Probabilistic characterisation of models of first-order theories

Abstract

We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms T and a structure M which we only know is a model of T, we are interested in the probability that M would satisfy a sentence ψ. Answering this question for all sentences in the language will give a probability distribution over the set of sentences which can be regarded as the probabilistic characterisation of the model M. We investigate defining these probabilistic characterisations as the limit of probability functions imposed on the set of finite models of T. We show how a symmetry axiom can uniquely specify the probability function over finite models and will study the existence of the limit in terms of the quantifier complexity of T.