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Symmetric Weighted First-Order Model Counting

3 December 2014
P. Beame
Guy Van den Broeck
Eric Gribkoff
Dan Suciu
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Abstract

The FO Model Counting problem (FOMC) is the following: given a sentence Φ\PhiΦ in FO and a number nnn, compute the number of models of Φ\PhiΦ over a domain of size nnn; the Weighted variant (WFOMC) generalizes the problem by associating a weight to each tuple and defining the weight of a model to be the product of weights of its tuples. In this paper we study the complexity of the symmetric WFOMC, where all tuples of a given relation have the same weight. Our motivation comes from an important application, inference in Knowledge Bases with soft constraints, like Markov Logic Networks, but the problem is also of independent theoretical interest. We study both the data complexity, and the combined complexity of FOMC and WFOMC. For the data complexity we prove the existence of an FO3^{3}3 formula for which FOMC is #P1_11​-complete, and the existence of a Conjunctive Query for which WFOMC is #P1_11​-complete. We also prove that all γ\gammaγ-acyclic queries have polynomial time data complexity. For the combined complexity, we prove that, for every fragment FOk^{k}k, k≥2k\geq 2k≥2, the combined complexity of FOMC (or WFOMC) is #P-complete.

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