In formal language theory, a context-free language (CFL) is a language generated by a context-free grammar (CFG).

Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars.

Background

Context-free grammar

Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.

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 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

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

${\displaystyle {\begin{aligned}\delta (q_{0},a,z)&=(q_{0},az)\\\delta (q_{0},a,a)&=(q_{0},aa)\\\delta (q_{0},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},\varepsilon ,z)&=(q_{f},\varepsilon )\end{aligned))}$

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 ${\displaystyle \{a^{n}b^{m}c^{m}d^{n}|n,m>0\))$ with ${\displaystyle \{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 ${\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\))$ which is the intersection of these two languages.^[1]

Dyck language

The language of all properly matched parentheses is generated by the grammar ${\displaystyle S\to SS~|~(S)~|~\varepsilon }$.

Properties

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 ${\displaystyle w}$, determine whether ${\displaystyle w\in L(G)}$ where ${\displaystyle L}$ is the language generated by a given grammar ${\displaystyle G}$; 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^2.3728596).^{[2][note 2]} 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.^[3]

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.^[4]

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

Closure properties

The class of context-free languages is 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 ${\displaystyle L\cup P}$ of L and P^[5]
• the reversal of L^[6]
• the concatenation ${\displaystyle L\cdot P}$ of L and P^[5]
• the Kleene star ${\displaystyle L^{*))$ of L^[5]
• the image ${\displaystyle \varphi (L)}$ of L under a homomorphism ${\displaystyle \varphi }$^[7]
• the image ${\displaystyle \varphi ^{-1}(L)}$ of L under an inverse homomorphism ${\displaystyle \varphi ^{-1))$^[8]
• the circular shift of L (the language ${\displaystyle \{vu:uv\in L\))$)^[9]
• the prefix closure of L (the set of all prefixes of strings from L)^[10]
• the quotient L/R of L by a regular language R^[11]

Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages ${\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\))$ and ${\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\))$, which are both context-free.^{[note 3]} Their intersection is ${\displaystyle 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. As a consequence, context-free languages cannot be closed under complementation, as for any languages A and B, their intersection can be expressed by union and complement: ${\displaystyle A\cap B={\overline ((\overline {A))\cup {\overline {B))))}$. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: ${\displaystyle {\overline {L))=\Sigma ^{*}\setminus L}$.^[12]

However, if L is a context-free language and D is a regular language then both their intersection ${\displaystyle L\cap D}$ and their difference ${\displaystyle L\setminus D}$ are context-free languages.^[13]

Decidability

In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.

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

• Equivalence: is ${\displaystyle L(A)=L(B)}$?^[14]
• Disjointness: is ${\displaystyle L(A)\cap L(B)=\emptyset }$ ?^[15] However, the intersection of a context-free language and a regular language is context-free,^[16][17] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
• Containment: is ${\displaystyle L(A)\subseteq L(B)}$ ?^[18] Again, the variant of the problem where B is a regular grammar is decidable,^{[citation needed]} while that where A is regular is generally not.^[19]
• Universality: is ${\displaystyle L(A)=\Sigma ^{*))$?^[20]
• Regularity: is ${\displaystyle L(A)}$ a regular language?^[21]
• Ambiguity: is every grammar for ${\displaystyle L(A)}$ ambiguous?^[22]

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

• Emptiness: Given a context-free grammar A, is ${\displaystyle L(A)=\emptyset }$ ?^[23]
• Finiteness: Given a context-free grammar A, is ${\displaystyle L(A)}$ finite?^[24]
• Membership: Given a context-free grammar G, and a word ${\displaystyle w}$, does ${\displaystyle w\in L(G)}$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),^[25] 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^[26]

Languages that are not context-free

The set ${\displaystyle \{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.^[27] 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^[26] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[28]