Context-free language

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Short description: Formal language generated by context-free grammar

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

An example context-free language is [math]\displaystyle{ L = \{a^nb^n:n\geq1\} }[/math], 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 [math]\displaystyle{ S\to aSb ~|~ ab }[/math]. This language is not regular. It is accepted by the pushdown automaton [math]\displaystyle{ M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\}) }[/math] where [math]\displaystyle{ \delta }[/math] is defined as follows:[note 1]

[math]\displaystyle{ \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} }[/math]

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 [math]\displaystyle{ \{a^n b^m c^m d^n | n, m \gt 0\} }[/math] with [math]\displaystyle{ \{a^n b^n c^m d^m | n, m \gt 0\} }[/math]. 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 [math]\displaystyle{ \{a^n b^n c^n d^n | n \gt 0\} }[/math] which is the intersection of these two languages.[1]

Dyck language

The language of all properly matched parentheses is generated by the grammar [math]\displaystyle{ S\to SS ~|~ (S) ~|~ \varepsilon }[/math].

Properties

Context-free parsing

Main page: 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 [math]\displaystyle{ w }[/math], determine whether [math]\displaystyle{ w \in L(G) }[/math] where [math]\displaystyle{ L }[/math] is the language generated by a given grammar [math]\displaystyle{ G }[/math]; 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(n2.3728596).[2][note 2] Conversely, Lillian Lee has shown O(n3−ε) boolean matrix multiplication to be reducible to O(n3−3ε) 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 [math]\displaystyle{ L \cup P }[/math] of L and P[5]
  • the reversal of L[6]
  • the concatenation [math]\displaystyle{ L \cdot P }[/math] of L and P[5]
  • the Kleene star [math]\displaystyle{ L^* }[/math] of L[5]
  • the image [math]\displaystyle{ \varphi(L) }[/math] of L under a homomorphism [math]\displaystyle{ \varphi }[/math][7]
  • the image [math]\displaystyle{ \varphi^{-1}(L) }[/math] of L under an inverse homomorphism [math]\displaystyle{ \varphi^{-1} }[/math][8]
  • the circular shift of L (the language [math]\displaystyle{ \{vu : uv \in L \} }[/math])[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 [math]\displaystyle{ A = \{a^n b^n c^m \mid m, n \geq 0 \} }[/math] and [math]\displaystyle{ B = \{a^m b^n c^n \mid m,n \geq 0\} }[/math], which are both context-free.[note 3] Their intersection is [math]\displaystyle{ A \cap B = \{ a^n b^n c^n \mid n \geq 0\} }[/math], 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: [math]\displaystyle{ A \cap B = \overline{\overline{A} \cup \overline{B}} }[/math]. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: [math]\displaystyle{ \overline{L} = \Sigma^* \setminus L }[/math].[12]

However, if L is a context-free language and D is a regular language then both their intersection [math]\displaystyle{ L\cap D }[/math] and their difference [math]\displaystyle{ L\setminus D }[/math] 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 [math]\displaystyle{ L(A)=L(B) }[/math]?[14]
  • Disjointness: is [math]\displaystyle{ L(A) \cap L(B) = \emptyset }[/math] ?[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 [math]\displaystyle{ L(A) \subseteq L(B) }[/math] ?[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 [math]\displaystyle{ L(A)=\Sigma^* }[/math]?[20]
  • Regularity: is [math]\displaystyle{ L(A) }[/math] a regular language?[21]
  • Ambiguity: is every grammar for [math]\displaystyle{ L(A) }[/math] ambiguous?[22]

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

  • Emptiness: Given a context-free grammar A, is [math]\displaystyle{ L(A) = \emptyset }[/math] ?[23]
  • Finiteness: Given a context-free grammar A, is [math]\displaystyle{ L(A) }[/math] finite?[24]
  • Membership: Given a context-free grammar G, and a word [math]\displaystyle{ w }[/math], does [math]\displaystyle{ w \in L(G) }[/math] ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),[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 [math]\displaystyle{ \{a^n b^n c^n d^n | n \gt 0\} }[/math] 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

  1. ↑ meaning of [math]\displaystyle{ \delta }[/math]'s arguments and results: [math]\displaystyle{ \delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push}) }[/math]
  2. ↑ In Valiant's paper, O(n2.81) was the then-best known upper bound. See Matrix multiplication for bound improvements since then.
  3. ↑ 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

  1. ↑ Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
  2. ↑ Valiant, Leslie G. (April 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. https://figshare.com/articles/journal_contribution/General_context-free_recognition_in_less_than_cubic_time/6605915/1/files/12096398.pdf. 
  3. ↑ Lee, Lillian (January 2002). "Fast Context-Free Grammar Parsing Requires Fast Boolean Matrix Multiplication". J ACM 49 (1): 1–15. doi:10.1145/505241.505242. http://www.cs.cornell.edu/home/llee/papers/bmmcfl-jacm.pdf. 
  4. ↑ 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. 
  5. ↑ 5.0 5.1 5.2 Hopcroft & Ullman 1979, p. 131, Corollary of Theorem 6.1.
  6. ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4d.
  7. ↑ Hopcroft & Ullman 1979, p. 131-132, Corollary of Theorem 6.2.
  8. ↑ Hopcroft & Ullman 1979, p. 132, Theorem 6.3.
  9. ↑ Hopcroft & Ullman 1979, p. 142-144, Exercise 6.4c.
  10. ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4b.
  11. ↑ Hopcroft & Ullman 1979, p. 142, Exercise 6.4a.
  12. ↑ Stephen Scheinberg (1960). "Note on the Boolean Properties of Context Free Languages". Information and Control 3 (4): 372–375. doi:10.1016/s0019-9958(60)90965-7. https://core.ac.uk/download/pdf/82210847.pdf. 
  13. ↑ 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". http://www.cs.umd.edu/~gasarch/BLOGPAPERS/cfg.pdf. 
  14. ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
  15. ↑ Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
  16. ↑ (Salomaa 1973), p. 59, Theorem 6.7
  17. ↑ Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
  18. ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
  19. ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
  20. ↑ Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
  21. ↑ Hopcroft & Ullman 1979, p. 205, Theorem 8.15.
  22. ↑ Hopcroft & Ullman 1979, p. 206, Theorem 8.16.
  23. ↑ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
  24. ↑ Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
  25. ↑ 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
  26. ↑ 26.0 26.1 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. 
  27. ↑ Hopcroft & Ullman 1979.
  28. ↑ "How to prove that a language is not context-free?". https://cs.stackexchange.com/q/265. 

Works cited

Further reading



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