The Promise Polynomial Hierarchy

TL;DR

This paper introduces the promise polynomial hierarchy, demonstrating its simplicity and resolving many open questions by establishing reductions and equivalences within promise problems.

Contribution

It presents the promise polynomial hierarchy as a simpler framework and proves key reductions, including that problems reducible to SAT are also reducible to UVAL2, and identifies a complete promise problem.

Findings

Promise polynomial hierarchy is simpler to analyze.

Weak reductions to SAT imply strong reductions to UVAL2.

Identifies a complete promise problem for UP intersect coUP.

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.