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

An example context-free language is ${\displaystyle L=\{a^{n}b^{n}:n\geq 1\}}$, 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 ${\displaystyle S\to aSb~|~ab}$. This language is not regular. It is accepted by the pushdown automaton ${\displaystyle M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},z,\{q_{f}\})}$ where ${\displaystyle \delta }$ is defined as follows:[note 1]

${\displaystyle {\begin{aligned}\delta (q_{0},a,z)&=(q_{0},az)\\\delta (q_{0},a,a)&=(q_{0},aa)\\\delta (q_{0},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},b,a)&=(q_{1},\varepsilon )\\\delta (q_{1},\varepsilon ,z)&=(q_{f},\varepsilon )\end{aligned}}}$

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

Dyck language

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

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 ${\displaystyle w}$, determine whether ${\displaystyle w\in L(G)}$ where ${\displaystyle L}$ is the language generated by a given grammar ${\displaystyle G}$; 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][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.[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

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 ${\displaystyle L\cup P}$ of L and P[5]
• the reversal of L[6]
• the concatenation ${\displaystyle L\cdot P}$ of L and P[5]
• the Kleene star ${\displaystyle L^{*}}$ of L[5]
• the image ${\displaystyle \varphi (L)}$ of L under a homomorphism ${\displaystyle \varphi }$[7]
• the image ${\displaystyle \varphi ^{-1}(L)}$ of L under an inverse homomorphism ${\displaystyle \varphi ^{-1}}$[8]
• the circular shift of L (the language ${\displaystyle \{vu:uv\in L\}}$)[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 ${\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}}$ and ${\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}}$, which are both context-free.[note 3] Their intersection is ${\displaystyle A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}}$, 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: ${\displaystyle A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}}$. In particular, context-free language cannot be closed under difference, since complement can be expressed by difference: ${\displaystyle {\overline {L}}=\Sigma ^{*}\setminus L}$.[12]

However, if L is a context-free language and D is a regular language then both their intersection ${\displaystyle L\cap D}$ and their difference ${\displaystyle L\setminus D}$ 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.[14]

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

• Equivalence: is ${\displaystyle L(A)=L(B)}$?[15]
• Disjointness: is ${\displaystyle L(A)\cap L(B)=\emptyset }$ ?[16] However, the intersection of a context-free language and a regular language is context-free,[17][18] hence the variant of the problem where B is a regular grammar is decidable (see "Emptiness" below).
• Containment: is ${\displaystyle L(A)\subseteq L(B)}$ ?[19] Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.[20]
• Universality: is ${\displaystyle L(A)=\Sigma ^{*}}$ ?[21]

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

• Emptiness: Given a context-free grammar A, is ${\displaystyle L(A)=\emptyset }$ ?[22]
• Finiteness: Given a context-free grammar A, is ${\displaystyle L(A)}$ finite?[23]
• Membership: Given a context-free grammar G, and a word ${\displaystyle w}$, does ${\displaystyle w\in L(G)}$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

According to Hopcroft, Motwani, Ullman (2003),[24] 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[25]

Languages that are not context-free

The set ${\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\}}$ is a context-sensitive language, but there does not exist a context-free grammar generating this language.[26] 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[25] or a number of other methods, such as Ogden's lemma or Parikh's theorem.[27]

Notes

1. meaning of ${\displaystyle \delta }$'s arguments and results: ${\displaystyle \delta (\mathrm {state} _{1},\mathrm {read} ,\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )}$
2. In Valiant's paper, O(n^2.81) was the then-best known upper bound. See Matrix multiplication#Algorithms for efficient matrix multiplication and Coppersmith–Winograd algorithm 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. Archived from the original on 10 November 2014.
3. Lee, Lillian (January 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.
4. 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. Archived from the original (PDF) on 15 March 2012. Retrieved 29 May 2011.
5. 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" (PDF). Information and Control. 3: 372–375. doi:10.1016/s0019-9958(60)90965-7.
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" (PDF). University of Maryland Department of Computer Science. Retrieved June 6, 2020.
14. Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1138. ISBN 1-57955-008-8.
15. Hopcroft & Ullman 1979, p. 203, Theorem 8.12(1).
16. Hopcroft & Ullman 1979, p. 202, Theorem 8.10.
17. Salomaa (1973), p. 59, Theorem 6.7
18. Hopcroft & Ullman 1979, p. 135, Theorem 6.5.
19. Hopcroft & Ullman 1979, p. 203, Theorem 8.12(2).
20. Hopcroft & Ullman 1979, p. 203, Theorem 8.12(4).
21. Hopcroft & Ullman 1979, p. 203, Theorem 8.11.
22. Hopcroft & Ullman 1979, p. 137, Theorem 6.6(a).
23. Hopcroft & Ullman 1979, p. 137, Theorem 6.6(b).
24. 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
25. 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.
26. Hopcroft & Ullman 1979.
27. How to prove that a language is not context-free?

Works cited

• Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley.
• 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". In G. Rozenberg; A. Salomaa (eds.). Handbook of Formal Languages (PDF). 1. Springer-Verlag. pp. 111–174.
• 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.