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In this paper we study lifted inference for the Weighted First-Order ModelCounting problem (WFOMC), which counts the assignments that satisfy a givensentence in first-order logic (FOL); it has applications in StatisticalRelational Learning (SRL) and Probabilistic Databases (PDB). We present severalresults. First, we describe a lifted inference algorithm that generalizes priorapproaches in SRL and PDB. Second, we provide a novel dichotomy result for anon-trivial fragment of FO CNF sentences, showing that for each sentence theWFOMC problem is either in PTIME or #P-hard in the size of the input domain; weprove that, in the first case our algorithm solves the WFOMC problem in PTIME,and in the second case it fails. Third, we present several properties of thealgorithm. Finally, we discuss limitations of lifted inference for symmetricprobabilistic databases (where the weights of ground literals depend only onthe relation name, and not on the constants of the domain), and prove theimpossibility of a dichotomy result for the complexity of probabilisticinference for the entire language FOL.

Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting