# Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers

arXiv: https://arxiv.org/abs/2007.05619 · HTML: https://mlsanity.com/paper/2007.05619/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2020.

## Built on by (most cited first, 16 of 16)

- [Lifted Inference with Linear Order Axiom](https://mlsanity.com/paper/2211.01164/) (2022)
- [Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula](https://mlsanity.com/paper/2110.05992/) (2021)
- [Lifted Algorithms for Symmetric Weighted First-Order Model Sampling](https://mlsanity.com/paper/2308.08828/) (2023)
- [Lifted Inference beyond First-Order Logic](https://mlsanity.com/paper/2308.11738/) (2023)
- [Bridging Weighted First Order Model Counting and Graph Polynomials](https://mlsanity.com/paper/2407.11877/) (2024)
- [On Exact Sampling in the Two-Variable Fragment of First-Order Logic](https://mlsanity.com/paper/2302.02730/) (2023)
- [Faster Lifting for Ordered Domains with Predecessor Relations](https://mlsanity.com/paper/2507.19182/) (2025)
- [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)
- [Polynomial-Time Relational Probabilistic Inference in Open Universes](https://mlsanity.com/paper/2505.04115/) (2025)
- [Model Enumeration of Two-Variable Logic with Quadratic Delay Complexity](https://mlsanity.com/paper/2505.19648/) (2025)
- [Tractable Weighted First-Order Model Counting with Bounded Treewidth Binary Evidence](https://mlsanity.com/paper/2511.09174/) (2025)
- [A Fast Model Counting Algorithm for Two-Variable Logic with Counting and Modulo Counting Quantifiers](https://mlsanity.com/paper/2605.03391/) (2026)
- [Solving Combinatorial Counting Problems with Weighted First-Order Model Counting](https://mlsanity.com/paper/2605.24845/) (2026)
- [CombEval: A Framework for Evaluating Combinatorial Counting in Large Language Models](https://mlsanity.com/paper/2606.19788/) (2026)

## Abstract (arXiv)

It is known due to the work of Van den Broeck et al [KR, 2014] that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in the number of domain elements. In this paper we extend this result to the two-variable fragment with counting quantifiers.