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arXiv · 1807.01794

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

Svetlana Popova·Maksim Zhukovskii
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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.

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Svetlana Popova, Maksim Zhukovskii. 2018-07-04. Existential monadic second order logic of undirected graphs: a disproof of the Le Bars conjecture. https://arxiv.org/abs/1807.01794

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