Logical Definability of Counting Functions

Abstract

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 logicL, #Lis the class of functions on finite structures counting the tuples (T, ) satisfying a given formulaψ(T, ) inL. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures #FO=#P (where FO denotes first-order logic) and that every function in #Σ1has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in #Σ1. 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 #Σ1. It is shown that on ordered structures #Σ11captures the previously studied class spanP. On unordered structures #FO is a proper subclass of #P and #Σ11is a proper subclass of spanP; in fact, no class #Lcontains 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 #Sgr;11and spanP. Finally, we discuss the closure properties of #FO under arithmetical operations.