A Computational Framework for the Study of Partition Functions and Graph Polynomials

Partition functions and graph polynomials have found many applications in combinatorics, physics, biology and even the mathematics of finance. Studying their complexity poses some problems. To capture the complexity of their combinatorial nature, the Turing model of computation and Valiant's notion of counting complexity classes seem most natural. To capture the algebraic and numeric nature of partition functions as real or complex valued functions, the Blum-Shub-Smale ($\BSS$) model of computation seems more natural. As a result many papers use a naive hybrid approach in discussing their complexity or restrict their considerations to sub-fields of $\C$ which can be coded in a way to allow dealing with Turing computability. In this paper we propose a unified natural framework for the study of computability and complexity of partition functions and graph polynomials and show how classical results can be cast in this framework.