SO(∀∃*) Sentences and Their Asymptotic Probabilities

Abstract

We prove a 0-1 law for the fragment of second order logic SO(∀∃*) over parametric classes of finite structures which allow only one unary atomic type. This completes the investigation of 0-1 laws for fragments of second order logic defined in terms of first order quantifier prefixes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for Σ14(∀∃*) without any restriction on the number of unary types.