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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]].
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In terms of '''[[DSPACE]]''' and '''[[NSPACE]]''',
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:<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&ndash;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.&nbsp;313&ndash;317. Demonstrates that determining equivalence of regular expressions with exponentiation is EXPSPACE-complete.
 
{{ComplexityClasses}}
 
[[Category:Complexity classes]]

Latest revision as of 07:36, 8 August 2014

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