Relativizing Function Classes

The operators min?, max?, and #? translate classes of the polynomial-time hierarchy to function classes. Although the inclusion relationships between these function classes have been studied in depth, some questions concerning separations remained open. We provide oracle constructions that answer most of these open questions in the relativized case. As a typical instance for the type of results of this paper, we construct a relativized world where min_P #?NP, thus giving evidence for the hardness of proving min?P #?NP in the unrelativized case. The strongest results, proved in the paper, are the constructions of oracles D and E, such that min_coNPD #?PD NPD coNPD and UPE = NPE min?PE #?PE

Relativizing Function Classes

The relationships of min, max, and # classes on the polynomial time hierarchy have been studied by various authors during the recent years. In general, inclusions between such classes are equivalent with inclusions between suitable complexity classes. In some cases, no or no satisfactory comparison result is known. For instance, it is not known how minP relates to #NP (see [HW97]). In this paper we provide relativized worlds for most of those inclusion relationships that are not known to be true. As a typical instance for the type of results of this paper, we construct a relativized world for minP #NP and another one for minP 6 #NP, thus giving evidence for the problem of comparing these two classes in the unrelativized case to be hard. The strongest results, proved in the paper, are the constructions of oracles D and E, such that mincoNP D #P D ^ NP D 6= coNP D and UP E = NP E ^ minP E 6 #P E . 1 Introduction This paper deals with classes of the fo..