Context-free languages

In formal language theory, a context-free language (CFL) 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 = \$, the language of all non-empty even-length strings, the entire first halves of which are 's, and the entire second halves of which are 's. is generated by the grammar $S\to aSb ~, ~ ab$.
This language is not regular.
It is accepted by the pushdown automaton $M=(\, \, \, \delta, q_0, z, \)$ where $\delta$ is defined as follows:meaning of $\delta$'s arguments and results: $\delta(\mathrm_1, \mathrm, \mathrm) = (\mathrm_2, \mathrm)$

:$\begin \delta(q_0, a, z) &= (q_0, az) \\ \delta(q_0, a, a) &= (q_0, aa) \\ \delta(q_0, b, a) &= (q_1, \varepsilon) \\ \delta(q_1, b, a) &= (q_1, \varepsilon) \\ \delta(q_1, \varepsilon, z) &= (q_f, \varepsilon) \end$

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 $\$ with $\$. 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 $\$ which is the intersection of these two languages.

Dyck language

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

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).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.
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 $L \cup P$ 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 $\varphi^(L)$ of ''L'' under an inverse homomorphism $\varphi^$
*the circular shift of ''L'' (the language $\$)
*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 $A = \$ and $B = \$, which are both context-free.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. Their intersection is $A \cap B = \$, 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: $A \cap B = \overline$. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: $\overline = \Sigma^* \setminus L$.

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.

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 $L(A)=L(B)$?
*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 $L(A)=\Sigma^*$?
*Regularity: is $L(A)$ a regular language?
*Ambiguity: is every grammar for $L(A)$ ambiguous?

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

Languages that are not context-free

The set $\$ 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.

Notes

References

Works cited

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Further reading

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{{Formal languages and grammars

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