Context-free language Totally Explained

Everything about Context-free Language totally explained

A context-free language is a formal language that can be defined by a 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 =

, which can be shown to be non-context-free by the pumping lemma for context-free languages.

Decidability properties

The following problems for context-free languages are undecidable:

Equivalence: given two context-free grammars A and B, is

L(A)=L(B)

L(A) cap L(B) = emptyset

L(A)=Sigma^*

L(A) subseteq L(B)

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

L(A)=emptyset

L(A)

finite?

Membership: given any word w, does

w in L(A)

? (membership problem is even polynomially decidable - see CYK 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 regular 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 isn't context-free, one may employ the pumping lemma for context-free languages.Further Information

Get more info on 'Context-free Language'.

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