Context-free language explained

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 's, and the entire second halves of which are 's. 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:^[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.

Dyck language

The language of all properly matched parentheses is generated by the grammar

S\to SS ~|~ (S) ~|~ \varepsilon .

Properties

Context-free parsing

See main article: 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] [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.^[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 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:

L \cup P of L and P

• the reversal of L

L ⋅ P of L and PL ^{*} 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 circular shift of L (the language

\{vu : uv \in L \} )

• the prefix closure of L (the set of all prefixes of strings from L)
• the quotient L/R of L by a regular language R

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.^[6] 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 .^[7]

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.^[8]

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) ?

• Disjointness: is

L(A) \cap L(B) = \emptyset ? However, the intersection of a context-free language and a regular language is context-free,^[9] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).

• Containment: is

L(A) \subseteq L(B) ? Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.

• Universality: is

L(A) = \Sigma ^{*} ?

• Regularity: is

L(A) a regular language?

• Ambiguity: is every grammar for

L(A) ambiguous?

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),^[10] 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^[11]

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. 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^[11] or a number of other methods, such as Ogden's lemma or Parikh's theorem.^[12]

References

Works cited

• Book: Hopcroft. John E. . Ullman . Jeffrey D. . Introduction to Automata Theory, Languages, and Computation . registration . Addison-Wesley . 1st . 1979. 9780201029888 .
• Book: Salomaa, Arto . Formal Languages . ACM Monograph Series . 1973.

Further reading

• Book: Jean-Michel . Autebert . Jean . Berstel . Luc . Boasson . Handbook of Formal Languages . https://web.archive.org/web/20110516030515/http://www-igm.univ-mlv.fr/%7Eberstel/Articles/1997CFLPDA.pdf . 2011-05-16 . live . Context-Free Languages and Push-Down Automata . G. Rozenberg . A. Salomaa . 1 . Springer-Verlag . 1997 . 111–174.
• Book: Ginsburg, Seymour . Seymour Ginsburg . The Mathematical Theory of Context-Free Languages . 1966 . McGraw-Hill . New York, NY, USA.
• Book: Sipser, Michael . Michael Sipser . 1997 . Introduction to the Theory of Computation . PWS Publishing . 0-534-94728-X . registration . 91–122 . 2: Context-Free Languages.

Notes and References

1. meaning of \delta 's arguments and results: \delta( state _{1,} read , pop ) = ( state _{2,} push )
2. Leslie G. . Valiant . General context-free recognition in less than cubic time . Journal of Computer and System Sciences . April 1975 . 10 . 2 . 308–315 . 10.1016/s0022-0000(75)80046-8 . free .
3. 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.
4. Lillian . Lee . Lillian Lee (computer scientist) . Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication . J ACM . January 2002 . 49 . 1 . 1–15 . https://web.archive.org/web/20030427152836/http://www.cs.cornell.edu/home/llee/papers/bmmcfl-jacm.pdf . 2003-04-27 . live . 10.1145/505241.505242 . cs/0112018. 1243491 .
5. Knuth . D. E. . Donald Knuth . On the translation of languages from left to right . 10.1016/S0019-9958(65)90426-2 . Information and Control . 8 . 6 . 607–639 . July 1965 .
6. 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.
7. Note on the Boolean Properties of Context Free Languages . https://web.archive.org/web/20181126005901/https://core.ac.uk/download/pdf/82210847.pdf . 2018-11-26 . live . Stephen Scheinberg . Information and Control . 3 . 372 - 375 . 1960 . 4 . 10.1016/s0019-9958(60)90965-7. free .
8. Web site: 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. https://web.archive.org/web/20141212060332/http://www.cs.umd.edu/~gasarch/BLOGPAPERS/cfg.pdf . 2014-12-12 . live. June 6, 2020. University of Maryland Department of Computer Science.
9. , p. 59, Theorem 6.7
10. Book: John E. Hopcroft . Rajeev Motwani . Jeffrey D. Ullman . Introduction to Automata Theory, Languages, and Computation. 2003. Addison Wesley. Here: Sect.7.6, p.304, and Sect.9.7, p.411
11. Yehoshua Bar-Hillel . Micha Asher Perles . Eli Shamir . On Formal Properties of Simple Phrase-Structure Grammars. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung. 1961. 14. 2. 143–172.
12. Web site: How to prove that a language is not context-free?.