300:
579:
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
802:
595:
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
114:…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.
940:
620:
1109:
1028:
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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
1354:
1435:
1319:
476:
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
340:
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:
583:
Nevertheless, removing grammar ambiguity may produce a deterministic context-free grammar and thus allow for more efficient parsing. Compiler generators such as
1193:
More examples, and a general review of techniques for proving inherent ambiguity of context-free languages, are found given by
Bassino and Nicaud (2011).
813:
106:
The simplest example is the following ambiguous grammar (with start symbol A) for the trivial language that consists of only the empty string:
468:
This is resolved in various ways in different languages. Sometimes the grammar is modified so that it is unambiguous, such as by requiring an
306:
The language that it generates, however, is not inherently ambiguous; the following is a non-ambiguous grammar generating the same language:
17:
337:
statement is optional, which results in nested conditionals having multiple ways of being recognized in terms of the context-free grammar.
66:
are always unambiguous, and are an important subclass of unambiguous grammars; there are non-deterministic unambiguous grammars, however.
1841:
334:
1660:
797:{\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\}}
492:
The existence of multiple derivations of the same string does not suffice to indicate that the grammar is ambiguous; only multiple
1234:
58:
admits an ambiguous grammar by introducing e.g. a duplicate rule. A language that only admits ambiguous grammars is called an
1376:
1244:
1111:
is inherently ambiguous. This set is context-free, since the union of two context-free languages is always context-free. But
1033:
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1507:
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include features for resolving some kinds of ambiguity, such as by using the precedence and associativity constraints.
549:
118:
63:
299:
1764:
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1444:
1360:
1328:
1844:- tool for analyzing context-free grammars with respect to language universality, ambiguity, and similar properties.
1118:
129:…meaning that the unique production can produce only the empty string, which is the unique string in the language.
1774:
Brabrand, Claus; Giegerich, Robert; Møller, Anders (March 2010). "Analyzing
Ambiguity of Context-Free Grammars".
553:
1714:
1115:
give a proof that any context-free grammar for this union language cannot unambiguously parse strings of form
1819:
1345:
509:
172:
tree (for the unambiguous grammar) or allowing both left- and right- association. This is elaborated below.
538:
230:→ A + A + A (First A is replaced by A+A. Replacement of the second A would yield a similar derivation)
943:
1857:
78:
43:
575:
Unambiguous context-free grammars can be nondeterministic. For example, the language of even-length
1788:
1756:
132:
In the same way, any grammar for a non-empty language can be made ambiguous by adding duplicates.
77:
problem. If present, these ambiguities are generally resolved by adding precedence rules or other
548:
The efficiency of parsing a context-free grammar is determined by the automaton that accepts it.
90:
1783:
1654:"Philippe Flajolet & Analytic Combinatorics: Inherent Ambiguity of Context-Free Languages"
81:
parsing rules, so the overall phrase grammar is unambiguous. Some parsing algorithms (such as
611:
561:
484:. In this latter case the grammar is unambiguous, but the context-free grammar is ambiguous.
86:
55:
1748:
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70:
39:
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parsers) can generate sets of parse trees (or "parse forests") from strings that are
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is ambiguous since there are two leftmost derivations for the string a + a + a:
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935:{\displaystyle \{a^{n}b^{m}c^{m}|m,n\geq 1\}\cup \{a^{m}b^{m}c^{n}|m,n\geq 1\}}
542:
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The existence of inherently ambiguous context-free languages was proven with
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82:
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derivations (or, equivalently, multiple parse trees) indicate ambiguity.
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1202:
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325:
A common example of ambiguity in computer programming languages is the
291:
51:
117:
This language also has an unambiguous grammar, consisting of a single
73:, the reference grammar is often ambiguous, due to issues such as the
1535:. Quarterly Progress Report, Research Laboratory of Electronics, MIT.
557:
1651:
1395:(July 1965). "On the translation of languages from left to right".
