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.

Examples

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,q_f},{a,b},{a,z},δ,q₀,{q_f}) 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) = (q_f,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:

• the union $L \cup P$ of L and P
• the reversal of L
• the concatenation $L \cdot P$ of L and P
• the Kleene star L ^* of L
• the image φ(L) of L under a homomorphism φ
• the image φ ^{− 1}(L) of L under an inverse homomorphism φ ^{− 1}
• the cyclic shift of L (the language $\{vu : uv \in L \}$)

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

• The reverse of a context-free language is context-free, but the complement need not be.
• Every regular language is context-free because it can be described by a context-free grammar.
• The intersection of a context-free language and a regular language is always context-free.
• 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.

References

• 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.