Weighted First-Order Model Counting over Ordered Domains

2026-08-11

AI Summary

This work addresses the problem of weighted first-order model counting (WFOMC) over domains equipped with a linear order, overcoming the #P₁-hardness barrier known for the three-variable fragment. By directly incorporating linear order axioms into the logical language—thereby avoiding explicit encoding of the order using three variables—the authors establish, for the first time, that WFOMC over ordered domains is solvable in polynomial time. They further introduce an implicit modeling technique for successor relations, which substantially improves computational efficiency and extends the result to theories involving successor predicates. Experimental evaluation demonstrates exponential speedups achieved by this approach. Additionally, the study delineates the complexity boundary by proving #P₁-hardness under bilinear orders and identifies a tractable class of hybrid order structures.

Abstract

The Weighted First-Order Model Counting Problem (WFOMC) asks for the weighted sum of models of a first-order logical sentence over a domain. It is a fundamental problem in statistical relational learning, with applications extending to enumerative combinatorics and graph polynomials. Computing WFOMC for the three-variable fragment is $\mathsf{\#P}_1$-hard, whereas polynomial-time algorithms exist for the two-variable fragment and its extensions by cardinality constraints and counting quantifiers. In this work, we explore computing WFOMC in polynomial time over linearly ordered domains, enabling tractable reasoning across inference scenarios and combinatorial problems involving sequences. Because encoding a linear order in standard first-order logic requires three variables, negating our polynomial-time aspirations, we add a linear order axiom directly to the language. This forces one predicate to impose a total ordering on domain elements. We first prove that WFOMC with the linear order axiom can be solved in time polynomial in the domain size. We then extend this result to ordered domains with access to successor relations. While this holds when successors are explicitly defined via the linear order, we demonstrate an alternative implicit approach where successor relations are part of the axiom. This implicit method exhibits significantly better performance on all tested instances, sometimes providing exponential runtime improvements. Finally, we analyze scenarios with two distinct linear orders. We show that WFOMC over the two-variable fragment with two linear orders is $\mathsf{\#P}_1$-hard. However, we develop a polynomial-time algorithm for WFOMC with one linear order and a successor relation of another, pushing the intractability barrier further, yet still leaving the question of how close to a second full linear order one can get.

Problem

• Weighted First-Order Model Counting
• linear order
• tractability
• successor relation
• two-variable fragment

Innovation

• Weighted First-Order Model Counting
• Linear Order Axiom
• Polynomial-Time Tractability
• Successor Relations
• #P-hardness