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 <math>L = \{a^nb^n:n\geq1\}</math>, the language of all non-empty even-length strings, the entire first halves of which are Template:Mvar's, and the entire second halves of which are Template:Mvar's. Template:Mvar is generated by the grammar <math>S\to aSb ~|~ ab</math>. This language is not regular. It is accepted by the pushdown automaton <math>M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\})</math> where <math>\delta</math> is defined as follows:<ref group="note">meaning of <math>\delta</math>'s arguments and results: <math>\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})</math></ref>

<math>\begin{align}

\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{align}</math>

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 <math>\{a^n b^m c^m d^n | n, m > 0\}</math> with <math>\{a^n b^n c^m d^m | n, m > 0\}</math>. 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 <math>\{a^n b^n c^n d^n | n > 0\}</math> which is the intersection of these two languages.Template:Sfn

Dyck language

The language of all properly matched parentheses is generated by the grammar <math>S\to SS ~|~ (S) ~|~ \varepsilon</math>.

Properties

Context-free parsing

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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 <math>w</math>, determine whether <math>w \in L(G)</math> where <math>L</math> is the language generated by a given grammar <math>G</math>; 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).<ref>Template:Cite journal</ref><ref group=note>In Valiant's paper, O(n^2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity for bound improvements since then.</ref> 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.<ref>Template:Cite journal</ref>

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.<ref>Template:Cite journal</ref>

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 <math>L \cup P</math> of L and PTemplate:Sfn
• the reversal of LTemplate:Sfn
• the concatenation <math>L \cdot P</math> of L and PTemplate:Sfn
• the Kleene star <math>L^*</math> of LTemplate:Sfn
• the image <math>\varphi(L)</math> of L under a homomorphism <math>\varphi</math>Template:Sfn
• the image <math>\varphi^{-1}(L)</math> of L under an inverse homomorphism <math>\varphi^{-1}</math>Template:Sfn
• the circular shift of L (the language <math>\{vu : uv \in L \}</math>)Template:Sfn
• the prefix closure of L (the set of all prefixes of strings from L)Template:Sfn
• the quotient L/R of L by a regular language RTemplate:Sfn

Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages <math>A = \{a^n b^n c^m \mid m, n \geq 0 \}</math> and <math>B = \{a^m b^n c^n \mid m,n \geq 0\}</math>, which are both context-free.<ref group=note>A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S → Sc | aTb | ε; T → aTb | ε. The grammar for B is analogous.</ref> Their intersection is <math>A \cap B = \{ a^n b^n c^n \mid n \geq 0\}</math>, 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: <math>A \cap B = \overline{\overline{A} \cup \overline{B}} </math>. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: <math>\overline{L} = \Sigma^* \setminus L</math>.<ref name="Scheinberg.1960">Template:Cite journal</ref>

However, if L is a context-free language and D is a regular language then both their intersection <math>L\cap D</math> and their difference <math>L\setminus D</math> are context-free languages.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 <math>L(A)=L(B)</math>?Template:Sfn
• Disjointness: is <math>L(A) \cap L(B) = \emptyset </math> ?Template:Sfn However, the intersection of a context-free language and a regular language is context-free,<ref>Template:Harvtxt, p. 59, Theorem 6.7</ref>Template:Sfn hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
• Containment: is <math>L(A) \subseteq L(B)</math> ?Template:Sfn Again, the variant of the problem where B is a regular grammar is decidable,Template:Citation needed while that where A is regular is generally not.Template:Sfn
• Universality: is <math>L(A)=\Sigma^*</math>?Template:Sfn
• Regularity: is <math>L(A)</math> a regular language?Template:Sfn
• Ambiguity: is every grammar for <math>L(A)</math> ambiguous?Template:Sfn

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

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

According to Hopcroft, Motwani, Ullman (2003),<ref>Template:Cite book Here: Sect.7.6, p.304, and Sect.9.7, p.411</ref> 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<ref name="Bar-Hillel.Perles.Shamir.1961">Template:Cite journal</ref>

Languages that are not context-free

The set <math>\{a^n b^n c^n d^n | n > 0\}</math> 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<ref name="Bar-Hillel.Perles.Shamir.1961"/> or a number of other methods, such as Ogden's lemma or Parikh's theorem.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

References

Works cited

• Template:Hopcroft and Ullman 1979Template:Sfn whitelist
• Template:Cite book

Further reading

• Template:Cite book
• Template:Cite book
• Template:Sipser 1997