Markov Logic in Infinite Domains

TL;DR

This paper extends Markov logic to infinite domains using Gibbs measures, allowing for a broader range of logical and probabilistic modeling, including systems with phase transitions and non-unique measures.

Contribution

It introduces a framework for Markov logic in infinite domains, characterizing conditions for existence and uniqueness of measures, and linking satisfiability to measure properties.

Findings

MLNs admit Gibbs measures when each atom has finitely many neighbors

Unique measures exist if non-unit clause weights are small enough

MLNs can model phenomena with phase transitions and multiple measures

Abstract

Combining first-order logic and probability has long been a goal of AI. Markov logic (Richardson & Domingos, 2006) accomplishes this by attaching weights to first-order formulas and viewing them as templates for features of Markov networks. Unfortunately, it does not have the full power of first-order logic, because it is only defined for finite domains. This paper extends Markov logic to infinite domains, by casting it in the framework of Gibbs measures (Georgii, 1988). We show that a Markov logic network (MLN) admits a Gibbs measure as long as each ground atom has a finite number of neighbors. Many interesting cases fall in this category. We also show that an MLN admits a unique measure if the weights of its non-unit clauses are small enough. We then examine the structure of the set of consistent measures in the non-unique case. Many important phenomena, including systems with phase transitions, are represented by MLNs with non-unique measures. We relate the problem of satisfiability in first-order logic to the properties of MLN measures, and discuss how Markov logic relates to previous infinite models.