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. In this paper, we reconstruct the closed-form formula for polynomial-time First Order Model Counting (FOMC) in the universally quantified fragment of FO\(^2\), earlier proposed by Beame et al.. We then expand this closed-form to incorporate cardinality constraints and existential quantifiers. Our approach requires a constant time (w.r.t the previous linear time result) for handling equality and allows us to handle cardinality constraints in a completely combinatorial fashion. 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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Notes
- 1.
The list includes atoms of the form P(x, x) for binary predicate P.
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We thank Alessandro Daniele for the fruitful insights and for reviewing the proofs.
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Malhotra, S., Serafini, L. (2022). A Combinatorial Approach to Weighted Model Counting in the Two-Variable Fragment with Cardinality Constraints. In: Bandini, S., Gasparini, F., Mascardi, V., Palmonari, M., Vizzari, G. (eds) AIxIA 2021 – Advances in Artificial Intelligence. AIxIA 2021. Lecture Notes in Computer Science(), vol 13196. Springer, Cham. https://doi.org/10.1007/978-3-031-08421-8_10
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