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:
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 language of all properly matched parentheses is generated by the grammar .
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). 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.
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.
See also parsing expression grammar as an alternative approach to grammar and parser.
The class of context-free languages is 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 circular shift of L (the language )
- the prefix closure of L (the set of all prefixes of strings from L)
- the quotient L/R of L by a regular language R
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. 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.
- Equivalence: is ?
- 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.
- Universality: is ?
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
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. 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 or a number of other methods, such as Ogden's lemma or Parikh's theorem.
- meaning of 's arguments and results:
- In Valiant's papers, O(n2.81) given, the then best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm for bound improvements since then.
- A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S → Sc | aTb | ε; T → aTb | ε. The grammar for B is analogous.
- 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.
- Leslie G. Valiant (1975). "General context-free recognition in less than cubic time". Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8.
- Lillian Lee (2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication" (PDF). J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242.
- Knuth, D. E. (July 1965). "On the translation of languages from left to right" (PDF). Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Archived from the original (PDF) on 15 March 2012. Retrieved 29 May 2011.
- Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
- Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
- Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
- Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
- Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
- Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3: 372–375. doi:10.1016/s0019-9958(60)90965-7.
- Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1138. 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
- 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.
- Arto Salomaa (1973). Formal Languages. ACM Monograph Series.
- Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. 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.