On the Completeness of First-Order Knowledge Compilation for Lifted Probabilistic Inference

Probabilistic logics are receiving a lot of attention today because of their expressive power for knowledge representation and learning. However, this expressivity is detrimental to the tractability of inference, when done at the propositional level. To solve this problem, various lifted inference algorithms have been proposed that reason at the first-order level, about groups of objects as a whole. Despite the existence of various lifted inference approaches, there are currently no completeness results about these algorithms. The key contribution of this paper is that we introduce a formal definition of lifted inference that allows us to reason about the completeness of lifted inference algorithms relative to a particular class of probabilistic models. We then show how to obtain a completeness result using a first-order knowledge compilation approach for theories of formulae containing up to two logical variables.