Context-sensitive language
Language defined by context-sensitive grammar From Wikipedia, the free encyclopedia
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.
Computational properties
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.[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]
Examples
Summarize
Perspective
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.) and is also context-sensitive.[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:
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 (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 and shows). Because of the commutative property 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.[6]
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 .[7]
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.
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.[9]
- The complement of a context-sensitive language is itself context-sensitive[10] 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.
See also
- Linear bounded automaton
- List of parser generators for context-sensitive languages
- Chomsky hierarchy
- Indexed languages – a strict subset of the context-sensitive languages
- Weir hierarchy
References
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