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

arXiv: https://arxiv.org/abs/2110.05992 · HTML: https://mlsanity.com/paper/2110.05992/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 12 Oct 2021.

## Builds on (most cited first, 6 of 6)

- [Symmetric Weighted First-Order Model Counting](https://mlsanity.com/paper/1412.1505/) (2014)
- [Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers](https://mlsanity.com/paper/2007.05619/) (2020)
- [Skolemization for Weighted First-Order Model Counting](https://mlsanity.com/paper/1312.5378/) (2013)
- [New Liftable Classes for First-Order Probabilistic Inference](https://mlsanity.com/paper/1610.08445/) (2016)
- [Weighted model counting beyond two-variable logic](https://mlsanity.com/paper/1804.10185/) (2018)
- [On the uniform one-dimensional fragment](https://mlsanity.com/paper/1604.01673/) (2016)

## Built on by (most cited first, 8 of 8)

- [Lifted Inference with Linear Order Axiom](https://mlsanity.com/paper/2211.01164/) (2022)
- [On Projectivity in Markov Logic Networks](https://mlsanity.com/paper/2204.04009/) (2022)
- [Lifted Inference beyond First-Order Logic](https://mlsanity.com/paper/2308.11738/) (2023)
- [Weighted First Order Model Counting with Directed Acyclic Graph Axioms](https://mlsanity.com/paper/2302.09830/) (2023)
- [Synthesising Recursive Functions for First-Order Model Counting: Challenges, Progress, and Conjectures](https://mlsanity.com/paper/2306.04189/) (2023)
- [Towards Practical First-Order Model Counting](https://mlsanity.com/paper/2502.12278/) (2025)
- [The generalised distribution semantics and projective families of distributions](https://mlsanity.com/paper/2211.06751/) (2022)
- [A Fast Model Counting Algorithm for Two-Variable Logic with Counting and Modulo Counting Quantifiers](https://mlsanity.com/paper/2605.03391/) (2026)

## Abstract (arXiv)

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.