Lifted inference with tree axioms

Timothy van Bremen, Ondrej Kuzelka

Abstract: We consider the problem of weighted first-order model counting (WFOMC): given a first-order sentence ϕ and domain size n∈N<math><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, determine the weighted sum of models of ϕ over the domain {1,…,n}<math><mo stretchy="false" is="true">{</mo><mn is="true">1</mn><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><mi is="true">n</mi><mo stretchy="false" is="true">}</mo></math>. 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<math><msup is="true"><mrow is="true"><mi mathvariant="bold" is="true">C</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math>, the two-variable fragment with counting quantifiers [3]. In this paper, we further expand this result to any C2<math><msup is="true"><mrow is="true"><mi mathvariant="bold" is="true">C</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math> 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.