Difference hierarchy

Abstract: Chang and Kadin have shown that if the difference hierarchy over NP collapses to level $k$, then the polynomial hierarchy (PH) is equal to the $k$th level of the difference hierarchy over $\Sigma_{2}^{p}$. We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level $k$, then PH = $\left(P_{(k-1)-tt}^{NP}\right)^{NP}$. We also extend the result to classes other than NP: For any class $C$ that has $\leq_{m}^{p}$-complete sets and is closed under $\leq_{conj}^{p}$and $\leq_{m}^{NP}$-reductions, if the difference hierarchy over $C$ collapses to level $k$, then $PH^{C} = $\left(P_{(k-1)-tt}^{NP}\right)^{C}$. Then we show that the exact counting class $C_{=}P$ is closed under $\leq_{disj}^{p}$and $\leq_{m}^{co-NP}$-reductions. Consequently, if the difference hierarchy over $C_{=}P$ collapses to level $k$ then $PH^{PP}$ is equal to $\left(P_{(k-1)-tt}^{NP}\right)^{PP}$. In contrast, the difference hierarchy over the closely related class PP is known to collapse. Finally, we consider two ways of relativizing the bounded query class $P_{k-tt}^{NP}$: the restricted relativization $P_{k-tt}^{NP^{C}}$, and the full relativization $\left(P_{k-tt}^{NP}\right)^{C}$. If $C$ is NP-hard, then we show that the two relativizations are different unless $PH^{C}$ collapses.

Abstract: We introduce and study two classifications refining the polynomial hierarchy. Both extend the difference hierarchy over NP and are analogs of some hierarchies from recursion theory. We answer some natural questions on the introduced classifications, e.g. we extend the result of J.Kadin that the difference hierarchy over NP does not collapse (if the polynomial hierarchy does not collapse).

Abstract: This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that a multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2k-1 nonadaptive queries to NPMV; a characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP; unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV are different than k+1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness, and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial 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.