Markov Logic in Infinite Domains

Abstract

Markov logic combines logic and probability by attaching weights to
first-order formulas, and viewing them as templates for features of Markov
networks. Unfortunately, in its original formulation it does not have the
full power of first-order logic, because it applies only to finite domains.
Recently, we have extended Markov logic to infinite domains, by casting it
in the framework of Gibbs measures. In this talk I will summarize our main
results to date, including sufficient conditions for the existence and
uniqueness of a Gibbs measure consistent with an infinite MLN, and
properties of the set of consistent measures in the non-unique case.
(Many important phenomena, like phase transitions, are modeled by
non-unique MLNs.) Under the conditions for existence, we have extended
to infinite domains the result in Richardson and Domingos (2006) that
first-order logic is the limiting case of Markov logic when all weights
tend to infinity. I will also discuss some fundamental limitations of
Herbrand interpretations (and representations based on them) for
probabilistic modeling of infinite domains, and how to get around them.
Finally, I will discuss some of the surprising insights for learning
and inference in large finite domains that result from considering the
infinite limit.