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Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.
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.
The set of all context-free languages is identical to the set of languages accepted by pushdown automata, 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.
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 inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of with . 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 which is the intersection of these two languages.
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).[note 2]
Conversely, Lillian Lee has shown O(n3-?) boolean matrix multiplication to be reducible to O(n3-3?) CFG parsing, thus establishing some kind of lower bound for the latter.
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.
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. As a consequence, context-free languages cannot be closed under complementation, as for any languages A and B, their intersection can be expressed by union and complement: . In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: .
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.
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 with a different grammar.
Disjointness: is ? However, the intersection of a context-free language and a regular language is context-free, hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
Containment: is ? Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.
The following problems are decidable for arbitrary context-free languages:
Emptiness: Given a context-free grammar A, is ?
Finiteness: Given a context-free grammar A, is finite?
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),
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
Salomaa, Arto (1973). Formal Languages. ACM Monograph Series.
Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata". In G. Rozenberg; A. Salomaa (eds.). Handbook of Formal Languages(PDF). 1. Springer-Verlag. pp. 111-174.
Ginsburg, Seymour (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill.