Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

TL;DR

This paper explores the relationships between Boolean hierarchies, unambiguous polynomial time (UP), and sparse Turing-complete sets, revealing new equalities, separations, and implications for complexity class collapses.

Contribution

It proves that Boolean hierarchies over classes closed under intersection, like UP, are equal, and establishes new consequences of sparse Turing-complete sets for UP and the unambiguous polynomial hierarchy.

Findings

Boolean hierarchies over classes closed under intersection are equal

Hausdorff and nested difference hierarchies do not capture UP in some relativized worlds

Sparse Turing-complete sets for UP imply UP is in Low_2 and hierarchy collapses

Abstract

It is known that for any class C closed under union and intersection, the Boolean closure of C, the Boolean hierarchy over C, and the symmetric difference hierarchy over C all are equal. We prove that these equalities hold for any complexity class closed under intersection; in particular, they thus hold for unambiguous polynomial time (UP). In contrast to the NP case, we prove that the Hausdorff hierarchy and the nested difference hierarchy over UP both fail to capture the Boolean closure of UP in some relativized worlds. Karp and Lipton proved that if nondeterministic polynomial time has sparse Turing-complete sets, then the polynomial hierarchy collapses. We establish the first consequences from the assumption that unambiguous polynomial time has sparse Turing-complete sets: (a) UP is in Low_2, where Low_2 is the second level of the low hierarchy, and (b) each level of the unambiguous polynomial hierarchy is contained one level lower in the promise unambiguous polynomial hierarchy than is otherwise known to be the case.