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 $L=\{a^{n}b^{n}:n\geq 1\}$ , 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 $S\to aSb~|~ab$ . 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{aligned}\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{aligned}}

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)^[10]
the quotient L/R of L by a regular language R^[11]

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: ${\overline {L}}=\Sigma ^{*}\setminus L$ .^[12]

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.^[13]

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 $L(A)=L(B)$ ?^[14]
Disjointness: is $L(A)\cap L(B)=\emptyset$ ?^[15] However, the intersection of a context-free language and a regular language is context-free,^[16]^[17] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
Containment: is $L(A)\subseteq L(B)$ ?^[18] 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.^[19]
Universality: is $L(A)=\Sigma ^{*}$ ?^[20]
Regularity: is $L(A)$ a regular language?^[21]
Ambiguity: is every grammar for $L(A)$ ambiguous?^[22]

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

Emptiness: Given a context-free grammar A, is $L(A)=\emptyset$ ?^[23]
Finiteness: Given a context-free grammar A, is $L(A)$ finite?^[24]
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),^[25] 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.^[26]

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.^[27] 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^[26] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[28]

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, p. 100, Theorem 4.7.
^ Valiant 1975.
^ Lee 2002.
^ Knuth 1965.
^ ^a ^b ^c Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
^ Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
^ Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
^ Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
^ Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
^ Scheinberg 1960.
^ Beigel & Gasarch.
^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
^ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
^ Salomaa 1973, p. 59, Theorem 6.7.
^ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
^ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
^ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
^ Hopcroft & Ullman 1979, p. 205, Theorem 8.15.
^ Hopcroft & Ullman 1979, p. 206, Theorem 8.16.
^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
^ Hopcroft, Motwani & Ullman 2006, See Section 7.6 for properties of context-free languages and Section 9.7 for related exercises.
^ ^a ^b 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" (PDF). University of Maryland Department of Computer Science. Archived (PDF) from the original on 12 December 2014. Retrieved 6 June 2020.
Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 0-201-02988-X. (accessible to patrons with print disabilities)
Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2006) [1979]. Introduction to Automata Theory, Languages, and Computation (3rd ed.). Addison-Wesley. ISBN 0-321-45536-3.
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" (PDF). J ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241.505242. S2CID 1243491. Archived (PDF) from the original on 27 April 2003.
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" (PDF). Information and Control. 3 (4): 372–375. doi:10.1016/s0019-9958(60)90965-7. Archived (PDF) from the original on 26 November 2018.
Valiant, Leslie G. (April 1975). "General context-free recognition in less than cubic time" (PDF). Journal of Computer and System Sciences. 10 (2): 308–315. doi:10.1016/s0022-0000(75)80046-8.