A Generalized Quantifier Concept in Computational Complexity Theory
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We investigate function classes (#P)f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., (#P)f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every rela- tivization if and only if f is a finite sum of binomial coeficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrela- tivized case. For closure properties f of #P, we have (#P)f = #I?. The other end of the range is marked by operations f for which (#P)f corresponds to the counting hierarch,y. We call these operations count- ing hard and give general criteria for hardness. For many operations f we show that (#P)f corresponds to some subclass C oaf the counting hierarchy. This will then imply that #P is closed under f if and only if UP = C; and on the other hand, f is counting hard if and only if C contains the counting hierarchy.