Probabilities of First-Order Sentences about Unary Functions

Let be any fixed positive integer and a sentence in the first-order predicate calculus of unary functions. For positive integers , an -structure is a model with universe <!-- MATH $\{ 0,1, \ldots ,n - 1\}$ --> and unary functions, and <!-- MATH $\mu (n,\sigma )$ --> is the ratio of the number of -structures satisfying to , the number of -structures. We show that <!-- MATH ${\lim _{n \to \infty }}\mu (n,\sigma )$ --> exists for all such , and its value is given by an expression consisting of integer constants and the operators <!-- MATH $+ , - , \cdot ,/$ --> , and .