On Using 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 k ? 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 is diierent than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the diierence hierarchy over NPMV are also studied. The diierence hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the diierence and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses.