Logical Definability of Counting Functions

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is #P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, #L is the class of functions on nite structures counting the tuples (T ; c) satisfying a given formula (T ; c) in L. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures #FO = #P (where FO denotes rst-order logic) and that every function in ## 1 has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in ## 1. A consequence is that functions counting the number of cliques, the number of Hamilton cycles, and the number of pairs with distance greater than two in a graph, are not contained in ## 1. It is shown that on ordered structures ## 1 1 captures the previously studied class spanP. On unordered structures #FO is a proper subclass of #P and ## 1 1 is a proper subclass of spanP; in fact, no class #L contains all polynomial-time computable functions on unordered structures. However, it is shown that on unordered structures every function in #P is identical almost everywhere with some function #FO, and similarly for ## 1 1 and spanP. Finally, we discuss the closure properties of #FO under arithmetical operations.