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 finite structures (of a fixed signature) counting the tuples (T¯, c¯) satisfying a given formula ψ(T¯, c¯) in (L. Saluja et al., 1992) showed that on classes of ordered structures FO= P (where FO denotes first-order logic) and that every function in Σ₁ has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in Σ₁ (on unordered structures). 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 Σ₁ captures the previously studied class span P. On unordered structures FO is a proper subclass of P and Σ₁ 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 Σ₁ and spanP. Finally, it is shown that FO is closed under various operations under which P is closed, but that FO is not closed under other operations under which P would be closed only if certain generally believed assumptions in complexity theory failed