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P versus NP problem
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==Harder problems== {{See also|Complexity class}} Although it is unknown whether P = NP, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class [[EXPTIME]] is the set of all decision problems that have ''exponential'' running time. In other words, any problem in EXPTIME is 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''. A decision problem is [[EXPTIME#EXPTIME-complete|EXPTIME-complete]] if it is in EXPTIME, and every problem in EXPTIME has a [[polynomial-time many-one reduction]] to it. A number of problems are known to be EXPTIME-complete. Because it can be shown that P β EXPTIME, these problems are outside P, and so require more than polynomial time. In fact, by the [[time hierarchy theorem]], they cannot be solved in significantly less than exponential time. Examples include finding a perfect strategy for [[chess]] positions on an ''N'' Γ ''N'' board<ref name="Fraenkel1981">{{Cite journal| author = [[Aviezri Fraenkel]] and D. Lichtenstein| title = Computing a perfect strategy for ''n'' Γ ''n'' chess requires time exponential in ''n''| journal = [[Journal of Combinatorial Theory]] | series=Series A | volume = 31| issue = 2| year = 1981| pages = 199β214 | doi = 10.1016/0097-3165(81)90016-9| doi-access = }}</ref> and similar problems for other board games.<ref>{{Cite web|title=Computational Complexity of Games and Puzzles |url=http://www.ics.uci.edu/~eppstein/cgt/hard.html |author=[[David Eppstein]]}}</ref> The problem of deciding the truth of a statement in [[Presburger arithmetic]] requires even more time. [[Michael J. Fischer|Fischer]] and [[Michael O. Rabin|Rabin]] proved in 1974<ref>{{cite journal | first1=Michael J. | last1=Fischer | author-link1=Michael J. Fischer | first2=Michael O. | last2=Rabin | author-link2=Michael O. Rabin | date=1974 | title=Super-Exponential Complexity of Presburger Arithmetic | url=http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps | journal=Proceedings of the SIAM-AMS Symposium in Applied Mathematics | volume=7 | pages=27β41| access-date=15 October 2017 | archive-url=https://web.archive.org/web/20060915010325/http://www.lcs.mit.edu/publications/pubs/ps/MIT-LCS-TM-043.ps | archive-date=15 September 2006 }}</ref> that every algorithm that decides the truth of Presburger statements of length ''n'' has a runtime of at least <math>2^{2^{cn}}</math> for some constant ''c''. Hence, the problem is known to need more than exponential run time. Even more difficult are the [[undecidable problem]]s, such as the [[halting problem]]. They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all. It is also possible to consider questions other than decision problems. One such class, consisting of counting problems, is called [[Sharp-P#P|#P]]: whereas an NP problem asks "Are there any solutions?", the corresponding #P problem asks "How many solutions are there?". Clearly, a #P problem must be at least as hard as the corresponding NP problem, since a count of solutions immediately tells if at least one solution exists, if the count is greater than zero. Surprisingly, some #P problems that are believed to be difficult correspond to easy (for example linear-time) P problems.<ref>{{cite journal |author=Valiant, Leslie G. |title=The complexity of enumeration and reliability problems |journal=SIAM Journal on Computing |volume=8 |issue=3 |year=1979 |pages=410β421 |doi=10.1137/0208032}}</ref> For these problems, it is very easy to tell whether solutions exist, but thought to be very hard to tell how many. Many of these problems are [[Sharp-P-complete|#P-complete]], and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems.
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