487:
810:
can be used to prove that certain context-free languages, such as
1508:"formal languages - Can regular expressions be made unambiguous?"
1346:"Analyzing Context-Free Grammars Using an Incremental SAT Solver"
290:
As another example, the grammar is ambiguous since there are two
1355:
International
Colloquium on Automata, Languages and Programming
1835:
1236:
An
Introduction to the Theory of Formal Languages and Automata
1205:, a type of parser for ambiguous and nondeterministic grammars
369:
some ambiguous phrase structures can appear. The expression
1728:
Introduction to
Automata Theory, Languages, and Computation
1436:
Introduction to automata theory, languages, and computation
1323:(2nd ed.). Addison-Wesley. Theorem 9.20, pp. 405–406.
1320:
Introduction to automata theory, languages, and computation
584:
1751:
Introduction to
Automata Theory, Languages and Computation
1747:
Hopcroft, John; Motwani, Rajeev; Ullman, Jeffrey (2001).
1652:
Fredérique
Bassino and Cyril Nicaud (December 16, 2011).
1344:
Axelsson, Roland; Heljanko, Keijo; Lange, Martin (2008).
568:
and can be parsed in polynomial time, for example by the
1773:
1460:
Book, R.; Even, S.; Greibach, S.; Ott, G. (Feb 1971).
1746:
1421:
1363:. Vol. 5126. Springer-Verlag. pp. 410–422.
1343:
1305:
1294:
An efficient augmented-context-free parsing algorithm
1121:
1104:{\displaystyle \{a^{n}b^{m}c^{m}d^{n}\mid n,m>0\}}
1036:
1023:{\displaystyle \{a^{n}b^{n}c^{m}d^{m}\mid n,m>0\}}
955:
816:
623:
603:
There are no inherently ambiguous regular languages.
537:
because it can be shown that it is equivalent to the
556:
and can be parsed in linear time, for example by an
1459:
1232:
59:
1725:
1439:(2nd ed.). Addison-Wesley. pp. 249–253.
1296:." Computational linguistics 13.1-2 (1987): 31-46.
1182:
1103:
1022:
934:
796:
545:for detecting ambiguity of context-free grammars.
1598:"A helpful result for proving inherent ambiguity"
590:
524:
1849:
1266:Electronic Notes in Theoretical Computer Science
533:of whether an arbitrary grammar is ambiguous is
488:An unambiguous grammar with multiple derivations
1677:"Inherent ambiguity of minimal linear grammars"
1211:, another type of parser for ambiguous grammars
1724:Hopcroft, John E.; Ullman, Jeffrey D. (1979).
1723:
1112:
541:. At least, there are tools implementing some
521:Only the former derivation is a leftmost one.
1183:{\displaystyle a^{n}b^{n}c^{n}d^{n},(n>0)}
1262:"SPPF-Style Parsing From Earley Recognizers"
1098:
1037:
1017:
956:
929:
876:
870:
817:
791:
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366:Statement | ... Condition → ...
175:
144:of unary strings of a given character, say
1787:
1755:(2nd ed.). Addison Wesley. pp.
1692:
1563:
1277:
1226:
1701:
14:
1850:
1707:Introduction to Formal Language Theory
1545:
1530:
1674:
1595:
1462:"Ambiguity in Graphs and Expressions"
1391:
1259:
160:…but also has the ambiguous grammar:
550:Deterministic context-free grammars
101:
64:Deterministic context-free grammars
24:
1260:Scott, Elizabeth (April 1, 2008).
739:
671:
580:lengths of a semi-parsed string.
560:. They are a strict subset of the
343:In a grammar containing the rules
25:
1874:
1829:
1675:Gross, Maurice (September 1964).
1361:Lecture Notes in Computer Science
1732:(1st ed.). Addison-Wesley.
1666:from the original on 2022-09-25.
