Context-sensitive language

In formal language theory, a context-sensitive language is a language that can be defined by a context-sensitive grammar (and equivalently by a noncontracting grammar). Context-sensitive is one of the four types of grammars in the Chomsky hierarchy.

Computational properties[edit]

Computationally, a context-sensitive language is equivalent to a linear bounded nondeterministic Turing machine, also called a linear bounded automaton. That is a non-deterministic Turing machine with a tape of only $kn$ cells, where $n$ is the size of the input and $k$ is a constant associated with the machine. This means that every formal language that can be decided by such a machine is a context-sensitive language, and every context-sensitive language can be decided by such a machine.

This set of languages is also known as NLINSPACE or NSPACE(O(n)), because they can be accepted using linear space on a non-deterministic Turing machine.^[1] The class LINSPACE (or DSPACE(O(n))) is defined the same, except using a deterministic Turing machine. Clearly LINSPACE is a subset of NLINSPACE, but it is not known whether LINSPACE=NLINSPACE.^[2]

Examples[edit]

One of the simplest context-sensitive but not context-free languages is $L=\{a^{n}b^{n}c^{n}:n\geq 1\}$ : the language of all strings consisting of $n$ occurrences of the symbol "a", then $n$ "b"'s, then $n$ "c"'s (abc, aabbcc, aaabbbccc, etc.). A superset of this language, called the Bach language,^[3] is defined as the set of all strings where "a", "b" and "c" (or any other set of three symbols) occurs equally often (aabccb, baabcaccb, etc.) and is also context-sensitive.^[4]^[5]

$L$ can be shown to be a context-sensitive language by constructing a linear bounded automaton which accepts $L$ . The language can easily be shown to be neither regular nor context free by applying the respective pumping lemmas for each of the language classes to $L$ .

Similarly:

$L_{Cross}=\{a^{m}b^{n}c^{m}d^{n}:m\geq 1,n\geq 1\}$ is another context-sensitive language; the corresponding context-sensitive grammar can be easily projected starting with two context-free grammars generating sentential forms in the formats $a^{m}C^{m}$ and $B^{n}d^{n}$ and then supplementing them with a permutation production like $CB\rightarrow BC$ , a new starting symbol and standard syntactic sugar.

$L_{MUL3}=\{a^{m}b^{n}c^{mn}:m\geq 1,n\geq 1\}$ is another context-sensitive language (the "3" in the name of this language is intended to mean a ternary alphabet); that is, the "product" operation defines a context-sensitive language (but the "sum" defines only a context-free language as the grammar $S\rightarrow aSc|R$ and $R\rightarrow bRc|bc$ shows). Because of the commutative property of the product, the most intuitive grammar for $L_{MUL3}$ is ambiguous. This problem can be avoided considering a somehow more restrictive definition of the language, e.g. $L_{ORDMUL3}=\{a^{m}b^{n}c^{mn}:1<m<n\}$ . This can be specialized to $L_{MUL1}=\{a^{mn}:m>1,n>1\}$ and, from this, to $L_{m^{2}}=\{a^{m^{2}}:m>1\}$ , $L_{m^{3}}=\{a^{m^{3}}:m>1\}$ , etc.

$L_{REP}=\{w^{|w|}:w\in \Sigma ^{*}\}$ is a context-sensitive language. The corresponding context-sensitive grammar can be obtained as a generalization of the context-sensitive grammars for $L_{Square}=\{w^{2}:w\in \Sigma ^{*}\}$ , $L_{Cube}=\{w^{3}:w\in \Sigma ^{*}\}$ , etc.

$L_{EXP}=\{a^{2^{n}}:n\geq 1\}$ is a context-sensitive language.^[6]

$L_{PRIMES2}=\{w:|w|{\mbox{ is prime }}\}$ is a context-sensitive language (the "2" in the name of this language is intended to mean a binary alphabet). This was proved by Hartmanis using pumping lemmas for regular and context-free languages over a binary alphabet and, after that, sketching a linear bounded multitape automaton accepting $L_{PRIMES2}$ .^[7]

$L_{PRIMES1}=\{a^{p}:p{\mbox{ is prime }}\}$ is a context-sensitive language (the "1" in the name of this language is intended to mean an unary alphabet). This was credited by A. Salomaa to Matti Soittola by means of a linear bounded automaton over an unary alphabet^[8] (pages 213-214, exercise 6.8) and also to Marti Penttonen by means of a context-sensitive grammar also over an unary alphabet (See: Formal Languages by A. Salomaa, page 14, Example 2.5).

