The Promise Polynomial Hierarchy

Abstract: The polynomial hierarchy is a grading of problems by difficulty, including P, NP and coNP as the best known classes. The promise polynomial hierarchy is similar, but extended to include promise problems. It turns out that the promise polynomial hierarchy is considerably simpler to work with, and many open questions about the polynomial hierarchy can be resolved in the promise polynomial hierarchy. Our main theorem is that, in the world of promise problems, if phi has a weak (Turing, Cook) reduction to SAT then phi has a strong (Karp, many-one) reduction to UVAL2, where UVAL2(f) is the promise problem of finding the unique x such that f(x,y)=1 for all y. We also give a complete promise problem for the promise problem equivalent of UP intersect coUP.