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.

Computational properties

Computationally the context-sensitive languages are equivalent with linear bounded non-deterministic Turing machines. 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.

Examples

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

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 context-sensitive.
• Every context-free language is context-sensitive.

See also: Chomsky hierarchy