Generalizations of Opt P to the polynomial hierarchy

Abstract

AbstractThe author defined Opt P as a generalization of NP by considering problems as functions that compute their optimal value. An Opt P function is computed by applying the max (or min) operator to the branches of a nondeterministic machine.In this paper, we show that Opt P has a natural extension to the polynomial hierarchy by considering alternating Turing machines with the max and min operators. We show an equivalence between k alternations of max and min and functions computable with an oracle for the kth level in the polynomial hierarchy. Then we give natural complete problems for two and three alternations and show how to extend these results to give complete problems for Δp3 and Δp4. We show a further equivalence between unbounded alternations of max and min and functions computable in polynomial space and give complete functions for polynomial space