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In [[computational complexity theory|complexity theory]], '''EXPSPACE''' is the [[Set (mathematics)|set]] of all [[decision problem]]s solvable by a deterministic [[Turing machine]] in [[big O notation|O]](2<sup>''p''(''n'')</sup>) space, where ''p''(''n'') is a polynomial function of ''n''. (Some authors restrict ''p''(''n'') to be a [[linear function]], but most authors instead call the resulting class '''[[ESPACE]]'''.) If we use a nondeterministic machine instead, we get the class '''NEXPSPACE''', which is equal to '''EXPSPACE''' by [[Savitch's theorem]]. | |||
In terms of '''[[DSPACE]]''' and '''[[NSPACE]]''', | |||
:<math>\mbox{EXPSPACE} = \bigcup_{k\in\mathbb{N}} \mbox{DSPACE}(2^{n^k}) = \bigcup_{k\in\mathbb{N}} \mbox{NSPACE}(2^{n^k})</math> | |||
A decision problem is '''EXPSPACE-complete''' if it is in '''EXPSPACE''', and every problem in '''EXPSPACE''' has a [[polynomial-time many-one reduction]] to it. In other words, there is a polynomial-time [[algorithm]] that transforms instances of one to instances of the other with the same answer. '''EXPSPACE-complete''' problems might be thought of as the hardest problems in '''EXPSPACE'''. | |||
'''EXPSPACE''' is a strict superset of '''[[PSPACE]]''', '''[[NP (complexity)|NP]]''', and '''[[P (complexity)|P]]''' and is believed to be a strict superset of '''[[EXPTIME]]'''. | |||
An example of an '''EXPSPACE-complete''' problem is the problem of recognizing whether two [[regular expression]]s represent different languages, where the expressions are limited to four operators: union, [[concatenation]], the [[Kleene star]] (zero or more copies of an expression), and squaring (two copies of an expression).<ref>Meyer, A.R. and [[Larry Stockmeyer|L. Stockmeyer]]. [http://people.csail.mit.edu/meyer/rsq.pdf The equivalence problem for regular expressions with squaring requires exponential space]. ''13th IEEE Symposium on Switching and Automata Theory'', Oct 1972, pp.125–129.</ref> | |||
If the Kleene star is left out, then that problem becomes '''[[NEXPTIME]]-complete''', which is like '''EXPTIME-complete''', except it is defined in terms of [[non-deterministic Turing machine]]s rather than deterministic. | |||
It has also been shown by L. Berman in 1980 that the problem of verifying/falsifying any [[first-order logic|first-order]] statement about [[real number]]s that involves only [[addition]] and comparison (but no [[multiplication]]) is in '''EXPSPACE'''. | |||
==See also== | |||
*[[Game complexity]] | |||
== References == | |||
<references /> | |||
* L. Berman ''The complexity of logical theories'', Theoretical Computer Science 11:71-78, 1980. | |||
* {{cite book|author = [[Michael Sipser]] | year = 1997 | title = Introduction to the Theory of Computation | publisher = PWS Publishing | isbn = 0-534-94728-X}} Section 9.1.1: Exponential space completeness, pp. 313–317. Demonstrates that determining equivalence of regular expressions with exponentiation is EXPSPACE-complete. | |||
{{ComplexityClasses}} | |||
[[Category:Complexity classes]] | |||
Revision as of 19:52, 10 September 2013
In complexity theory, EXPSPACE is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) space, where p(n) is a polynomial function of n. (Some authors restrict p(n) to be a linear function, but most authors instead call the resulting class ESPACE.) If we use a nondeterministic machine instead, we get the class NEXPSPACE, which is equal to EXPSPACE by Savitch's theorem.
In terms of DSPACE and NSPACE,
A decision problem is EXPSPACE-complete if it is in EXPSPACE, and every problem in EXPSPACE has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. EXPSPACE-complete problems might be thought of as the hardest problems in EXPSPACE.
EXPSPACE is a strict superset of PSPACE, NP, and P and is believed to be a strict superset of EXPTIME.
An example of an EXPSPACE-complete problem is the problem of recognizing whether two regular expressions represent different languages, where the expressions are limited to four operators: union, concatenation, the Kleene star (zero or more copies of an expression), and squaring (two copies of an expression).[1]
If the Kleene star is left out, then that problem becomes NEXPTIME-complete, which is like EXPTIME-complete, except it is defined in terms of non-deterministic Turing machines rather than deterministic.
It has also been shown by L. Berman in 1980 that the problem of verifying/falsifying any first-order statement about real numbers that involves only addition and comparison (but no multiplication) is in EXPSPACE.
See also
References
- ↑ Meyer, A.R. and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. 13th IEEE Symposium on Switching and Automata Theory, Oct 1972, pp.125–129.
- L. Berman The complexity of logical theories, Theoretical Computer Science 11:71-78, 1980.
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Section 9.1.1: Exponential space completeness, pp. 313–317. Demonstrates that determining equivalence of regular expressions with exponentiation is EXPSPACE-complete.
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