Existential monadic second order logic of undirected graphs: a disproof of the Le Bars conjecture

TL;DR

This paper disproves Le Bars' conjecture that the zero-one law holds for EMSO sentences with two first-order variables in undirected graphs, by showing a counterexample where probabilities do not converge.

Contribution

It provides the first counterexample to Le Bars' conjecture, demonstrating that the zero-one law fails even for EMSO sentences with two variables.

Findings

Counterexample shows non-convergence of probabilities

Disproves Le Bars' conjecture for 2-variable EMSO sentences

Zero-one law does not hold in this case

Abstract

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) about undirected graphs. He proved that there exists an EMSO sentence $\phi$ such that ${\sf P}(G_n \models\phi)$ does not converge as $n\to\infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1, \dots, n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.