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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 ...
If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. [21] Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai, runs in quasi-polynomial ...
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 ...
The theory has emerged as a result of (still failing) attempts to resolve the first and still the most important question of this kind, the P = NP problem.Most of the research is done basing on the assumption of P not being equal to NP and on a more far-reaching conjecture that the polynomial time hierarchy of complexity classes is infinite.
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In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: = PH was first defined by Larry Stockmeyer. [1] It is a special case of hierarchy of bounded alternating Turing machine.
BRUSSELS (Reuters) -The European Commission could fine Microsoft if it doesn't provide adequate information on risks stemming from generative AI features in search engine Bing by May 27.