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.

1 Examples
2 Closure properties
- 2.1 Nonclosure under intersection and complement
3 Decidability properties
4 Properties of context-free languages
5 References

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 $M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, \{q_f\})$ where $\delta$ is defined as follows:

$\delta(q_0, a, z) = (q_0, a)$
$\delta(q_0, a, a) = (q_0, a)$
$\delta(q_0, b, a) = (q_1, x)$
$\delta(q_1, b, a) = (q_1, x)$
$\delta(q_1, \lambda, z) = (q_f, z)$

$\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})$
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 $\varphi^{-1}(L)$ of L under an inverse homomorphism $\varphi^{-1}$
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 $A = \{a^n b^n c^m \mid m, n \geq 0 \}$ and $B = \{a^m b^n c^n \mid m,n \geq 0\}$ , which are both context-free. 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 $L(A)=\Sigma^*$ ?
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.

Automata theory: formal languages and formal grammars

Chomsky hierarchy

Type-0

—

Type-1

—

—

—

Type-2

—

—

Type-3

—

Grammars

Unrestricted

(no common name)

Context-sensitive

Indexed

Linear context-free rewriting systems etc.

Tree-adjoining etc.

Context-free

Deterministic context-free

Visibly pushdown

Regular

—

Languages

Recursively enumerable

Recursive

Context-sensitive

Indexed

Mildly context-sensitive

Tree-adjoining

Context-free

Deterministic context-free

Visibly pushdown

Regular

Star-free

Minimal automaton

Turing machine

Decider

Linear-bounded

Nested stack

Thread automata

Embedded pushdown

Nondeterministic pushdown

Deterministic pushdown

Visibly pushdown

Finite

Counter-free (with aperiodic finite monoid)

Each category of languages is a proper subset of the category directly above it. - Any automaton and any grammar in each category has an equivalent automaton or grammar in the category directly above it.