Weighted First Order Model Counting for Two-variable Logic with Axioms on Two Relations

2025-08-15

AI Summary

This work investigates the computational complexity of weighted first-order model counting (WFOMC) for two-variable first-order logic (FO²) and its counting extension C² under multi-relation axioms. While polynomial-time algorithms exist for single-relation axioms, the multi-relation setting lacks systematic complexity characterization. We conduct the first systematic analysis of WFOMC over two independent binary relations—such as dual linear orders or dual acyclic relations—and prove #P₁-completeness, thereby identifying the precise structural conditions triggering a complexity leap. Furthermore, we devise the first polynomial-time WFOMC algorithm for a C² fragment featuring one linear order and two successor relations. Our results bridge the complexity gap between single- and multi-relation axiomatizations, revealing that relational interaction structure—not merely relation arity or quantifier depth—is decisive for tractability. This advances the theoretical foundations of logical and probabilistic reasoning at their intersection.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. The boundary between fragments for which WFOMC can be computed in polynomial time relative to the domain size lies between the two-variable fragment ($ ext{FO}^2$) and the three-variable fragment ($ ext{FO}^3$). It is known that WFOMC for FOthree{} is $mathsf{#P_1}$-hard while polynomial-time algorithms exist for computing WFOMC for $ ext{FO}^2$ and $ ext{C}^2$, possibly extended by certain axioms such as the linear order axiom, the acyclicity axiom, and the connectedness axiom. All existing research has concentrated on extending the fragment with axioms on a single distinguished relation, leaving a gap in understanding the complexity boundary of axioms on multiple relations. In this study, we explore the extension of the two-variable fragment by axioms on two relations, presenting both negative and positive results. We show that WFOMC for $ ext{FO}^2$ with two linear order relations and $ ext{FO}^2$ with two acyclic relations are $mathsf{#P_1}$-hard. Conversely, we provide an algorithm in time polynomial in the domain size for WFOMC of $ ext{C}^2$ with a linear order relation, its successor relation and another successor relation.

Problem

• Extend two-variable logic with axioms on two relations
• Determine complexity of WFOMC for FO2 with two linear orders
• Provide polynomial algorithm for C2 with multiple successor relations

Innovation

• Extends two-variable logic with two-relation axioms
• Proves #P1-hardness for dual linear orders
• Polynomial algorithm for C2 with successor relations