Weighed Model Counting in FO$^2$ with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

Weighted First Order Model Counting (WFOMC) computes the weighted sum of the models of a first order theory on a domain of a given finite size. WFOMC has emerged as a fundamental tool for probabilistic inference. Algorithms for WFOMC that run in polynomial time w.r.t. the domain size are called lifted inference algorithms. Such algorithms have been developed for multiple extensions of FO$^2$(the fragment of First Order Logic with two variables) for the special case of symmetric weight functions. In this paper, instead of developing a specific algorithm, we derive a closed form formula for WFOMC in FO$^2$. The three key advantages of our proposal are: (i) it deals with existential quantifiers without introducing negative weights; (ii) it easily extends to FO$^2$ with cardinality constraints and counting quantifiers (aka C$^2$); finally, (iii) it supports WFOMC for a class of weight functions strictly larger than symmetric weight functions, which can model count distributions, without introducing complex or negative weights.