On the Power of Number-Theoretic Operations with Respect to Counting

We investigate function classes h#Pi f which are deened as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summa-tion over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi 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 nite sum of binomial coeecients 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 h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of 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.