Context-free language

In formal language theory, a context-free language (CFL) is a language generated by some context-free grammar (CFG). Different CF grammars can generate the same CF language. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).

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

Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar $S\to SS~|~(S)~|~\varepsilon$. Also, most arithmetic expressions are generated by context-free grammars.

Examples

An archetypal context-free language is $L=\{a^{n}b^{n}:n\geq 1\}$, 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 $S\to aSb~|~ab$. This language is not regular. It is accepted by the pushdown automaton $M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},z,\{q_{f}\})$ where $\delta$ is defined as follows:

$\delta (q_{0},a,z)=(q_{0},az)$
$\delta (q_{0},a,a)=(q_{0},aa)$
$\delta (q_{0},b,a)=(q_{1},\varepsilon )$
$\delta (q_{1},b,a)=(q_{1},\varepsilon )$

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.Template:Sfn

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.Template:Sfn 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.

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 $\varphi (L)$ of L under a homomorphism $\varphi$
• 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 and difference

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

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

Decidability properties

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

• Equivalence: Given two context-free grammars A and B, is $L(A)=L(B)$?
• Intersection Emptiness: Given two context-free grammars A and B, is $L(A)\cap L(B)=\emptyset$ ? However, the intersection of a context-free language and a regular language is context-free, and the variant of the problem where B is a regular grammar is decidable.
• Containment: Given a context-free grammar A, is $L(A)\subseteq L(B)$ ? Again, the variant of the problem where B is a regular grammar is decidable.
• Universality: Given a context-free grammar A, is $L(A)=\Sigma ^{*}$ ?

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

• Emptiness: Given a context-free grammar A, is $L(A)=\emptyset$ ?
• Finiteness: Given a context-free grammar A, is $L(A)$ finite?
• Membership: Given a context-free grammar G, and a word $w$, does $w\in L(G)$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

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

Parsing

Determining an instance of the membership problem; i.e. given a string $w$, determine whether $w\in L(G)$ 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 the 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.

See also

• Deterministic context-free language
• Parsing

References

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|CitationClass=book }} 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.