An example of recursive language that is not context-sensitive is any recursive language whose decision is an EXPSPACE-hard problem, say, the set of pairs of equivalent regular expressions with exponentiation.

Properties of context-sensitive languages[edit]

The union, intersection, concatenation of two context-sensitive languages is context-sensitive, also the Kleene plus of a context-sensitive language is context-sensitive.^[9]
The complement of a context-sensitive language is itself context-sensitive^[10] a result known as the Immerman–Szelepcsényi theorem.
Membership of a string in a language defined by an arbitrary context-sensitive grammar, or by an arbitrary deterministic context-sensitive grammar, is a PSPACE-complete problem.

References[edit]

^ Rothe, Jörg (2005), Complexity theory and cryptology, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, p. 77, ISBN 978-3-540-22147-0, MR 2164257.
^ Odifreddi, P. G. (1999), Classical recursion theory. Vol. II, Studies in Logic and the Foundations of Mathematics, 143, Amsterdam: North-Holland Publishing Co., p. 236, ISBN 978-0-444-50205-6, MR 1718169.
^ Pullum, Geoffrey K. (1983). Context-freeness and the computer processing of human languages. Proc. 21st Annual Meeting of the ACL.
^ Bach, E. (1981). "Discontinuous constituents in generalized categorial grammars" Archived 2014-01-21 at the Wayback Machine. NELS, vol. 11, pp. 1–12.
^ Joshi, A.; Vijay-Shanker, K.; and Weir, D. (1991). "The convergence of mildly context-sensitive grammar formalisms". In: Sells, P., Shieber, S.M. and Wasow, T. (Editors). Foundational Issues in Natural Language Processing. Cambridge MA: Bradford.
^ Example 9.5 (p. 224) of Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley
^ J. Hartmanis and H. Shank (Jul 1968). "On the Recognition of Primes by Automata" (PDF). Journal of the ACM. 15 (3): 382–389. doi:10.1145/321466.321470.
^ Salomaa, Arto (1969), Theory of Automata, ISBN 978-0-08-013376-8, Pergamon, 276 pages. doi:10.1016/C2013-0-02221-9
^ John E. Hopcroft; Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.; Exercise 9.10, p.230. In the 2000 edition, the chapter on context-sensitive languages has been omitted.
^ Immerman, Neil (1988). "Nondeterministic space is closed under complementation" (PDF). SIAM J. Comput. 17 (5): 935–938. CiteSeerX 10.1.1.54.5941. doi:10.1137/0217058.

Sipser, M. (1996), Introduction to the Theory of Computation, PWS Publishing Co.

[1] Rothe, Jörg (2005), Complexity theory and cryptology, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, p. 77, ISBN 978-3-540-22147-0, MR 2164257.

[2] Odifreddi, P. G. (1999), Classical recursion theory. Vol. II, Studies in Logic and the Foundations of Mathematics, 143, Amsterdam: North-Holland Publishing Co., p. 236, ISBN 978-0-444-50205-6, MR 1718169.

[3] Pullum, Geoffrey K. (1983). Context-freeness and the computer processing of human languages. Proc. 21st Annual Meeting of the ACL.

[4] Bach, E. (1981). "Discontinuous constituents in generalized categorial grammars" Archived 2014-01-21 at the Wayback Machine. NELS, vol. 11, pp. 1–12.

[5] Joshi, A.; Vijay-Shanker, K.; and Weir, D. (1991). "The convergence of mildly context-sensitive grammar formalisms". In: Sells, P., Shieber, S.M. and Wasow, T. (Editors). Foundational Issues in Natural Language Processing. Cambridge MA: Bradford.

[6] Example 9.5 (p. 224) of Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley

[7] J. Hartmanis and H. Shank (Jul 1968). "On the Recognition of Primes by Automata" (PDF). Journal of the ACM. 15 (3): 382–389. doi:10.1145/321466.321470.

[8] Salomaa, Arto (1969), Theory of Automata, ISBN 978-0-08-013376-8, Pergamon, 276 pages. doi:10.1016/C2013-0-02221-9

[9] John E. Hopcroft; Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.; Exercise 9.10, p.230. In the 2000 edition, the chapter on context-sensitive languages has been omitted.

[10] Immerman, Neil (1988). "Nondeterministic space is closed under complementation" (PDF). SIAM J. Comput. 17 (5): 935–938. CiteSeerX 10.1.1.54.5941. doi:10.1137/0217058.

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