942:, are inherently ambiguous. See
499:For example, the simple grammar
329:problem. In many languages, the
314:
298:
294:for the string a + a − a:
168:These correspond to producing a
152:), has the unambiguous grammar:
1838:- a grammar ambiguity analyzer.
1776:Science of Computer Programming
1645:
1636:
1589:
1546:Parikh, Rohit J. (1966-10-01).
1539:
1524:
1500:
554:deterministic pushdown automata
135:
1812:
1531:Parikh, Rohit (January 1961).
1466:IEEE Transactions on Computers
1453:
1415:
1385:
1357:(ICALP'08), Reykjavik, Iceland
1337:
1299:
1286:
1253:
1177:
1165:
910:
851:
631:
591:Inherently ambiguous languages
525:Recognizing ambiguous grammars
27:Type of a context-free grammar
13:
1:
1694:10.1016/S0019-9958(64)90422-X
1409:10.1016/S0019-9958(65)90426-2
1239:. John Benjamins Publishing.
1220:
457:is associated with the first
60:inherently ambiguous language
18:Inherently ambiguous language
1369:10.1007/978-3-540-70583-3_34
1233:Willem J. M. Levelt (2008).
1113:Hopcroft & Ullman (1979)
614:in an MIT research report.
46:that can have more than one
7:
1818:The following example uses
1798:10.1016/j.scico.2009.11.002
1602:Mathematical Systems Theory
1596:Ogden, William (Sep 1968).
1548:"On Context-Free Languages"
1533:Language-generating devices
1279:10.1016/j.entcs.2008.03.044
1196:
539:Post correspondence problem
193:
96:
10:
1879:
518:S ⇒ A + A ⇒ A + 0 ⇒ 0 + 0
318:
804:is inherently ambiguous.
453:depending on whether the
279:
261:
243:
225:
206:
42:for which there exists a
1805:
1782:(3). Elsevier: 176–191.
1353:Proceedings of the 35th
564:, which are accepted by
393:can be parsed as either
354:Statement |
271:
253:
235:
217:
197:
176:Addition and subtraction
148:(the regular expression
1681:Information and Control
1478:10.1109/t-c.1971.223204
1397:Information and Control
543:semi-decision procedure
91:syntactically ambiguous
1184:
1105:
1024:
936:
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512:A + A ⇒ 0 + A ⇒ 0 + 0
1565:10.1145/321356.321364
1185:
1106:
1025:
937:
799:
562:context-free grammars
310:A → A + a | A − a | a
187:A → A + A | A − A | a
71:programming languages
56:context-free language
1119:
1034:
953:
814:
621:
598:inherently ambiguous
502:S → A + A A → 0 | 1
472:statement or making
182:context free grammar
40:context-free grammar
1215:Syntactic ambiguity
48:leftmost derivation
1709:. Addison-Wesley.
1642:p.99-103, Sect.4.7
1614:10.1007/bf01694004
1552:Journal of the ACM
1180:
1101:
1020:
932:
794:
54:. Every non-empty
1703:Michael, Harrison
1378:978-3-540-70582-6
1292:Tomita, Masaru. "
1246:978-90-272-3250-2
752:
751: where
691:
566:pushdown automata
480:with the nearest
288:
287:
170:right-associative
79:context-sensitive
36:ambiguous grammar
16:(Redirected from
1870:
1858:Formal languages
1836:dk.brics.grammar
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1586:Here: Theorem 3.
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164:A → aA | Aa | ε
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1423:Hopcroft, John
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949:The union of
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808:Ogden's lemma
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617:The language
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315:Dangling else
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69:For computer
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1515:. Retrieved
1512:MathOverflow
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1393:Knuth, D. E.
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1272:(2): 53–67.
