Symmetric Weighted First-Order Model Counting

TL;DR

This paper investigates the complexity of symmetric weighted first-order model counting, revealing computational hardness results and identifying tractable cases, with implications for probabilistic inference in knowledge bases.

Contribution

It characterizes the complexity of symmetric WFOMC for various fragments and query types, providing new hardness results and identifying polynomial-time cases.

Findings

FOMC is #P1-complete for some FO^3 formulas.

WFOMC is #P1-complete for some conjunctive queries.

All γ-acyclic queries have polynomial data complexity.

Abstract

The FO Model Counting problem (FOMC) is the following: given a sentence $\Phi$ in FO and a number $n$, compute the number of models of $\Phi$ 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 $\gamma$-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.