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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 ...
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]
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
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]
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The NP-complete problems represent the hardest problems in NP. If some NP-complete problem has a polynomial time algorithm, all problems in NP do. The set of NP-complete problems is often denoted by NP-C or NPC. Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly.