Context-free language

Template:Short description Template:Use dmy dates 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 $L=\{a^n{b}^n:n\ge 1\}$, 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 $S\to aSb|ab$. 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:

$\begin{array}{rl}\delta (q_0,a,z)& =(q_0,az)\\ \delta (q_0,a,a)& =(q_0,aa)\\ \delta (q_0,b,a)& =(q_1,\epsilon )\\ \delta (q_1,b,a)& =(q_1,\epsilon )\\ \delta (q_1,\epsilon ,z)& =(q_f,\epsilon )\end{array}$

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

Dyck language

The language of all properly matched parentheses is generated by the grammar $S\to SS|(S)|\epsilon$.

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).Template:Sfn 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.Template:Sfn

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.Template:Sfn

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 PTemplate:Sfn
• the reversal of LTemplate:Sfn
• the concatenation $L\cdot P$ of L and PTemplate:Sfn
• the Kleene star $L^{*}$ of LTemplate:Sfn
• the image $\phi (L)$ of L under a homomorphism $\phi$Template:Sfn
• the image ${\phi }^{-1}(L)$ of L under an inverse homomorphism ${\phi }^{-1}$Template:Sfn
• the circular shift of L (the language $\{vu:uv\in L\}$)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 $A=\{a^n{b}^n{c}^m\mid m,n\ge 0\}$ and $B=\{a^m{b}^n{c}^n\mid m,n\ge 0\}$, which are both context-free. Their intersection is $A\cap B=\{a^n{b}^n{c}^n\mid n\ge 0\}$, 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: $A\cap B=\overline{\stackrel{‾}{A}\cup \stackrel{‾}{B}}$. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: $\bar{L}={\mathrm{\Sigma }}^{*}\setminus L$.Template:Sfn

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.Template:Sfn

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

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

• Emptiness: Given a context-free grammar A, is $L(A)=\mathrm{\varnothing }$ ?Template:Sfn
• Finiteness: Given a context-free grammar A, is $L(A)$ finite?Template:Sfn
• 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 (2006),Template:Sfn 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.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 languagesTemplate:Sfn or a number of other methods, such as Ogden's lemma or Parikh's theorem.

References

Works cited

• Template:Hopcroft and Ullman 1979Template:Sfn whitelist
• Template:Hopcroft, Motwani, and Ullman 2006Template:Sfn whitelist

Further reading

• Template:Sipser 1997