## On the Power of Number-Theoretic Operations with Respect to Counting (1995)

Venue: | IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY |

Citations: | 32 - 9 self |

### BibTeX

@INPROCEEDINGS{Hertrampf95onthe,

author = {Ulrich Hertrampf and Heribert Vollmer and Klaus W. Wagner},

title = {On the Power of Number-Theoretic Operations with Respect to Counting},

booktitle = {IN PROCEEDINGS 10TH STRUCTURE IN COMPLEXITY THEORY},

year = {1995},

pages = {299--314},

publisher = {}

}

### OpenURL

### Abstract

We investigate function classes h#Pi 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., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized 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 ...