Markov Logic in Infinite Domains
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.
💡 Research Summary
The paper tackles a fundamental limitation of Markov Logic Networks (MLNs): their original formulation is confined to finite domains, which prevents the full expressive power of first‑order logic from being exploited in many realistic settings where the universe of objects is unbounded. To overcome this, the authors recast MLNs within the mathematical framework of Gibbs measures, a well‑established theory for defining probability distributions on infinite graphs.
The core technical contribution is a set of sufficient conditions under which an infinite‑domain MLN admits at least one Gibbs measure. The authors introduce the notion of “local finiteness”: every ground atom must have only finitely many neighboring atoms (i.e., atoms that appear together in some ground clause). Under this condition, each atom’s conditional distribution depends on a finite set of other atoms, allowing the construction of a consistent family of conditional probabilities that satisfy the Dobrushin–Lanford–Ruelle (DLR) equations. Consequently, a global Gibbs measure exists even though the underlying set of random variables is infinite.
Having established existence, the paper investigates uniqueness. By adapting Dobrushin’s influence‑coefficient criterion, the authors prove that if the absolute values of all non‑unit clause weights are sufficiently small relative to the maximum degree of the dependency graph, the Gibbs measure is unique. Intuitively, weak interactions prevent the system from supporting multiple macroscopic states, guaranteeing that standard Gibbs sampling will converge to a single stationary distribution.
The authors then turn to the complementary regime where the uniqueness condition is violated. In this “non‑unique” case, multiple Gibbs measures can coexist, mirroring phase‑transition phenomena familiar from statistical physics (e.g., the Ising model). They illustrate how the same set of logical formulas can give rise to distinct probability distributions depending on the magnitude of the weights, thereby capturing rich behaviours such as symmetry breaking and metastability. The paper analyses the structure of the set of consistent measures, showing that it forms a convex polytope whose extreme points correspond to the pure phases of the system.
A particularly insightful part of the work connects logical satisfiability to the existence of Gibbs measures. The authors prove two complementary statements: (1) if an MLN admits no Gibbs measure, the underlying first‑order theory is unsatisfiable (i.e., it contains a contradiction); (2) if at least one Gibbs measure exists, the theory is satisfiable. This establishes a probabilistic analogue of the classic SAT/UNSAT dichotomy and suggests that techniques from statistical physics could be leveraged for reasoning about logical consistency in infinite domains.
Finally, the paper situates infinite‑domain MLNs among related infinite‑model formalisms such as infinite Markov random fields and infinite Bayesian networks. While those models capture probabilistic dependencies, they lack the explicit logical template mechanism that MLNs provide. Conversely, traditional logical formalisms lack a built‑in notion of uncertainty. By marrying Gibbs‑measure theory with weighted first‑order clauses, the authors deliver a unified framework that preserves the expressive power of first‑order logic, supports rich probabilistic semantics, and remains mathematically well‑grounded even when the domain is unbounded.
The conclusion outlines practical guidelines for building infinite‑domain MLNs: ensure local finiteness through careful domain design, control clause weights to stay within the uniqueness regime when a single stationary distribution is desired, and be prepared to interpret multiple Gibbs measures as representing distinct macroscopic behaviours when phase transitions are intentional. Future research directions include developing scalable inference algorithms for infinite Gibbs measures, criteria for selecting among multiple measures, and applying infinite MLNs to large‑scale knowledge bases and relational data streams.