Context free language

Context-free grammar

Different CF grammars can generate the same CF language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar.

Automata

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

Examples

A model context-free language is L = { a n b n : n ≥ 1 } , 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 → a S b   |   a b . This language is not regular. It is accepted by the pushdown automaton M = ( { q 0 , q 1 , q f } , { a , b } , { a , z } , δ , q 0 , z , { q f } ) where δ is defined as follows:

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.

Dyck language

The language of all properly matched parentheses is generated by the grammar S → S S   |   ( S )   |   ε .

Context-free parsing

The context-free nature of the language makes it simple to parse with a pushdown automaton.

Determining an instance of the membership problem; i.e. given a string w , determine whether w ∈ 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(n^2.3728639). Conversely, Lillian Lee has shown O(n^3−ε) boolean matrix multiplication to be reducible to O(n^3−3ε) CFG parsing, thus establishing some kind of lower bound for the latter.

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.

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

Closure

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. This was proved by Scheinberg in 1960. However, if L is a context-free language and D is a regular language then both their intersection L ∩ D and their difference L ∖ D are context-free languages.

Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages A = { a n b n c m ∣ m , n ≥ 0 } and B = { a m b n c n ∣ m , n ≥ 0 } , which are both context-free. Their intersection is A ∩ B = { a n b n c n ∣ n ≥ 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 ∩ B = A ¯ ∪ B ¯ ¯ .

Context-free language are also not closed under difference: L^C = Σ^* L.

Decidability

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

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

According to Hopcroft, Motwani, Ullman (2003), 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

Languages that are not context-free

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. 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 or a number of other methods, such as Ogden's lemma or Parikh's theorem.