Context-free language

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^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:^{[note 1]}

\begin{align}
\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{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 \{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.^[1]

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).^[2]^{[note 2]} 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.^[3]

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.^[4]

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^[5]
the reversal of L^[6]
the concatenation L \cdot P of L and P^[5]
the Kleene star L^* of L^[5]
the image \varphi(L) of L under a homomorphism \varphi^[7]
the image \varphi^{-1}(L) of L under an inverse homomorphism \varphi^{-1}^[8]
the circular shift of L (the language \{vu : uv \in L \})^[9]
the prefix closure of L (the set of all prefixes of strings from L)^[6]
the quotient L/R of L by a regular language R^[6]

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.^{[note 3]} 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. 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{\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^* \setminus L.^[10]

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.^[11]

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)?^[12]
Disjointness: is L(A) \cap L(B) = \emptyset ?^[13] However, the intersection of a context-free language and a regular language is context-free,^[14]^[15] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
Containment: is L(A) \subseteq L(B) ?^[12] Again, the variant of the problem where B is a regular grammar is decidable,^{[citation needed]} while that where A is regular is generally not.^[12]
Universality: is L(A)=\Sigma^*?^[12]
Regularity: is L(A) a regular language?^[16]
Ambiguity: is every grammar for L(A) ambiguous?^[17]

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

Emptiness: Given a context-free grammar A, is L(A) = \emptyset ?^[18]
Finiteness: Given a context-free grammar A, is L(A) finite?^[18]
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),^[19] 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.^[20]

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.^[21] 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^[20] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[22]

Notes

^ meaning of \delta's arguments and results: \delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})
^ 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.
^ 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.

References

^ Hopcroft & Ullman 1979, 100.
^ Valiant 1975.
^ Lee 2002.
^ Knuth 1965.
^ Hopcroft & Ullman 1979, 131.
^ Hopcroft & Ullman 1979, 142.
^ Hopcroft & Ullman 1979, pp. 131–132.
^ Hopcroft & Ullman 1979, 132.
^ Hopcroft & Ullman 1979, pp. 142–144.
^ Scheinberg 1960.
^ Beigel & Gasarch.
^ Hopcroft & Ullman 1979, 203.
^ Hopcroft & Ullman 1979, 202.
^ Salomaa 1973, 59.
^ Hopcroft & Ullman 1979, 135.
^ Hopcroft & Ullman 1979, 205.
^ Hopcroft & Ullman 1979, 206.
^ Hopcroft & Ullman 1979, 137.
^ Hopcroft, Motwani & Ullman 2006, pp. See Section 7.6 for properties of context-free languages and Section 9.7 for related exercises.
^ Bar-Hillel, Perles & Shamir 1961.
^ Hopcroft & Ullman 1979.
^ Stack Exchange. "How to prove that a language is not context-free?"

Works cited

Bar-Hillel, Yehoshua; Perles, Micha Asher; Shamir, Eli (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.
Beigel, Richard & Gasarch, William. "A Proof that if L = L1 ∩ L2 where L1 is CFL and L2 is Regular then L is Context Free Which Does Not use PDA's". University of Maryland Department of Computer Science. Archived 2014-12-12 at the Wayback Machine. Retrieved 2020-06-06.
Knuth, D. E. (July 1965). "On the translation of languages from left to right". Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2
Lee, Lillian (January 2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication". J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242. S2CID 1243491. Archived 2003-04-27 at the Wayback Machine.
Salomaa, Arto (1973). Formal Languages. ACM Monograph Series. New York: Academic Press. ISBN 978-0126157505.
Scheinberg, Stephen (1960). "Note on the Boolean Properties of Context Free Languages". Information and Control. 3 (4): 372-375. doi:10.1016/s0019-9958(60)90965-7. Archived 2018-11-26 at the Wayback Machine.
Valiant, Leslie G. (April 1975). "General context-free recognition in less than cubic time". Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8