# Weighted model counting beyond two-variable logic

arXiv: https://arxiv.org/abs/1804.10185 · HTML: https://mlsanity.com/paper/1804.10185/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2018.

## Built on by (most cited first, 13 of 13)

- [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)
- [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)
- [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)
- [Polynomial-Time Relational Probabilistic Inference in Open Universes](https://mlsanity.com/paper/2505.04115/) (2025)
- [Weighted First Order Model Counting for Two-variable Logic with Axioms on Two Relations](https://mlsanity.com/paper/2508.11515/) (2025)
- [Solving Combinatorial Counting Problems with Weighted First-Order Model Counting](https://mlsanity.com/paper/2605.24845/) (2026)

## Abstract (arXiv)

It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO2 is in polynomial time, while it is Sharp-P1 complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form "phi and forallexists^=1 psi" with phi and psi in FO2, and to sentences formulated in the uniform one-dimensional fragment of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. Note that the former generalizes the extension of FO2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or Sharp-P1 complete.