Oracles that Compute Values

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 1. a multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2 − 1 nonadaptive queries to NPMV; 2. 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; 3. 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.