On Exact Sampling in the Two-Variable Fragment of First-Order Logic
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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 𝐅𝐎^2 (𝐔𝐅𝐎^2) to the entire fragment of 𝐅𝐎^2. Specifically, we prove the domain-liftability under sampling of 𝐅𝐎^2, meaning that there exists a sampling algorithm for 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 ∀ x∃_=k y: φ(x,y) and ∃_=k x∀ y: φ(x,y), for some quantifier-free formula φ(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.
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