Context-free language

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.

## Contents

* 1 Background
* 1.1 Context-free grammar
* 1.2 Automata
* 2 Examples
* 2.1 Dyck language
* 3 Properties
* 3.1 Context-free parsing
* 3.2 Closure properties
* 3.2.1 Nonclosure under intersection, complement, and difference
* 3.3 Decidability
* 3.4 Languages that are not context-free
* 4 Notes
* 5 References
* 5.1 Works cited
* 6 Further reading

## 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={anbn:n≥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->aSb|ab. This language is not regular. It is accepted by the pushdown automaton M=({q0,q1,qf},{a,b},{a,z},δ,q0,z,{qf}) where δ is defined as follows:[note 1]

δ(q0,a,z)=(q0,az)δ(q0,a,a)=(q0,aa)δ(q0,b,a)=(q1,ε)δ(q1,b,a)=(q1,ε)δ(q1,ε,z)=(qf,ε)

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 {anbmcmdn|n,m>0} with {anbncmdm|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 {anbncndn|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->SS|(S)|ε.

## Properties

### Context-free parsing

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∈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(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 L∪P of L and P[5]
* the reversal of L[6]
* the concatenation L⋅P of L and P[5]
* the Kleene star L* of L[5]
* the image φ(L) of L under a homomorphism φ[7]
* the image φ−1(L) of L under an inverse homomorphism φ−1[8]
* the circular shift of L (the language {vu:uv∈L})[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 A={anbncm∣m,n≥0} and B={ambncn∣m,n≥0}, which are both context-free.[note 3] Their intersection is A∩B={anbncn∣n≥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∩B=A‾∪B‾‾. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: L‾=Σ*∖L.[12]

However, if L is a context-free language and D is a regular language then both their intersection L∩D and their difference L∖D 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 L(A)=L(B)?[14]
* Disjointness: is L(A)∩L(B)=∅ ?[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 L(A)⊆L(B) ?[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 L(A)=Σ*?[20]
* Regularity: is L(A) a regular language?[21]
* Ambiguity: is every grammar for L(A) ambiguous?[22]

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

* Emptiness: Given a context-free grammar A, is L(A)=∅ ?[23]
* Finiteness: Given a context-free grammar A, is L(A) finite?[24]
* Membership: Given a context-free grammar G, and a word w, does w∈L(G) ? 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 {anbncndn|n>0} 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 δ's arguments and results: δ(state1,read,pop)=(state2,push)
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

* Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 9780201029888. https://archive.org/details/introductiontoau00hopc.
* Salomaa, Arto (1973). Formal Languages. ACM Monograph Series.

## Further reading

* Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata". Handbook of Formal Languages. 1. Springer-Verlag. pp. 111–174. http://www-igm.univ-mlv.fr/~berstel/Articles/1997CFLPDA.pdf.
* Ginsburg, Seymour (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill.
* Sipser, Michael (1997). "2: Context-Free Languages". Introduction to the Theory of Computation. PWS Publishing. pp. 91–122. ISBN 0-534-94728-X. https://archive.org/details/introductiontoth00sips.

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Automata theory: formal languages and formal grammars

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