1269:
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1209:Chart parser
1192:
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612:Rohit Parikh
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346:Statement →
342:
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305:
289:
284:→ a + a + a
276:→ a + a + a
266:→ a + a + A
258:→ a + a + A
248:→ a + A + A
240:→ a + A + A
190:
179:
167:
159:
139:
136:Unary string
131:
128:
116:
113:
105:
68:
35:
29:
1842:CFGAnalyzer
610:in 1961 by
577:palindromes
535:undecidable
292:parse trees
1852:Categories
1716:0201029553
1517:2023-02-23
1221:References
1203:GLR parser
461:or second
362:Statement
358:Condition
350:Condition
156:A → aA | ε
52:parse tree
1863:Ambiguity
1784:CiteSeerX
1622:0025-5661
1574:0004-5411
1486:0018-9340
1081:∣
1000:∣
944:this page
924:≥
874:∪
865:≥
786:≥
740:′
672:′
558:LR parser
110:A → A | ε
1705:(1978).
1661:Archived
1630:13197551
1582:12263468
1494:20676251
1433:(2001).
1317:(2001).
1197:See also
782:′
765:′
494:leftmost
222:→ a + A
212:→ A + A
203:→ A + A
97:Examples
1822:syntax
1820:Pascal
1786:
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1327:
1243:
423:or as
333:in an
83:Earley
44:string
1806:Notes
1664:(PDF)
1657:(PDF)
1626:S2CID
1578:S2CID
1490:S2CID
1349:(PDF)
1030:with
470:endif
434:begin
404:begin
125:A → ε
38:is a
34:, an
1761:ISBN
1734:ISBN
1711:ISBN
1618:ISSN
1570:ISSN
1482:ISSN
1470:C-20
1441:ISBN
1373:ISBN
1325:ISBN
1241:ISBN
1172:>
1093:>
1012:>
585:YACC
529:The
515:and
478:else
474:else
455:else
445:else
441:then
431:then
418:else
411:then
401:then
388:else
384:then
377:then
364:else
360:then
352:then
331:else
180:The
140:The
1794:doi
1757:217
1689:doi
1610:doi
1560:doi
1474:doi
1405:doi
1365:doi
1274:doi
1270:203
572:.
449:end
447:s2
420:s2
415:end
390:s2
146:'a'
87:GLR
85:or
50:or
30:In
1854::
1792:.
1780:75
1778:.
1759:.
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1600:.
1576:.
1568:.
1556:13
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1268:.
1264:.
1190:.
600:.
508:S
482:if
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463:if
459:if
443:s
439:b
437:if
429:a
427:if
413:s
409:b
407:if
399:a
397:if
386:s
382:b
380:if
375:a
373:if
356:if
348:if
150:a*
121::
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62:.
1800:.
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1769:.
1742:.
1719:.
1697:.
1691::
1685:7
1632:.
1612::
1606:2
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1476::
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1411:.
1407::
1401:8
1381:.
1367::
1333:.
1282:.
1276::
1249:.
1178:)
1175:0
1169:n
1166:(
1163:,
1158:n
1154:d
1148:n
1144:c
1138:n
1134:b
1128:n
1124:a
1099:}
1096:0
1090:m
1087:,
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1062:c
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1052:b
1046:n
1042:a
1038:{
1018:}
1015:0
1009:m
1006:,
1003:n
995:m
991:d
985:m
981:c
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971:b
965:n
961:a
957:{
930:}
927:1
921:n
918:,
915:m
911:|
905:n
901:c
895:m
891:b
885:m
881:a
877:{
871:}
868:1
862:n
859:,
856:m
852:|
846:m
842:c
836:m
832:b
826:n
822:a
818:{
792:}
789:1
779:m
775:,
772:m
769:,
762:n
758:,
755:n
747:,
736:m
731:b
725:n
721:a
715:m
711:b
705:n
701:a
697:=
694:x
684:m
680:b
668:n
663:a
657:m
653:b
647:n
643:a
639:=
636:x
632:|
628:x
625:{
510:⇒
209:A
200:A
20:)
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