Lifted inference with tree axioms

Abstract

We consider the problem of weighted first-order model counting (WFOMC): given a first-order sentence ϕ and domain size n∈N, determine the weighted sum of models of ϕ over the domain {1,…,n}. Past work has shown that any sentence using at most two logical variables admits an algorithm for WFOMC that runs in time polynomial in the given domain size [1,2]. The same property was later also shown to hold for C2, the two-variable fragment with counting quantifiers [3]. In this paper, we further expand this result to any C2 sentence ϕ with the addition of a tree axiom, stating that some distinguished binary relation in ϕ forms a tree in the graph-theoretic sense.