Context-sensitive language

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

Computational properties

Computationally, a context-sensitive language is equivalent with 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

One of the simplest context-sensitive, but not context-free languages is $L = \{ a^nb^nc^n : n \ge 1 \}$: 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.

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

• The union, intersection, concatenation of two context-sensitive languages is context-sensitive; the Kleene plus of a context-sensitive language is context-sensitive.^[6]
• The complement of a context-sensitive language is itself context-sensitive^[7] 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.

See also

• Linear bounded automaton
• Chomsky hierarchy
• Indexed languages – a strict subset of the context-sensitive languages
• Weir hierarchy

References

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 0-444-50205-X, 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". 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. John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.; Exercise 9.10, p.230. In the 2003 edition, the chapter on context-sensitive languages has been omitted.
7. Immerman, Neil (1988). "Nondeterministic space is closed under complementation" (PDF). SIAM J. Comput. 17 (5): 935–938. doi:10.1137/0217058.

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