Abstract
Our purpose is to draw the frontier line which separates logics with and without 0– 1 laws for the fragments of monadic existential second-order logic (MESO for short). We establish the failure of the 0– 1 law for monadic existential second-order logic on undirected graphs by giving a sentence in this logic which has no asymptotic probability. This improves a similar result by Kaufmann on a vocabulary of 4 directed binary relations. Both results require 9 first-order variables. In contrast, we conjecture that the 0– 1 law holds for MESO with two first-order variables (that is MESO 2 ) on undirected graphs. Our result is optimal with respect to the vocabulary. The report is self-contained.
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