The Promise Polynomial Hierarchy

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. Goldreich [3] argues that promise problems are a more natural object of study than non-promise problems, and our results would seem to confirm this. Our main theorem is that, in the world of promise problems, if φ ∝ T SAT then φ ∝ UVAL 2 , where UVAL 2 (f) is the promise problem of finding the unique x such that ∀y, f (x, y) = 1. We also give a complete promise problem for the promise problem equivalent of UP ∩ coUP, and prove the promise problem equivalents of P UP ∩ coUP = P UP and BPP UP ∩ coUP = BPP NP. Analagous results are known for NP and coNP [1]. as the most important, so (0, 1) < (1, 0). We write Σ * = ∞ n=1 Σ n for the set of finite strings from Σ. We write P + Σ = {{0}, {1}, Σ} for the set of non-empty subsets of Σ. We define ¬ : P + Σ → P + Σ by ¬{0} = {1}, ¬{1} = {0} and ¬Σ = Σ. 1.2. Promise problems. We extend the usual definition of a problem to a promise problem. Promise problems were first introduced by Even, Selman and Yacobi [1]. The concept of a promise problem encompasses two generalizations of a problem simultaneously. Firstly, a promise problem comes with a promise which we may assume is satisfied; if the promise is not satisfied, then either answer is valid. Secondly , a promise problem allows for more than one valid answer to a problem. Our approach is to reduce non-promise problems to promise problems and then work exclusively with promise problems in the sequel. A problem (or, for emphasis, a non-promise problem) is a function φ : Σ * → Σ. A function σ : Σ * → Σ solves φ when σ = φ. Let Φ be the set of all non-promise problems. A promise problem is a function φ : Σ * → P + Σ. A function σ : Σ …