Skolemization for Weighted First-Order Model Counting

TL;DR

This paper introduces a Skolemization method that removes existential quantifiers from first-order logic theories without altering their weighted model count, enabling more efficient probabilistic reasoning.

Contribution

It presents a Skolemization algorithm that preserves weighted model counts, extending lifted model counting to theories with existential quantifiers, which was previously infeasible.

Findings

Enables polynomial-time lifted model counting for certain first-order theories with existential quantifiers.

Maintains the weighted model count after Skolemization, ensuring correctness.

Simplifies the design of lifted model counting algorithms for probabilistic logic programs.

Abstract

First-order model counting emerged recently as a novel reasoning task, at the core of efficient algorithms for probabilistic logics. We present a Skolemization algorithm for model counting problems that eliminates existential quantifiers from a first-order logic theory without changing its weighted model count. For certain subsets of first-order logic, lifted model counters were shown to run in time polynomial in the number of objects in the domain of discourse, where propositional model counters require exponential time. However, these guarantees apply only to Skolem normal form theories (i.e., no existential quantifiers) as the presence of existential quantifiers reduces lifted model counters to propositional ones. Since textbook Skolemization is not sound for model counting, these restrictions precluded efficient model counting for directed models, such as probabilistic logic programs, which rely on existential quantification. Our Skolemization procedure extends the applicability of first-order model counters to these representations. Moreover, it simplifies the design of lifted model counting algorithms.