# 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 ${\displaystyle S\to SS~|~(S)~|~\varepsilon }$. Also, most arithmetic expressions are generated by context-free grammars.

## Examples

An archetypal 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 ${\displaystyle a}$'s, and the entire second halves of which are ${\displaystyle b}$'s. ${\displaystyle L}$ 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]

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.Template:Sfn

## 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.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[1] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[2]

## 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:

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 ${\displaystyle L\cap D}$ and their difference ${\displaystyle L\setminus D}$ 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 ${\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}}$ and ${\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}}$, which are both context-free.[note 2] 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.

Context-free languages are also not closed under complementation, as for any languages A and B: ${\displaystyle A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}}$.

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

## Decidability properties

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

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

According to Hopcroft, Motwani, Ullman (2003),[4] 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[1]

## Parsing

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(n2.3728639).[5][6][note 3] Conversely, Lillian Lee has shown O(n3-ε) boolean matrix multiplication to be reducible to O(n3-3ε) CFG parsing, thus establishing some kind of lower bound for the latter.[7]

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.[8]

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

## Notes

1. A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: SSc | aTb | ε; TaTb | ε. The grammar for B is analogous.
2. In Valiant's papers, O(n2.81) given, the then best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm for bound improvements since then.

## References

1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
2. How to prove that a language is not context-free?
3. Template:Harvtxt, p. 59, Theorem 6.7
4. {{#invoke:citation/CS1|citation |CitationClass=book }} Here: Sect.7.6, p.304, and Sect.9.7, p.411
5. Template:Cite techreport
6. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
7. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
8. Template:Cite doi
• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• {{#invoke:citation/CS1|citation

|CitationClass=book }} Chapter 2: Context-Free Languages, pp. 91–122.