Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of models a first-order logic theory on given finite domain. Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called liftable. We introduce concept lifted interpretations as tool for formulating closed forms WFOMC. Using interpretations, we reconstruct closed-form formula FOMC in universally quantified fragment FO2, earlier proposed by Beame et al. then expand this to incorporate constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, show obtained motivates natural definition family weight functions strictly larger than symmetric functions.