Context-free language

Examples

An example context-free language is ${\displaystyle L=\{a^n{b}^n:n\ge 1\}}$, 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 $M=(\{q_0,q_1,q_f\},\{a,b\},\{a,z\},\delta ,q_0,z,\{q_f\})$ where ${\displaystyle \delta }$ is defined as follows:

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

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 ${\displaystyle \{a^n{b}^m{c}^m{d}^n|n,m>0\}}$ 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.

Dyck language

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

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

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.

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 ${\displaystyle L(A)=L(B)}$?
• Disjointness: is ${\displaystyle L(A)\cap L(B)=\mathrm{\varnothing }}$ ? 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 ${\displaystyle L(A)\subseteq L(B)}$ ? 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 ${\displaystyle L(A)={\mathrm{\Sigma }}^{\ast }}$?
• Regularity: is ${\displaystyle L(A)}$ a regular language?
• Ambiguity: is every grammar for ${\displaystyle L(A)}$ ambiguous?

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

• Emptiness: Given a context-free grammar A, is ${\displaystyle L(A)=\mathrm{\varnothing }}$ ?
• Finiteness: Given a context-free grammar A, is ${\displaystyle L(A)}$ finite?
• 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 (2006), 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 ${\displaystyle \{a^n{b}^n{c}^n{d}^n|n>0\}}$ is a context-sensitive language, but there does not exist a context-free grammar generating this language. 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.