Limit laws and automorphism groups of random nonrigid structures

TL;DR

This paper explores the asymptotic behavior and logical limit laws of classes of finite structures with complex automorphism groups, extending known results about rigidity and zero-one laws.

Contribution

It introduces a hierarchy of nonrigid structures based on automorphism complexity and proves that each level satisfies a logical limit law but not a zero-one law, with convergence results for structure counts.

Findings

Hierarchy of structures with automorphism groups analyzed

Each hierarchy level satisfies a logical limit law

Ratios of structure counts converge to rational numbers or infinity

Abstract

A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the $k$th level of the hierarchy can consist of all structures having at least $k$ elements which are moved by some automorphism. Or we can consider, for any finite group $G$, all finite structures $\mathcal{M}$ such that $G$ is a subgroup of the group of autmorphisms of $\mathcal{M}$; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless $k = 0$ or $G$ is trivial). Moreover, the number of (labelled or unlabelled) $n$-element structures in one place of the hierarchy divided by the number of $n$-element structures in another place always converges to a rational number or to $\infty$ as $n \to \infty$. All instances of the respective result are proved by an essentially uniform argument.