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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

A model context-free language is $L=a^nb^n:ngeq 1$ , 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 $Sto aSb~|~ab$ .
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]}

displaystyle beginaligneddelta (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 )endaligned

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 $n,m>0$ with $n,m>0$ . 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 $a^nb^nc^nd^n$ which is the intersection of these two languages.^[1]

Dyck language

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

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 $win L(G)$ 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).^[2]^[3]^{[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.^[4]

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.^[5]

See also parsing expression grammar as an alternative approach to grammar and parser.

Closure

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 $Lcup P$ of L and P^[6]

the reversal of L^[7]

the concatenation $Lcdot P$ of L and P^[6]

the Kleene star $L^*$ of L^[6]

the image $varphi (L)$ of L under a homomorphism $varphi$ ^[8]

the image $varphi ^-1(L)$ of L under an inverse homomorphism $varphi ^-1$ ^[9]

the circular shift of L (the language $vu:uvin L$ )^[10]

the prefix closure of L (the set of all prefixes of strings from L)^[11]

the quotient L/R of L by a regular language R^[12]

Nonclosure under intersection, complement, and difference

The context-free languages are not closed under intersection. This can be seen by taking the languages $A=a^nb^nc^mmid m,ngeq 0$ and $B=a^mb^nc^nmid m,ngeq 0$ , which are both context-free.^{[note 3]} Their intersection is $Acap B=a^nb^nc^nmid ngeq 0$ , 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: $Acap B=overline overline Acup overline B$ . In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: $displaystyle overline L=Sigma ^*setminus L$ .^[13]

However, if L is a context-free language and D is a regular language then both their intersection $Lcap D$ and their difference $Lsetminus D$ are context-free languages.^{[citation needed]}

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.^[14]

The following problems are undecidable for arbitrarily given context-free grammars A and B:

Equivalence: is $L(A)=L(B)$ ?^[15]

Disjointness: is $L(A)cap L(B)=emptyset$ ?^[16] However, the intersection of a context-free language and a regular language is context-free,^[17]^[18] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).

Containment: is $L(A)subseteq L(B)$ ?^[19] 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.^[20]

Universality: is $L(A)=Sigma ^*$ ?^[21]

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

Emptiness: Given a context-free grammar A, is $L(A)=emptyset$ ?^[22]

Finiteness: Given a context-free grammar A, is $L(A)$ finite?^[23]

Membership: Given a context-free grammar G, and a word $w$ , does $win L(G)$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),^[24]
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^[25]

Languages that are not context-free

The set $a^nb^nc^nd^n$ is a context-sensitive language, but there does not exist a context-free grammar generating this language.^[26] 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^[25] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[27]

Notes

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

^ In Valiant's papers, O(n^2.81) given, the then best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm 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.

^ Leslie Valiant (Jan 1974). General context-free recognition in less than cubic time (Technical report). Carnegie Mellon University. p. 11..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

^ Leslie G. Valiant (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.

^ Lillian Lee (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.

^ Knuth, D. E. (July 1965). "On the translation of languages from left to right" (PDF). Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Retrieved 29 May 2011.

^ ^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.

^ Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3: 372&ndash, 375.

^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1138. ISBN 1-57955-008-8.

^ 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. 137, Theorem 6.6(a).

^ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).

^ John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411

^ ^a^b Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.

^ Hopcroft & Ullman 1979.

^ How to prove that a language is not context-free?

.mw-parser-output .refbeginfont-size:90%;margin-bottom:0.5em.mw-parser-output .refbegin-hanging-indents>ullist-style-type:none;margin-left:0.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>ddmargin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none.mw-parser-output .refbegin-100font-size:100%

Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.

Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley.

Arto Salomaa (1973). Formal Languages. ACM Monograph Series.

Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. 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.

[1] $delta$ 's arguments and results: $delta (mathrm state _1,mathrm read ,mathrm pop )=(mathrm state _2,mathrm push )$

[5] In Valiant's papers, O(n^2.81) given, the then best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm for bound improvements since then.

[15] 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.

[FOOTNOTEHopcroftUllman1979100Theorem_4.7-2] Hopcroft & Ullman 1979, p. 100, Theorem 4.7.

[3] Leslie Valiant (Jan 1974). General context-free recognition in less than cubic time (Technical report). Carnegie Mellon University. p. 11..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

[4] Leslie G. Valiant (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.

[6] Lillian Lee (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.

[7] Knuth, D. E. (July 1965). "On the translation of languages from left to right" (PDF). Information and Control. 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Retrieved 29 May 2011.

[FOOTNOTEHopcroftUllman1979131Corollary_of_Theorem_6.1-8] Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.

[FOOTNOTEHopcroftUllman1979142Exercise_6.4d-9] Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.

[FOOTNOTEHopcroftUllman1979131-132Corollary_of_Theorem_6.2-10] Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.

[FOOTNOTEHopcroftUllman1979132Theorem_6.3-11] Hopcroft & Ullman 1979, p. 132, Theorem 6.3.

[FOOTNOTEHopcroftUllman1979142-144Exercise_6.4c-12] Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.

[FOOTNOTEHopcroftUllman1979142Exercise_6.4b-13] Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.

[FOOTNOTEHopcroftUllman1979142Exercise_6.4a-14] Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.

[Scheinberg.1960-16] Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages" (PDF). Information and Control. 3: 372&ndash, 375.

[17] Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1138. ISBN 1-57955-008-8.

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(1)-18] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).

[FOOTNOTEHopcroftUllman1979202Theorem_8.10-19] Hopcroft & Ullman 1979, p. 202, Theorem 8.10.

[20] Salomaa (1973), p. 59, Theorem 6.7

[FOOTNOTEHopcroftUllman1979135Theorem_6.5-21] Hopcroft & Ullman 1979, p. 135, Theorem 6.5.

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(2)-22] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(4)-23] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).

[FOOTNOTEHopcroftUllman1979203Theorem_8.11-24] Hopcroft & Ullman 1979, p. 203, Theorem 8.11.

[FOOTNOTEHopcroftUllman1979137Theorem_6.6(a)-25] Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).

[FOOTNOTEHopcroftUllman1979137Theorem_6.6(b)-26] Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).

[27] John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411

[Bar-Hillel.Perles.Shamir.1961-28] Yehoshua Bar-Hillel; Micha Asher Perles; Eli Shamir (1961). "On Formal Properties of Simple Phrase-Structure Grammars". Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 14 (2): 143–172.

[FOOTNOTEHopcroftUllman1979-29] Hopcroft & Ullman 1979.

[30] How to prove that a language is not context-free?

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