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

Examples

An archetypal context-free language is $L = \{a^nb^n:n\geq1\}$, 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:

$\delta(q_0, a, z) = (q_0, az)$
$\delta(q_0, a, a) = (q_0, aa)$
$\delta(q_0, b, a) = (q_1, \varepsilon)$
$\delta(q_1, b, a) = (q_1, \varepsilon)$

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.

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.

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:

• 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 cyclic shift of L (the language $\{vu : uv \in L \}$)

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 $L\cap D$ and their difference $L\setminus D$ are context-free languages.

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.

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

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

Decidability properties

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 $L(A)=\Sigma^*$ ?

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

Parsing

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^2.3728639). 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.

See also

• Deterministic context-free language
• Parsing

References

• Chapter 2: Context-Free Languages, pp. 91–122.
• Jean-Michel Autebert, Jean Berstel, Luc Boasson, Context-Free Languages and Push-Down Automata, in: G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag, 1997, 111-174.