Lifted Inference beyond First-Order Logic

TL;DR

This paper extends the class of logical properties for which weighted first-order model counting can be performed efficiently, enabling probabilistic inference on complex structures like acyclic graphs and trees.

Contribution

It introduces a novel 'counting by splitting' methodology that preserves domain liftability for C2 logic with various graph properties.

Findings

C2 logic remains domain liftable with acyclic, connected, and tree relations.

The new methodology generalizes counting techniques for combinatorial structures.

Results expand the applicability of lifted inference to more complex relational models.

Abstract

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers ($\mathrm{C^2}$) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in $\mathrm{C^2}$, or first order logic in general. In this work, we expand the domain liftability of $\mathrm{C^2}$ with multiple such properties. We show that any $\mathrm{C^2}$ sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.