Lifted Inference beyond First-Order Logic

2023-08-22

arXiv.org

AI Summary

Weighted First-Order Model Counting (WFOMC) has been limited in domain-liftability by the expressive constraints of classical first-order logic, particularly for relational domains requiring complex graph-theoretic properties—such as acyclicity, connectivity, or tree/forest structure. Method: We extend domain-liftability to such structured constraints within the framework of Counting Logic C² (two-variable first-order logic with counting quantifiers). We introduce a generic “divide-and-count” paradigm that integrates logical semantic analysis, combinatorial enumeration, and algebraic reasoning to uniformly characterize model counting under structural constraints. Contribution/Results: We establish, for the first time, polynomial-time WFOMC tractability for directed acyclic graphs (DAGs), connected graphs, trees, and forests in C². This significantly broadens the class of logically expressible problems admitting efficient probabilistic inference. Our framework provides a novel theoretical foundation for exact probabilistic modeling and counting over discrete relational structures—including phylogenetic networks and citation networks.

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.

Problem

• Extend domain liftability of C^2 logic.
• Model real-world properties like acyclicity, connectivity.
• Provide framework for counting combinatorial structures.

Innovation

• Extends C^2 with graph properties
• Uses counting by splitting methodology
• Applies to combinatorial structures counting