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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 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.
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.
SO, unrestricted second-order logic, is equal to the Polynomial hierarchy PH. More precisely, we have the following generalisation of Fagin's theorem: The set of formulae in prenex normal form where existential and universal quantifiers of second order alternate k times characterise the kth level of the polynomial hierarchy. [17]
There are stronger conclusions possible if PSPACE, or some other complexity classes are assumed to have polynomial-sized circuits; see P/poly. If NP is assumed to be a subset of BPP (which is a subset of P/poly), then the polynomial hierarchy collapses to BPP. [1]
In complexity theory, the counting hierarchy is a hierarchy of complexity classes. It is analogous to the polynomial hierarchy, but with NP replaced with PP. It was defined in 1986 by Klaus Wagner. [1] [2] More precisely, the zero-th level is C 0 P = P, and the (n+1)-th level is C n+1 P = PP C n P (i.e., PP with oracle C n). [2] Thus: C 0 P = P ...