Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula
This work provides a theoretical advancement for automated reasoning and probabilistic inference in AI, enabling efficient handling of complex logical constraints, though it is incremental as it builds on prior domain-liftability results.
The paper tackles the problem of efficiently computing weighted first-order model counting (WFOMC) for first-order logic theories with cardinality constraints and counting quantifiers, achieving domain-liftability (polynomial-time computation) by deriving a closed-form formula that extends to these constraints.
Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed-forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.