In formal language theory, a context-sensitive language is a language that can be defined by a context-sensitive grammar (and equivalently by a noncontracting grammar).
Context-sensitive is known as type-1 in the Chomsky hierarchy of formal languages.
Computationally, a context-sensitive language is equivalent to a linear bounded nondeterministic Turing machine, also called a linear bounded automaton.
That is a non-deterministic Turing machine with a tape of only
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(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] One of the simplest context-sensitive but not context-free languages is
: 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.)
[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. Similarly:
{\displaystyle L_{\textit {Cross}}=\{a^{m}b^{n}c^{m}d^{n}:m\geq 1,n\geq 1\}}
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 then supplementing them with a permutation production like
, a new starting symbol and standard syntactic sugar.
{\displaystyle L_{MUL3}=\{a^{m}b^{n}c^{mn}:m\geq 1,n\geq 1\}}
is another context-sensitive language (the "3" in the name of this language is intended to mean a ternary alphabet); that is, the "product" operation defines a context-sensitive language (but the "sum" defines only a context-free language as the grammar
Because of the commutative property of the product, the most intuitive grammar for
This problem can be avoided considering a somehow more restrictive definition of the language, e.g.
is a context-sensitive language.
The corresponding context-sensitive grammar can be obtained as a generalization of the context-sensitive grammars for
is a context-sensitive language.
is a context-sensitive language (the "2" in the name of this language is intended to mean a binary alphabet).
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 (the "1" in the name of this language is intended to mean a unary alphabet).
This was credited by A. Salomaa to Matti Soittola by means of a linear bounded automaton over a unary alphabet[8] (pages 213-214, exercise 6.8) and also to Marti Penttonen by means of a context-sensitive grammar also over a unary alphabet (See: Formal Languages by A. Salomaa, page 14, Example 2.5).
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.