Context-free language

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In formal language theory, a context-free language (CFL) is a language generated by some context-free grammar (CFG). Different CF grammars can generate the same CF language. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).

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. Indeed, given a CFG, there is a direct way to produce a pushdown automaton for the grammar (and corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.

Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar S\to SS ~|~ (S) ~|~ \varepsilon. Also, most arithmetic expressions are generated by context-free grammars.


An archetypal context-free language is L = \{a^nb^n:n\geq1\}, 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 S\to aSb ~|~ ab. This language is not regular. It is accepted by the pushdown automaton M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\}) where \delta is defined as follows:[note 1]

\delta(q_0, a, z) = (q_0, az)
\delta(q_0, a, a) = (q_0, aa)
\delta(q_0, b, a) = (q_1, \varepsilon)
\delta(q_1, b, a) = (q_1, \varepsilon)

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 \{a^n b^m c^m d^n | n, m > 0\} with \{a^n b^n c^m d^m | n, m > 0\}. 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 \{a^n b^n c^n d^n | n > 0\} which is the intersection of these two languages.[1]

Languages that are not context-free[edit]

The set \{a^n b^n c^n d^n | n > 0\} is a context-sensitive language, but there does not exist a context-free grammar generating this language.[2] 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[3] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[4]

Closure properties[edit]

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:

Context-free languages are not closed under complement, intersection, or difference. However, if L is a context-free language and D is a regular language then both their intersection L\cap D and their difference L\setminus D are context-free languages.

Nonclosure under intersection, complement, and difference[edit]

The context-free languages are not closed under intersection. This can be seen by taking the languages A = \{a^n b^n c^m \mid m, n \geq 0 \} and B = \{a^m b^n c^n \mid m,n \geq 0\}, which are both context-free.[note 2] Their intersection is A \cap B = \{ a^n b^n c^n \mid n \geq 0\}, 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: A \cap B = \overline{\overline{A} \cup \overline{B}} .

Context-free language are also not closed under difference: LC = Σ* \ L

Decidability properties[edit]

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

  • Equivalence: is L(A)=L(B)?[5]
  • Disjointness: is L(A) \cap L(B) = \emptyset  ?[6] However, the intersection of a context-free language and a regular language is context-free,[7][8] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
  • Containment: is L(A) \subseteq L(B) ?[9] 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.[10]
  • Universality: is L(A)=\Sigma^* ?[11]

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

  • Emptiness: Given a context-free grammar A, is L(A) = \emptyset ?[12]
  • Finiteness: Given a context-free grammar A, is L(A) finite?[13]
  • Membership: Given a context-free grammar G, and a word w, does w \in L(G) ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),[14] 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[3]


Determining an instance of the membership problem; i.e. given a string w, determine whether w \in L(G) where L is the language generated by a given grammar G; 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).[15][16][note 3] 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.[17]

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.[18]

See also parsing expression grammar as an alternative approach to grammar and parser.

See also[edit]


  1. ^ meaning of \delta's arguments and results: \delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})
  2. ^ A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: SSc | aTb | ε; TaTb | ε. The grammar for B is analogous.
  3. ^ 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.


  1. ^ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
  2. ^ Hopcroft & Ullman 1979.
  3. ^ 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. 
  4. ^ How to prove that a language is not context-free?
  5. ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
  6. ^ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
  7. ^ Salomaa (1973), p. 59, Theorem 6.7
  8. ^ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
  9. ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
  10. ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
  11. ^ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
  12. ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
  13. ^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
  14. ^ 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
  15. ^ Leslie Valiant (Jan 1974). General context-free recognition in less than cubic time (Technical report). Carnegie Mellon University. p. 11. 
  16. ^ 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. 
  17. ^ Lillian Lee (2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication" (PDF). JACM 49 (1): 1–15. doi:10.1145/505241.505242. 
  18. ^ 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. Retrieved 29 May 2011. 
  • 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.