The complexity of sets formed by boolean operations (union, intersection, and complement) on NP sets are studied, showing that in some relativized worlds the boolean hierarchy is infinite, and that for every k there is a relativization world in which the Boolean hierarchy extends exactly k levels.Expand

The authors prove P/sup EP(log)/ 25 PP, investigate the Boolean closure BC( EP) of EP, and give a relativization principle which allows them to completely separate BC(EP) in a suitable relativized world and to give simple proofs for known relativizing results.Expand

New machines whose computational power is bounded by that of alternating Turing machines making only one alternation are introduced whose polynomial time classes are exactly the levels of the Boolean closure of NP which can be defined in a natural way.Expand

We show that every set in the 0~ level of the polynomial hierarchy-every set polynomial time truth-table reducible to SAT~is accepted by a probabilistic polynomial-time Turing machine: pNP[log] ~… Expand

This work defines a general maximization operator max and a general minimization operator min for complexity classes and shows that there are other interesting optimization classes beside OptP and is able to characterize the polynomial hierarchy uniformly by three operators.Expand

It is shown that in many cases solution reduction is impossible unless the polynomial hierarchy collapses, and a sufficient condition is given for such solution reduction for finite cardinality types.Expand

It turns out that a general max and a general min operator for complexity classes yield a refinement of Krentel's hierarchy of optimization functions, and it is proved that this refinement is strict unless the polynomial hierarchy collapses and show that the refinement is useful to exactly classify optimization functions.Expand

It is proved that the constant resource bound functions yield an infinite hierarchy of complexity classes of the new measur~ that allows to prove lower time bounds on 2TM computations.Expand