3183:June 6,
2979:1243491
2480:, does
2453:finite?
2417: ?
2196:regular
1247:parsing
1088:Parsing
222:regular
57:(CFG).
49:, is a
45:Chomsky
3887:Syntax
3830:Finite
3762:Finite
3607:Type-3
3598:Type-2
3580:Type-1
3574:Type-0
3487:
3411:
2977:
1293:closed
1154:where
1058:
1050:
1038:
1030:
378:where
202:
194:
3788:PTIME
3453:(PDF)
3446:(PDF)
3177:(PDF)
3166:(PDF)
3148:(PDF)
3123:(PDF)
2986:(PDF)
2975:S2CID
2957:arXiv
2949:J ACM
2945:(PDF)
2906:(PDF)
2636:Notes
2616:is a
2424:, is
2382:, is
1309:union
838:with
34:, a
3535:and
3485:ISBN
3409:ISBN
3185:2020
2598:>
2519:and
1899:and
1737:and
1633:the
1563:the
1391:the
1385:and
1352:the
1340:and
1307:the
1299:and
1257:and
1006:The
980:>
903:>
820:>
167:'s.
3135:doi
3012:doi
2967:doi
2918:doi
2858:aTb
2846:aTb
2628:or
1644:of
1567:of
1524:of
1456:of
1421:of
1381:of
1336:of
40:CFL
30:In
3878::
3531::
3355:14
3353:.
3341:^
3172:.
3168:.
3143:.
3129:.
3125:.
3024:^
3006:.
2981:.
2973:.
2965:.
2953:49
2951:.
2947:.
2914:10
2912:.
2908:.
2860:|
2856:β
2852:;
2848:|
2844:|
2842:Sc
2840:β
2632:.
2010:.
1276:.
1261:.
1073:.
3723:β
3681:β
3648:β
3613:β
3610:β
3604:β
3601:β
3595:β
3592:β
3589:β
3586:β
3583:β
3577:β
3521:e
3514:t
3507:v
3493:.
3417:.
3383:.
3370:.
3187:.
3137::
3131:3
3018:.
3014::
3008:8
2969::
2959::
2926:.
2920::
2866:B
2862:Ξ΅
2854:T
2850:Ξ΅
2838:S
2834:S
2830:A
2813:n
2811:(
2809:O
2787:)
2783:h
2780:s
2777:u
2774:p
2770:,
2765:2
2760:e
2757:t
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2751:t
2748:s
2743:(
2740:=
2737:)
2733:p
2730:o
2727:p
2723:,
2719:d
2716:a
2713:e
2710:r
2706:,
2701:1
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2693:t
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2687:t
2684:s
2679:(
2604:}
2601:0
2595:n
2591:|
2585:n
2581:d
2575:n
2571:c
2565:n
2561:b
2555:n
2551:a
2547:{
2523:.
2503:)
2500:G
2497:(
2494:L
2488:w
2468:w
2458:G
2441:)
2438:A
2435:(
2432:L
2422:A
2402:=
2399:)
2396:A
2393:(
2390:L
2380:A
2354:)
2351:A
2348:(
2345:L
2322:)
2319:A
2316:(
2313:L
2302:?
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2277:)
2274:A
2271:(
2268:L
2255:A
2251:B
2237:)
2234:B
2231:(
2228:L
2222:)
2219:A
2216:(
2213:L
2200:B
2179:=
2176:)
2173:B
2170:(
2167:L
2161:)
2158:A
2155:(
2152:L
2141:?
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2126:B
2123:(
2120:L
2117:=
2114:)
2111:A
2108:(
2105:L
2061:D
2055:L
2035:D
2029:L
2019:D
2015:L
1998:L
1982:=
1974:L
1941:B
1928:A
1920:=
1917:B
1911:A
1901:B
1897:A
1879:}
1876:0
1870:n
1862:n
1858:c
1852:n
1848:b
1842:n
1838:a
1834:{
1831:=
1828:B
1822:A
1802:}
1799:0
1793:n
1790:,
1787:m
1779:n
1775:c
1769:n
1765:b
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1751:{
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1745:B
1725:}
1722:0
1716:n
1713:,
1710:m
1702:m
1698:c
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1324:P
1318:L
1301:P
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1024:S
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986:}
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967:n
963:d
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953:c
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909:}
906:0
900:m
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864:n
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854:n
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826:}
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814:,
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807:|
801:n
797:d
791:m
787:c
781:m
777:b
771:n
767:a
763:{
732:)
726:,
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717:q
713:(
710:=
703:)
700:z
697:,
691:,
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678:(
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662:,
657:1
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649:(
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639:)
636:a
633:,
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627:,
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614:(
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147:}
144:1
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120:n
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112:{
109:=
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38:(
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