Context-free language

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 $L=\{a^{n}b^{n}:n\geq 1\}$ , 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^2.3728639).^[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 $L\cup P$ 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 $\varphi ^{-1}(L)$ of L under an inverse homomorphism $\varphi ^{-1}$
the cyclic shift of L (the language $\{vu:uv\in L\}$ )

Context-free languages are not closed under complement, intersection, or difference. This was proved by Scheinberg in 1960.^[6] 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.

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.

Context-free languages are also not closed under complementation, as for any languages A and B: $A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}$ .

Context-free language are also not closed under difference: L^C = Σ^* \ L.^[6]

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.^[7]

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

Equivalence: is $L(A)=L(B)$ ?^[8]
Disjointness: is $L(A)\cap L(B)=\emptyset$ ?^[9] However, the intersection of a context-free language and a regular language is context-free,^[10]^[11] 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, while that where A is regular is generally not.^[13]
Universality: is $L(A)=\Sigma ^{*}$ ?^[14]

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

Emptiness: Given a context-free grammar A, is $L(A)=\emptyset$ ?^[15]
Finiteness: Given a context-free grammar A, is $L(A)$ finite?^[16]
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),^[17] 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^[18]

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.^[19] 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^[18] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[20]

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.
↑ 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.
1 2 Stephen Scheinberg, Note on the Boolean Properties of Context Free Languages, Information and Control, 3, 372-375 (1960)
↑ 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
1 2 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?

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.

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[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.

[9] 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.

[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.

[Scheinberg.1960-8] 1 2 Stephen Scheinberg, Note on the Boolean Properties of Context Free Languages, Information and Control, 3, 372-375 (1960)

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

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(1)-11] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).

[FOOTNOTEHopcroftUllman1979202Theorem_8.10-12] Hopcroft & Ullman 1979, p. 202, Theorem 8.10.

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

[FOOTNOTEHopcroftUllman1979135Theorem_6.5-14] Hopcroft & Ullman 1979, p. 135, Theorem 6.5.

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(2)-15] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).

[FOOTNOTEHopcroftUllman1979203Theorem_8.12(4)-16] Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).

[FOOTNOTEHopcroftUllman1979203Theorem_8.11-17] Hopcroft & Ullman 1979, p. 203, Theorem 8.11.

[FOOTNOTEHopcroftUllman1979137Theorem_6.6(a)-18] Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).

[FOOTNOTEHopcroftUllman1979137Theorem_6.6(b)-19] Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).

[20] 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.1961-21] 1 2 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.

[FOOTNOTEHopcroftUllman1979-22] Hopcroft & Ullman 1979.

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