Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

• URL: http://arxiv.org/abs/2110.05992v1
• Date: Tue, 12 Oct 2021 13:28:50 GMT
• Title: Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula
• Authors: Sagar Malhotra and Luciano Serafini
• Abstract summary: Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a finite domain. We introduce the concept of lifted interpretations as a tool for formulating closed-forms for WFOMC. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability.
• Score: 4.18804572788063
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed-forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.

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