# Context-free language

In formal language theory, a **context-free language** (**CFL**) is a language generated by a context-free grammar (CFG).

Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.

## 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 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.

## 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 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.[1]

### 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*(*n*^{2.3728639}).[2][3][note 2]
Conversely, Lillian Lee has shown *O*(*n*^{3−ε}) boolean matrix multiplication to be reducible to *O*(*n*^{3−3ε}) 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

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*[6] - the reversal of
*L*[7] - the concatenation of
*L*and*P*[6] - the Kleene star of
*L*[6] - the image of
*L*under a homomorphism [8] - the image of
*L*under an inverse homomorphism [9] - the circular shift of
*L*(the language )[10] - the prefix closure of
*L*(the set of all prefixes of strings from*L*)[11] - the quotient
*L*/*R*of*L*by a regular language*R*[12]

#### 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: .[13]

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.

### Decidability

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.[14]

The following problems are undecidable for arbitrarily given context-free grammars A and B:

- Equivalence: is ?[15]
- Disjointness: is ?[16] However, the intersection of a context-free language and a
*regular*language is context-free,[17][18] hence the variant of the problem where*B*is a regular grammar is decidable (see "Emptiness" below). - Containment: is ?[19] Again, the variant of the problem where
*B*is a regular grammar is decidable, while that where*A*is regular is generally not.[20] - Universality: is ?[21]

The following problems are *decidable* for arbitrary context-free languages:

- Emptiness: Given a context-free grammar
*A*, is ?[22] - Finiteness: Given a context-free grammar
*A*, is finite?[23] - 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),[24] 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[25]

### 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.[26] 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[25] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[27]

## Notes

- meaning of 's arguments and results:
- In Valiant's papers,
*O*(*n*^{2.81}) 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.

## 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. - 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.