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In computational complexity theory, the complexity class #P (pronounced "number P" or, sometimes "sharp P" or "hash P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ''ƒ''(''x'')", where ''ƒ'' is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. An NP problem is often of the form "Are there any solutions that satisfy certain constraints?" For example: * Are there any subsets of a list of integers that add up to zero? (subset sum problem) * Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem) * Are there any variable assignments that satisfy a given CNF formula? (Boolean satisfiability problem) The corresponding #P problems ask "how many" rather than "are there any". For example: * How many subsets of a list of integers add up to zero? * How many Hamiltonian cycles in a given graph have cost less than 100? * How many variable assignments satisfy a given CNF formula? Clearly, a #P problem must be at least as hard as the corresponding NP problem. If it's easy to count answers, then it must be easy to tell whether there are any answers – just count them and see whether the count is greater than zero. One consequence of Toda's theorem is that a polynomial-time machine with a #P oracle (P#P) can solve all problems in PH, the entire polynomial hierarchy. In fact, the polynomial-time machine only needs to make one #P query to solve any problem in PH. This is an indication of the extreme difficulty of solving #P-complete problems exactly. Surprisingly, some #P problems that are believed to be difficult correspond to easy P problems. For more information on this, see #P-complete. The closest decision problem class to #P is PP, which asks whether a majority (more than half) of the computation paths accept. This finds the most significant bit in the #P problem answer. The decision problem class ⊕P instead asks for the least significant bit of the #P answer. The complexity class #P was first defined by Leslie Valiant in a 1979 article on the computation of the permanent, in which he proved that permanent is #P-complete. Larry Stockmeyer has proved that for every #P problem ''P'' there exists a randomized algorithm using oracle for SAT, which given an instance ''a'' of ''P'' and ''ε'' > 0 returns with high probability a number ''x'' such that . The runtime of the algorithm is polynomial in ''a'' and 1/''ε''. The algorithm is based on leftover hash lemma. == References == 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sharp-P」の詳細全文を読む スポンサード リンク
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