# Context-free language

In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, 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 $L=\{a^{n}b^{n}:n\geq 1\}$, 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:
    ${\begin{aligned}\delta (q_{0},a,z)&=(q_{0},az)\\\delta (q_{0},a,a)&=(q_{0},aa)\\\delta (q_{0},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},\varepsilon ,z)&=(q_{f},\varepsilon )\end{aligned}}$
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 $\{a^{n}b^{m}c^{m}d^{n}|n,m>0\}$ 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 $S\to SS~|~(S)~|~\varepsilon$.

## 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 $w$, 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$^{2.3728596}$). 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 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 $L\cup P$ 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 circular shift of L (the language $\{vu:uv\in L\}$)
* the prefix closure of L (the set of all prefixes of strings from L)
* the quotient L/R of L by a regular language R

#### Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages $A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}$ 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. 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: $A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}$. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: ${\overline {L}}=\Sigma ^{*}\setminus L$.
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.

### 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 $L(A)=L(B)$?
* Disjointness: is $L(A)\cap L(B)=\emptyset$ ? However, the intersection of a context-free language and a regular language is context-free, hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
* Containment: is $L(A)\subseteq L(B)$ ? Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.
* Universality: is $L(A)=\Sigma ^{*}$?
* Regularity: is $L(A)$ a regular language?
* Ambiguity: is every grammar for $L(A)$ ambiguous?
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

### 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.