Convergence and Nonconvergence Laws for Random Expansions of Product Structures

Abstract

We prove (non)convergence laws for random expansions of product structures. More precisely, we ask which structures ${\mathfrak A}$ admit a limit law, saying that the probability that a randomly chosen expansion of ${\mathfrak A}^n$ satisfies a fixed first-order sentence always converges when n approaches infinity. For the groups ${\mathbb Z}_p$, where p is prime, we do indeed have such a limit law, even for the infinitary logic $L^\omega _{\infty \omega }$, and these probabilities always converge to dyadic rational numbers, whose denominator only depends on the expansion vocabulary. This can be used to prove that the Abelian group summation problem is not definable in $L^\omega _{\infty \omega }$. Further examples for structures with such a limit law are permutation structures and structures whose vocabulary only consists of monadic relations. As a negative example, we prove that the very simple structure $(\{0,1\}, \le )$ does not have a limit law. Furthermore, we develop a method based on positive primitive interpretations that allows transferring (non)convergence results to other structures. Using this method, we are able to prove that structures with binary function symbols or unary functions that are not interpreted by permutations do not have a limit law in general.

For Yuri Gurevich on the occasion of his 80th birthday