Context-sensitive language
Clash Royale CLAN TAG#URR8PPP 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.
Contents
1 Computational properties
2 Examples
3 Properties of context-sensitive languages
4 See also
5 References
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 kndisplaystyle kn
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
One of the simplest context-sensitive but not context-free languages is L=anbncn:n≥1displaystyle L=a^nb^nc^n:ngeq 1
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:
LCross=ambncmdn:m≥1,n≥1displaystyle L_Cross=a^mb^nc^md^n:mgeq 1,ngeq 1
amCmdisplaystyle a^mC^m
and
Bndndisplaystyle B^nd^n
and then supplementing them with a permutation production like
CB→BCdisplaystyle CBrightarrow BC
LMUL3=ambncmn:m≥1,n≥1displaystyle L_MUL3=a^mb^nc^mn:mgeq 1,ngeq 1
LMUL1=amn:m>1,n>1displaystyle L_MUL1=a^mn:m>1,n>1
LREP=:w∈Σ∗displaystyle L_REP=w^w:win Sigma ^*
LEXP=a2n:n≥1displaystyle L_EXP=a^2^n:ngeq 1
LPRIMES2=wdisplaystyle L_PRIMES2=w:
LPRIMES1=ap:p is prime displaystyle L_PRIMES1=a^p:pmbox is prime 
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
^ Rothe, Jörg (2005), Complexity theory and cryptology, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, p. 77, ISBN 978-3-540-22147-0, MR 2164257.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.
^ Odifreddi, P. G. (1999), Classical recursion theory. Vol. II, Studies in Logic and the Foundations of Mathematics, 143, Amsterdam: North-Holland Publishing Co., p. 236, ISBN 0-444-50205-X, MR 1718169.
^ Pullum, Geoffrey K. (1983). Context-freeness and the computer processing of human languages. Proc. 21st Annual Meeting of the ACL.
^ Bach, E. (1981). "Discontinuous constituents in generalized categorial grammars". NELS, vol. 11, pp. 1–12.
^ Joshi, A.; Vijay-Shanker, K.; and Weir, D. (1991). "The convergence of mildly context-sensitive grammar formalisms". In: Sells, P., Shieber, S.M. and Wasow, T. (Editors). Foundational Issues in Natural Language Processing. Cambridge MA: Bradford.
^ Example 9.5 (p. 224) of Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley
^ J. Hartmanis and H. Shank (Jul 1968). "On the Recognition of Primes by Automata". Journal of the ACM. 15 (3): 382–389. doi:10.1145/321466.321470.
^ Salomaa, Arto (1969), Theory of Automata,
ISBN 978-0-08-013376-8, Pergamon, 276 pages. doi:10.1016/C2013-0-02221-9
^ John E. Hopcroft; Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.; Exercise 9.10, p.230. In the 2000 edition, the chapter on context-sensitive languages has been omitted.
^ Immerman, Neil (1988). "Nondeterministic space is closed under complementation" (PDF). SIAM J. Comput. 17 (5): 935–938. doi:10.1137/0217058.
- Sipser, M. (1996), Introduction to the Theory of Computation, PWS Publishing Co.