Context-free language

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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 n b n : n ≥ 1 } {\displaystyle L=\{a^{n}b^{n}:n\geq 1\}} {\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 S → a S b   |   a b {\displaystyle S\to aSb~|~ab} {\displaystyle 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 } , δ , q 0 , z , { q f } ) {\textstyle M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},z,\{q_{f}\})} {\textstyle M=(\{q_{0},q_{1},q_{f}\},\{a,b\},\{a,z\},\delta ,q_{0},z,\{q_{f}\})} where δ {\displaystyle \delta } {\displaystyle \delta } is defined as follows:[note 1]

δ ( q 0 , a , z ) = ( q 0 , a z ) δ ( q 0 , a , a ) = ( q 0 , a a ) δ ( q 0 , b , a ) = ( q 1 , ε ) δ ( q 1 , b , a ) = ( q 1 , ε ) δ ( q 1 , ε , z ) = ( q f , ε ) {\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}}} {\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 { a n b m c m d n | n , m > 0 } {\displaystyle \{a^{n}b^{m}c^{m}d^{n}|n,m>0\}} {\displaystyle \{a^{n}b^{m}c^{m}d^{n}|n,m>0\}} with { a n b n c m d m | n , m > 0 } {\displaystyle \{a^{n}b^{n}c^{m}d^{m}|n,m>0\}} {\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 { a n b n c n d n | n > 0 } {\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\}} {\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 S → S S   |   ( S )   |   ε {\displaystyle S\to SS~|~(S)~|~\varepsilon } {\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 w {\displaystyle w} {\displaystyle w}, determine whether w ∈ L ( G ) {\displaystyle w\in L(G)} {\displaystyle w\in L(G)} where L {\displaystyle L} {\displaystyle L} is the language generated by a given grammar G {\displaystyle G} {\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(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 {\displaystyle L\cup P} {\displaystyle L\cup P} of L and P[5]
  • the reversal of L[6]
  • the concatenation L ⋅ P {\displaystyle L\cdot P} {\displaystyle L\cdot P} of L and P[5]
  • the Kleene star L ∗ {\displaystyle L^{*}} {\displaystyle L^{*}} of L[5]
  • the image φ ( L ) {\displaystyle \varphi (L)} {\displaystyle \varphi (L)} of L under a homomorphism φ {\displaystyle \varphi } {\displaystyle \varphi }[7]
  • the image φ − 1 ( L ) {\displaystyle \varphi ^{-1}(L)} {\displaystyle \varphi ^{-1}(L)} of L under an inverse homomorphism φ − 1 {\displaystyle \varphi ^{-1}} {\displaystyle \varphi ^{-1}}[8]
  • the circular shift of L (the language { v u : u v ∈ L } {\displaystyle \{vu:uv\in L\}} {\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 A = { a n b n c m ∣ m , n ≥ 0 } {\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}} {\displaystyle A=\{a^{n}b^{n}c^{m}\mid m,n\geq 0\}} and B = { a m b n c n ∣ m , n ≥ 0 } {\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}} {\displaystyle B=\{a^{m}b^{n}c^{n}\mid m,n\geq 0\}}, which are both context-free.[note 3] Their intersection is A ∩ B = { a n b n c n ∣ n ≥ 0 } {\displaystyle A\cap B=\{a^{n}b^{n}c^{n}\mid n\geq 0\}} {\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: A ∩ B = A ¯ ∪ B ¯ ¯ {\displaystyle A\cap B={\overline {{\overline {A}}\cup {\overline {B}}}}} {\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: L ¯ = Σ ∗ ∖ L {\displaystyle {\overline {L}}=\Sigma ^{*}\setminus L} {\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 L ∩ D {\displaystyle L\cap D} {\displaystyle L\cap D} and their difference L ∖ D {\displaystyle L\setminus D} {\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.

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

  • Equivalence: is L ( A ) = L ( B ) {\displaystyle L(A)=L(B)} {\displaystyle L(A)=L(B)}?[14]
  • Disjointness: is L ( A ) ∩ L ( B ) = ∅ {\displaystyle L(A)\cap L(B)=\emptyset } {\displaystyle L(A)\cap L(B)=\emptyset } ?[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 ) {\displaystyle L(A)\subseteq L(B)} {\displaystyle L(A)\subseteq L(B)} ?[18] Again, the variant of the problem where B is a regular grammar is decidable, while that where A is regular is generally not.[19]
  • Universality: is L ( A ) = Σ ∗ {\displaystyle L(A)=\Sigma ^{*}} {\displaystyle L(A)=\Sigma ^{*}}?[20]
  • Regularity: is L ( A ) {\displaystyle L(A)} {\displaystyle L(A)} a regular language?[21]
  • Ambiguity: is every grammar for L ( A ) {\displaystyle L(A)} {\displaystyle L(A)} ambiguous?[22]

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

  • Emptiness: Given a context-free grammar A, is L ( A ) = ∅ {\displaystyle L(A)=\emptyset } {\displaystyle L(A)=\emptyset } ?[23]
  • Finiteness: Given a context-free grammar A, is L ( A ) {\displaystyle L(A)} {\displaystyle L(A)} finite?[24]
  • Membership: Given a context-free grammar G, and a word w {\displaystyle w} {\displaystyle w}, does w ∈ L ( G ) {\displaystyle w\in L(G)} {\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 (2006),[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 { a n b n c n d n | n > 0 } {\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n>0\}} {\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.[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 {\displaystyle \delta } {\displaystyle \delta }'s arguments and results: δ ( s t a t e 1 , r e a d , p o p ) = ( s t a t e 2 , p u s h ) {\displaystyle \delta (\mathrm {state} _{1},\mathrm {read} ,\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )} {\displaystyle \delta (\mathrm {state} _{1},\mathrm {read} ,\mathrm {pop} )=(\mathrm {state} _{2},\mathrm {push} )}
  2. In Valiant's paper, O(n2.81) was the then-best known upper bound. See Matrix multiplication#Computational complexity 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 1975.
  3. Lee 2002.
  4. Knuth 1965.
  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. Scheinberg 1960.
  13. Beigel & Gasarch.
  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. Bar-Hillel, Perles & Shamir 1961.
  26. Hopcroft & Ullman 1979.
  27. Stack Exchange. "How to prove that a language is not context-free?".

Works cited

Further reading