Context-sensitive language

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In theoretical computer science, a context-sensitive language is a formal language that can be defined by a context-sensitive grammar. That is one of the four types of grammars in the Chomsky hierarchy. Of the four, this is the least often used, in both theory and practice.

Contents 1 Computational properties 2 Examples 3 Properties of context-sensitive languages 4 See also 5 References

[edit] Computational properties

Computationally the context-sensitive languages are equivalent with linear bounded non-deterministic Turing machines, 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 NLIN-SPACE, because they can be accepted using linear space on a non-deterministic Turing machine. The class LIN-SPACE is defined the same, except using a deterministic Turing machine. Clearly LIN-SPACE is a subset of NLIN-SPACE, but it is not known whether LIN-SPACE=NLIN-SPACE. It is widely suspected they are not equal.

[edit] Examples

An example of a context-sensitive language that is not context-free is L = { a^p : p is a prime number }. The easiest way to show this is using a linear bounded Turing machine. 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.

[edit] Properties of context-sensitive languages

• The union, intersection, and concatenation of two context-sensitive languages is context-sensitive.
• The complement of a context-sensitive language is itself context-sensitive.
• Every context-free language is context-sensitive.
• 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.

[edit] See also

• Linear bounded automaton
• Chomsky hierarchy
• Noncontracting grammars – generate exactly the context-sensitive languages
• Indexed languages – a strict subset of the context-sensitive languages

[edit] References

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

• Immerman, Neil (1988). "Nondeterministic space is closed under complementation". SIAM J. Comput. (17-5): 935-938. 

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Minimal automaton
Type-0	Unrestricted	Recursively enumerable	Turing machine
n/a	(no common name)	Recursive	Decider
Type-1	Context-sensitive	Context-sensitive	Linear-bounded
n/a	Indexed	Indexed	Nested stack
n/a	Tree-adjoining etc.	(Mildly context-sensitive)	Embedded pushdown
Type-2	Context-free	Context-free	Nondeterministic pushdown
n/a	Deterministic context-free	Deterministic context-free	Deterministic pushdown
Type-3	Regular	Regular	Finite
n/a		Star-free	Counter-Free
Each category of languages or grammars is a proper subset of the category directly above it, and any automaton in each category has an equivalent automaton in the category directly above it.