Computationally, a context-sensitive language is equivalent to a linear boundednondeterministic Turing machine, also called a linear bounded automaton. That is a non-deterministic Turing machine with a tape of only cells, where is the size of the input and 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, because they can be accepted using linear space on a non-deterministic Turing machine. The class LINSPACE 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.
Examples
One of the simplest context-sensitive but not context-free languages is : the language of all strings consisting of occurrences of the symbol "a", then "b"'s, then "c"'s. A superset of this language, called the Bach language, is defined as the set of all strings where "a", "b" and "c" occurs equally often and is also context-sensitive. can be shown to be a context-sensitive language by constructing a linear bounded automaton which accepts. The language can easily be shown to be neitherregular nor context free by applying the respective pumping lemmas for each of the language classes to. Similarly: 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 and and then supplementing them with a permutation production like , a new starting symbol and standard syntactic sugar. is another context-sensitive language ; that is, the "product" operation defines a context-sensitive language. Because of the commutativeproperty of the product, the most intuitive grammar for is ambiguous. This problem can be avoided considering a somehow more restrictive definition of the language, e.g.. This can be specialized to and, from this, to,, etc. is a context-sensitive language. The corresponding context-sensitive grammar can be obtained as a generalization of the context-sensitive grammars for,, etc. is a context-sensitive language. is a context-sensitive language. 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. is a context-sensitive language. This was credited by A. Salomaa to Matti Soittola by means of a linear bounded automaton over an unary alphabet and also to Marti Penttonen by means of a context-sensitive grammar also over an unary alphabet. 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, also the Kleene plus of a context-sensitive language is context-sensitive.
The complement of a context-sensitive language is itself context-sensitive 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.
