On Exact Sampling in the Two-Variable Fragment of First-Order Logic

TL;DR

This paper proves that efficient, polynomial-time sampling algorithms exist for the entire two-variable first-order logic fragment, including counting constraints, enabling practical model generation for various logical and probabilistic systems.

Contribution

It extends domain-liftability under sampling to the full FO^2 fragment with counting, providing a constructive method for efficient sampling algorithms.

Findings

Sampling algorithm runs in polynomial time in domain size.

Sampling remains feasible with counting constraints.

Applicable to probabilistic logic models and combinatorial structures.

Abstract

In this paper, we study the sampling problem for first-order logic proposed recently by Wang et al. -- how to efficiently sample a model of a given first-order sentence on a finite domain? We extend their result for the universally-quantified subfragment of two-variable logic $\mathbf{FO}^2$ ($\mathbf{UFO}^2$) to the entire fragment of $\mathbf{FO}^2$. Specifically, we prove the domain-liftability under sampling of $\mathbf{FO}^2$, meaning that there exists a sampling algorithm for $\mathbf{FO}^2$ that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of counting constraints, such as $\forall x\exists_{=k} y: \varphi(x,y)$ and $\exists_{=k} x\forall y: \varphi(x,y)$, for some quantifier-free formula $\varphi(x,y)$. Our proposed method is constructive, and the resulting sampling algorithms have potential applications in various areas, including the uniform generation of combinatorial structures and sampling in statistical-relational models such as Markov logic networks and probabilistic logic programs.