Context-free language

Context-free language

In formal language theory, a context-free language is a language generated by some context-free grammar. The set of all context-free languages is identical to the set of languages accepted by pushdown automata.



An archetypical 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, and is accepted by the pushdown automaton M = ({q0,q1,qf},{a,b},{a,z},δ,q0,{qf}) where δ is defined as follows:

δ(q0,a,z) = (q0,a)
δ(q0,a,a) = (q0,a)
δ(q0,b,a) = (q1,x)
δ(q1,b,a) = (q1,x)
δ(q1,λ,z) = (qf,z)

δ(state1,read,pop) = (state2,push)
where z is initial stack symbol and x means pop action.

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) ~|~ \lambda. Also, most arithmetic expressions are generated by context-free grammars.

Closure properties

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 and complement

The context-free languages are not closed under intersection. This can be seen by taking the languages A = \{a^m b^m c^n \mid m, n \geq 0 \} and B = \{a^m b^n c^n \mid m,n \geq 0\}, which are both context-free. 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}} .

Decidability properties

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

  • Equivalence: is L(A) = L(B)?
  • is L(A) \cap L(B) = \emptyset  ? (However, the intersection of a context-free language and a regular language is context-free, so if B were a regular language, this problem becomes decidable.)
  • is L(A) = Σ *  ?
  • is L(A) \subseteq L(B) ?

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

  • is L(A)=\emptyset ?
  • is L(A) finite?
  • Membership: given any word w, does w \in L(A) ? (membership problem is even polynomially decidable - see CYK algorithm and Early's Algorithm)

Properties of context-free languages


  • Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.. 
  • 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.

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