Context-free language explained

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=\{anbn:n\geq1\}

, 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\toaSb~|~ab

. This language is not regular. It is accepted by the pushdown automaton M=(\, \, \, \delta, q_0, z, \) where

\delta

is defined as follows:[1]

\begin{align} \delta(q0,a,z)&=(q0,az)\\ \delta(q0,a,a)&=(q0,aa)\\ \delta(q0,b,a)&=(q1,\varepsilon)\\ \delta(q1,b,a)&=(q1,\varepsilon)\\ \delta(q1,\varepsilon,z)&=(qf,\varepsilon) \end{align}

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

\{anbmcmdn|n,m>0\}

with

\{anbncmdm|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

\{anbncndn|n>0\}

which is the intersection of these two languages.

Dyck language

The language of all properly matched parentheses is generated by the grammar

S\toSS~|~(S)~|~\varepsilon

.

Properties

Context-free parsing

See main article: 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\inL(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(n2.3728596).[2] Conversely, Lillian Lee has shown O(n3βˆ’Ξ΅) Boolean matrix multiplication to be reducible to O(n3βˆ’3Ξ΅) 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:

L\cupP

of L and P

L ⋅ P

of L and P

L*

of L

\varphi(L)

of L under a homomorphism

\varphi

\varphi-1(L)

of L under an inverse homomorphism

\varphi-1

\{vu:uv\inL\}

)

Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages

A=\{anbncm\midm,n\geq0\}

and

B=\{ambncn\midm,n\geq0\}

, which are both context-free.[3] Their intersection is

A\capB=\{anbncn\midn\geq0\}

, 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\capB=\overline{\overline{A}\cup\overline{B}}

. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference:

\overline{L}=\Sigma*\setminusL

.

However, if L is a context-free language and D is a regular language then both their intersection

L\capD

and their difference

L\setminusD

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:

L(A)=L(B)

?

L(A)\capL(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).

L(A)\subseteqL(B)

? Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.

L(A)=\Sigma*

?

L(A)

a regular language?

L(A)

ambiguous?

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

L(A)=\emptyset

?

L(A)

finite?

w

, does

w\inL(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

\{anbncndn|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.[4]

References

Works cited

Further reading

Notes and References

  1. meaning of

    \delta

    's arguments and results:

    \delta(state1,read,pop)=(state2,push)

  2. In Valiant's paper, O(n2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity for bound improvements since then.
  3. A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: S β†’ Sc | aTb | Ξ΅; T β†’ aTb | Ξ΅. The grammar for B is analogous.
  4. Web site: How to prove that a language is not context-free?. Stack Exchange.

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