# Context-sensitive language

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.

## 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 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[citation needed].

## Examples

An example of a context-sensitive language that is not context-free is L = { ap : p is a prime number }. 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.

Simpler example would be $L = \{ a^nb^nc^n : n \ge 1 \}$.

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 and Kleene star 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.