Uniform Characterizations of Complexity Classes of Functions

We introduce a general framework for the definition of function classes. Our model, which is based on nondeterministic polynomial-time Turing transducers, allows uniform characterizations of FP, FPNP, FPNP[O (log n)], , counting classes (#·P, #·NP, #·coNP, GapP, GapPNP), optimization classes (max·P, min·P, max·NP, min·NP), promise classes (NPSV, #few·P, c#·P), multivalued classes (FewFP, NPMV), and many more. Each such class is defined in our model by a scheme how to evaluate computation trees of nondeterministic machines. We study a reducibility notion between such evaluation schemes, which leads to a necessary and sufficient criterion for relativizable inclusion between function classes. As it turns out, this criterion is easily applicable and we get as a consequence, e.g., that there is an oracle A, such that min·PA⊈#·NPA (note that no structural consequences are known to follow from the corresponding positive inclusion).

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