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In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. [1] Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines.
The class P #P consists of all the problems that can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle). Thus Toda's theorem implies that for any problem in the polynomial hierarchy there is a deterministic polynomial-time Turing reduction to a ...
The union of the classes in the polynomial hierarchy: P NP: Solvable in polynomial time with an oracle for a problem in NP; also known as Δ 2 P PP: Probabilistically Polynomial (answer is right with probability slightly more than 1/2) PPAD: Polynomial Parity Arguments on Directed graphs PR: Solvable by recursively building up arithmetic ...
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If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete, this is widely regarded as a sign that a ...
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Because determined effort has yet to find a polynomial-time solution to any of these problems, it is suspected to be difficult to show P=UP, or even P=(UP ∩ co-UP). The Valiant–Vazirani theorem states that NP is contained in RP Promise-UP , which means that there is a randomized reduction from any problem in NP to a problem in Promise-UP .
February 7, 2024 On today’s show, CNN 10 heads to California where that state agreed to a $2 billion settlement to assist students who fell behind in their education during the Covid pandemic.