Context-free language: Difference between revisions
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===Decidability=== |
===Decidability=== |
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In formal language theory, questions about regular languages are always 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 with a different grammar.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=1138|isbn=1-57955-008-8}}</ref> |
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The following problems are [[Undecidable problem|undecidable]] for arbitrarily given [[context-free grammar]]s A and B: |
The following problems are [[Undecidable problem|undecidable]] for arbitrarily given [[context-free grammar]]s A and B: |
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*Equivalence: is <math>L(A)=L(B)</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(1)}} |
*Equivalence: is <math>L(A)=L(B)</math>?{{sfn|Hopcroft|Ullman|1979|p=203|loc=Theorem 8.12(1)}} |
Revision as of 18:36, 4 April 2018
In
Context-free languages have many applications in
Background
Context-free grammar
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.
Automata
The set of all context-free languages is identical to the set of languages accepted by
Examples
A model context-free language is , the language of all non-empty even-length strings, the entire first halves of which are a's, and the entire second halves of which are b's. L is generated by the grammar . This language is not regular. It is accepted by the pushdown automaton where is defined as follows:[note 1]
Unambiguous CFLs are a proper subset of all CFLs: there are
Dyck language
The language of all properly matched parentheses is generated by the grammar .
Properties
Context-free parsing
The context-free nature of the language makes it simple to parse with a pushdown automaton.
Determining an instance of the membership problem; i.e. given a string , determine whether where is the language generated by a given grammar ; is also known as recognition. Context-free recognition for Chomsky normal form grammars was shown by Leslie G. Valiant to be reducible to boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728639).[2][3][note 2] Conversely, Lillian Lee has shown O(n3−ε) boolean matrix multiplication to be reducible to O(n3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.[4]
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 parsing. Known parsers have a time complexity that is cubic in the size of the string that is parsed.
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 CYK algorithm and Earley's Algorithm.
A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR(k) parser.[5]
See also parsing expression grammar as an alternative approach to grammar and parser.
Closure
Context-free languages are closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:
- the union of L and P
- the reversal of L
- the concatenation of L and P
- the Kleene star of L
- the image of L under a homomorphism
- the image of L under an inverse homomorphism
- the cyclic shiftof L (the language )
Context-free languages are not closed under complement, intersection, or difference. This was proved by Scheinberg in 1960.[6] However, if L is a context-free language and D is a regular language then both their intersection and their difference are context-free languages.
Nonclosure under intersection, complement, and difference
The context-free languages are not closed under intersection. This can be seen by taking the languages and , which are both context-free.[note 3] Their intersection is , which can be shown to be non-context-free by the pumping lemma for context-free languages.
Context-free languages are also not closed under complementation, as for any languages A and B: .
Context-free language are also not closed under difference: LC = Σ* \ L.[6]
Decidability
In formal language theory, questions about regular languages are always 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 with a different grammar.[7]
The following problems are undecidable for arbitrarily given context-free grammars A and B:
- Equivalence: is ?[8]
- Disjointness: is ?[9] However, the intersection of a context-free language and a regular language is context-free,[10][11] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
- Containment: is ?[12] Again, the variant of the problem where B is a regular grammar is decidable,[citation needed] while that where A is regular is generally not.[13]
- Universality: is ?[14]
The following problems are decidable for arbitrary context-free languages:
- Emptiness: Given a context-free grammar A, is ?[15]
- Finiteness: Given a context-free grammar A, is finite?[16]
- Membership: Given a context-free grammar G, and a word , does ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.
According to Hopcroft, Motwani, Ullman (2003),[17] many of the fundamental closure and (un)decidability properties of context-free languages were shown in the 1961 paper of Bar-Hillel, Perles, and Shamir[18]
Languages that are not context-free
The set is a context-sensitive language, but there does not exist a context-free grammar generating this language.[19] 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 pumping lemma for context-free languages[18] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[20]
Notes
References
- ^ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
- ^ Leslie Valiant (Jan 1974). General context-free recognition in less than cubic time (Technical report). Carnegie Mellon University. p. 11.
- .
- .
- )
- ^ a b Stephen Scheinberg, Note on the Boolean Properties of Context Free Languages, Information and Control, 3, 372-375 (1960)
- ISBN 1-57955-008-8.
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
- ^ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
- ^ Salomaa (1973) , p. 59, Theorem 6.7
- ^ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
- ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
- ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
- ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
- ^ John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411
- ^ a b Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.
- ^ Hopcroft & Ullman 1979.
- ^ How to prove that a language is not context-free?
- Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.
- Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley.
{{cite book}}
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(help) - Arto Salomaa (1973). Formal Languages. ACM Monograph Series.
- ISBN 0-534-94728-X. Chapter 2: Context-Free Languages, pp. 91–122.
- Jean-Michel Autebert, Jean Berstel, Luc Boasson, Context-Free Languages and Push-Down Automata, in: G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag, 1997, 111-174.