Weighted First-Order Model Counting over Ordered Domains
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.