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Concept
Context-sensitive language
Summary
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 one of the four types of grammars in the Chomsky hierarchy.
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 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(O(n)), because they can be accepted using linear space on a non-deterministic Turing machine. 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.
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, , , etc.). A superset of this language, called the Bach language, is defined as the set of all strings where "a", "b" and "c" (or any other set of three symbols) occurs equally often (, , etc.) and is also context-sensitive.
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:
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.
Official source
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