On functions computable with nonadaptive queries to NP

We study FP_||^NP, the class of functions that can be computed with nonadaptive queries to an NP oracle. We show that optimization problems stemming from the known NP complete sets, where the optimum is taken over a polynomially bounded range, are hard for FP_||^NP. This is related to (and, in some sense, extends) work of Z. Chen and S. Toda (1991). In addition, it turns out that these optimization problems are all equivalent under a certain functional reducibility. By studying the question whether these function classes are complete for FP_||^NP, i.e. whether it is possible to compute an optimal value for a given optimization problem in FP_||^NP, we show that this is exactly as hard as to compute membership proofs for NP complete sets in FP_||^NP. On the other hand, FP_|| ^NP can be characterized as the class of functions that reduces to the above mentioned optimization functions. We call this property quasi-completeness. A subclass of FP_||^NP is NPSV, the class of functions that can be computed by single-valued NP transducers, We exhibit function classes that are quasi-complete for NPSV but not complete unless the polynomial time hierarchy collapses