Context-sensitive language

In theoretical computer science, 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, a context-sensitive language is equivalent with a linear bounded nondeterministic Turing machine, also called a linear bounded automaton. 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 = { ap : p is a prime number }. 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.

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, and concatenation of two context-sensitive languages is context-sensitive.
• The complement of a context-sensitive language is itself context-sensitive.
• Every context-free language is context-sensitive.
• 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.

## References

• Sipser, M. (1996), Introduction to the Theory of Computation, PWS Publishing Co.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Mildly context-sensitive language — In formal grammar theory, mildly context sensitive languages are a class of formal languages which can be efficiently parsed, but still possess enough context sensitivity to allow the parsing of natural languages. The concept was first introduced …   Wikipedia

• Context-free language — In formal language theory, a context free language is a language generated by some context free grammar. The set of all context free languages is identical to the set of languages accepted by pushdown automata. Contents 1 Examples 2 Closure… …   Wikipedia

• Context-sensitive — is an adjective meaning depending on context or depending on circumstances . It may refer to: Context sensitive grammar Context sensitive language Context sensitive help Context sensitive user interface in computing This disambiguation page lists …   Wikipedia

• Context-sensitive grammar — A context sensitive grammar (CSG) is a formal grammar in which the left hand sides and right hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols. Context sensitive grammars are more general than… …   Wikipedia

• Deterministic context-free language — A deterministic context free language is a formal language which is defined by a deterministic context free grammar.[1] The set of deterministic context free languages is called DCFL[2] and is identical to the set of languages accepted by a… …   Wikipedia

• Context-free grammar — In formal language theory, a context free grammar (CFG) is a formal grammar in which every production rule is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals (w can be empty). The… …   Wikipedia

• Indexed language — Indexed languages are a class of formal languages discovered by Alfred Aho [ cite journal | last = Aho | first = Alfred | year = 1968 | title = Indexed grammars an extension of context free grammars | journal = Journal of the ACM | volume = 15 |… …   Wikipedia

• List of formal language and literal string topics — This is a list of formal language and literal string topics, by Wikipedia page. Contents 1 Formal languages 2 Literal strings 3 Classical cryptography Formal languages Abstract syntax tree …   Wikipedia

• Bach language — In theoretical computer science, the Bach language is the formal language over an alphabet of three distinct symbols containing all strings in which the three symbols occur equally often. The Bach language is a context sensitive language. Pullum… …   Wikipedia

• Recursive language — This article is about a class of formal languages as they are studied in mathematics and theoretical computer science. For computer languages that allow a function to call itself recursively, see Recursion (computer science). In mathematics,… …   Wikipedia