Symmetric Weighted First-Order Model Counting
This work addresses theoretical complexity bounds for inference in knowledge bases with soft constraints, such as Markov Logic Networks, but is incremental as it builds on existing model counting frameworks.
The paper tackles the complexity of symmetric weighted first-order model counting (WFOMC), proving that for data complexity, there exists an FO³ formula where FOMC is #P₁-complete and a conjunctive query where WFOMC is #P₁-complete, while all γ-acyclic queries have polynomial time data complexity, and for combined complexity, FOMC and WFOMC are #P-complete for every FOᵏ fragment with k≥2.
The FO Model Counting problem (FOMC) is the following: given a sentence $Φ$ in FO and a number $n$, compute the number of models of $Φ$ over a domain of size $n$; 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 FO$^{3}$ formula for which FOMC is #P$_1$-complete, and the existence of a Conjunctive Query for which WFOMC is #P$_1$-complete. We also prove that all $γ$-acyclic queries have polynomial time data complexity. For the combined complexity, we prove that, for every fragment FO$^{k}$, $k\geq 2$, the combined complexity of FOMC (or WFOMC) is #P-complete.