  Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the
models of a first-order theory on a given finite domain. 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 FO2(the fragment
of first-order logic with two variables) for the special case of symmetric
weight functions. We introduce the concept of lifted interpretations as a tool
for formulating polynomials for WFOMC. Using lifted interpretations, we
reconstruct the closed-form formula for polynomial-time FOMC in the universal
fragment of FO2, earlier proposed by Beame et al. We then expand this
closed-form to incorporate existential quantifiers and cardinality constraints
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.
