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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 ...
Pictorial representation of the polynomial time hierarchy. The arrows denote inclusion. In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes, rather than computational complexity of individual problems and algorithms. It involves the research ...
For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.
As Democrats head toward an uncertain future under a second Trump administration, the party’s search for a new leader to help rebuild the party also remains unsettled.
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.