Context-free language

In formal language theory, a context-free language (CFL) is a language generated by some context-free grammar (CFG). Different CF grammars can generate the same CF language. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).

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. Indeed, given a CFG, there is a direct way to produce a pushdown automaton for the grammar (and corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.

Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar $S\to SS ~|~ (S) ~|~ \varepsilon$ . Also, most arithmetic expressions are generated by context-free grammars.

Examples

An archetypal 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]}

$\delta(q_0, a, z) = (q_0, az)$
$\delta(q_0, a, a) = (q_0, aa)$
$\delta(q_0, b, a) = (q_1, \varepsilon)$
$\delta(q_1, b, a) = (q_1, \varepsilon)$

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]

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.^[2] 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^[3] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[4]

Closure properties

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. 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 2]} 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

Decidability properties

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

Equivalence: Given two context-free grammars A and B, is $L(A)=L(B)$ ?
Intersection Emptiness: Given two context-free grammars A and B, is $L(A) \cap L(B) = \emptyset$ ? However, the intersection of a context-free language and a regular language is context-free,^[5] and the variant of the problem where B is a regular grammar is decidable.
Containment: Given a context-free grammar A, is $L(A) \subseteq L(B)$ ? Again, the variant of the problem where B is a regular grammar is decidable.
Universality: Given a context-free grammar A, is $L(A)=\Sigma^*$ ?

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

Emptiness: Given a context-free grammar A, is $L(A) = \emptyset$ ?
Finiteness: Given a context-free grammar A, is $L(A)$ finite?
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),^[6] 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^[3]

Parsing

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).^[7]^[8]^{[note 3]} 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.^[9]

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 the 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.^[10]

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

Notes

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

References

↑ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
↑ Hopcroft & Ullman 1979.
↑ 3.0 3.1 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.
↑ How to prove that a language is not context-free?
↑ Salomaa (1973), p. 59, Theorem 6.7
↑ 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
↑ 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). JACM 49 (1): 1–15. 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.

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.

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Abstract machines

Type-0 — Type-1 — — — — — Type-2 — — Type-3 — —	Unrestricted (no common name) Context-sensitive Positive range concatenation Indexed — Linear context-free rewriting systems Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular — Non-recursive	Recursively enumerable Decidable Context-sensitive Positive range concatenation^* Indexed^* — Linear context-free rewriting language Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular Star-free Finite	Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton — Embedded pushdown Nondeterministic pushdown Deterministic pushdown Visibly pushdown Finite Counter-free (with aperiodic finite monoid) Acyclic finite

Each category of languages, except those marked by a ^*, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.