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| {{Infobox Complexity Class
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| |class=EXPTIME, also called EXP
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| |image=[[File:Complexity subsets pspace.svg|200px]]
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| |long-name=Exponential time
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| |description=
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| |wheredefined=
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| |external-urls=[http://qwiki.stanford.edu/index.php/Complexity_Zoo:E#exp Complexity Zoo]
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| |complete-class=[[EXPTIME-complete]]
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| |complement-class=''self''
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| |proper-supersets=[[EXPSPACE]]
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| }}
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| :''"EXP" redirects here; for other uses, see [[exp]].''
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| In [[computational complexity theory]], the [[complexity class]] '''EXPTIME''' (sometimes called '''EXP''' or '''DEXPTIME''') 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>) time, where ''p''(''n'') is a polynomial function of ''n''.
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| In terms of [[DTIME]],
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| :<math> \mbox{EXPTIME} = \bigcup_{k \in \mathbb{N} } \mbox{ DTIME } \left( 2^{ n^k } \right) . </math>
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| We know
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| :[[P (complexity)|P]] <math>\subseteq</math> [[NP (complexity)|NP]] <math>\subseteq</math> [[PSPACE]] <math>\subseteq</math> EXPTIME <math>\subseteq</math> [[NEXPTIME]] <math>\subseteq</math> [[EXPSPACE]]
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| and also, by the [[time hierarchy theorem]] and the [[space hierarchy theorem]], that
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| :P <math>\subsetneq</math> EXPTIME{{nbsp|2}} and{{nbsp|2}} NP <math>\subsetneq</math> NEXPTIME{{nbsp|2}} and{{nbsp|2}} PSPACE <math>\subsetneq</math> EXPSPACE
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| so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are. Most experts{{Who|date=December 2010}} believe all the inclusions are proper. It's also known that if {{nowrap|[[P = NP problem|P = NP]]}}, then {{nowrap|EXPTIME {{=}} [[NEXPTIME]]}}, the class of problems solvable in exponential time by a [[nondeterministic Turing machine]].<ref>{{cite book| author = Christos Papadimitriou| title = Computational Complexity| publisher = Addison-Wesley| year = 1994| isbn = 0-201-53082-1| authorlink = Christos Papadimitriou}} Section 20.1, page 491.</ref> More precisely, '''EXPTIME''' ≠ '''NEXPTIME''' if and only if there exist [[sparse language]]s in '''NP''' that are not in '''P'''.<ref>Juris Hartmanis, Neil Immerman, Vivian Sewelson. Sparse Sets in NP-P: EXPTIME versus NEXPTIME. ''Information and Control'', volume 65, issue 2/3, pp.158–181. 1985. [http://portal.acm.org/citation.cfm?id=808769 At ACM Digital Library]</ref>
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| EXPTIME can also be reformulated as the space class [[APSPACE]], the problems that can be solved by an [[alternating Turing machine]] in polynomial space. This is one way to see that PSPACE <math>\subseteq</math> EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.<ref>Papadimitriou (1994), section 20.1, corollary 3, page 495.</ref>
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| EXPTIME is one class in an [[exponential hierarchy]] of complexity classes with increasingly more complex oracles or quantifier alternations. The class [[2-EXPTIME]] is defined similarly to EXPTIME but with a [[Double exponential function|doubly exponential]] time bound <math>2^{2^n}</math>. This can be generalized to higher and higher time bounds.
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| ==EXPTIME-complete==
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| A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME 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. Problems that are EXPTIME-complete might be thought of as the hardest problems in EXPTIME. Notice that although we don't know if NP is equal to P or not, we do know that EXPTIME-complete problems are not in P; it has been proven that these problems cannot be solved in [[polynomial time]], by the [[time hierarchy theorem]].
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| In [[computability theory]], one of the basic undecidable problems is that of deciding whether a [[deterministic Turing machine]] (DTM) halts. One of the most fundamental EXPTIME-complete problems is a simpler version of this, which asks if a DTM halts in at most ''k'' steps. It is in EXPTIME because a trivial simulation requires O(''k'') time, and the input ''k'' is encoded using O(log ''k'') bits.<ref>{{cite web| author = Chris Umans| url = http://www.cs.caltech.edu/~umans/cs21/lec18.pdf| title = CS 21: Lecture 18 notes}} Slide 10.</ref> It is EXPTIME-complete because, roughly speaking, we can use it to determine if a machine solving an EXPTIME problem accepts in an exponential number of steps; it will not use more. The same problem with the number of steps written in unary is [[P-complete]].
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| Other examples of EXPTIME-complete problems include the problem of evaluating a position in [[generalized game|generalized]] [[chess]],<ref name="Fraenkel1981">{{cite journal| author = [[Aviezri Fraenkel]] and D. Lichtenstein| title = Computing a perfect strategy for n×n chess requires time exponential in n| journal = J. Comb. Th. A| issue = 31| year = 1981| pages = 199–214}}</ref> [[checkers]],<ref name="robson1984">{{cite journal| author = J. M. Robson| title = N by N checkers is Exptime complete| journal = SIAM Journal on Computing| volume = 13| issue = 2| pages = 252–267| year = 1984| doi = 10.1137/0213018}}</ref> or [[Go (board game)|Go]] (with Japanese ko rules).<ref>{{Cite book| author = J. M. Robson| chapter = The complexity of Go| title = Information Processing; Proceedings of IFIP Congress| year = 1983| pages = 413–417}}</ref> These games have a chance of being EXPTIME-complete because games can last for a number of moves that is exponential in the size of the board. In the Go example, the Japanese ko rule is sufficiently intractable to imply EXPTIME-completeness, but it is not known if the more tractable American or Chinese rules for the game are EXPTIME-complete.
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| By contrast, generalized games that can last for a number of moves that is polynomial in the size of the board are often [[PSPACE-complete]]. The same is true of exponentially long games in which non-repetition is automatic.
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| Another set of important EXPTIME-complete problems relates to [[succinct circuit]]s. Succinct circuits are simple machines used to describe some graphs in exponentially less space. They accept two vertex numbers as input and output whether there is an edge between them. For many natural [[P-complete]] graph problems, where the graph is expressed in a natural representation such as an [[adjacency matrix]], solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.<ref>Papadimitriou (1994), section 20.1, page 492.</ref>
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| ==References==
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| {{Reflist}}
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| {{ComplexityClasses}}
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| [[Category:Complexity classes]]
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