# A Fast Model Counting Algorithm for Two-Variable Logic with Counting and Modulo Counting Quantifiers

arXiv: https://arxiv.org/abs/2605.03391 · HTML: https://mlsanity.com/paper/2605.03391/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 47 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 21 of 21)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Scalable Approximate Model Counter](https://mlsanity.com/paper/1306.5726/) (2013)
- [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)
- [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)
- [Domain Recursion for Lifted Inference with Existential Quantifiers](https://mlsanity.com/paper/1707.07763/) (2017)
- [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)
- [Uniform generation of random regular graphs](https://mlsanity.com/paper/1511.01175/) (2015)
- [Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat](https://mlsanity.com/paper/2404.12905/) (2024)
- [Towards a more efficient approach for the satisfiability of two-variable logic](https://mlsanity.com/paper/2104.10621/) (2021)
- [Faster Lifting for Ordered Domains with Predecessor Relations](https://mlsanity.com/paper/2507.19182/) (2025)
- [Regular graphs and the spectra of two-variable logic with counting](https://mlsanity.com/paper/1304.0829/) (2013)
- [Modulo Counting on Words and Trees](https://mlsanity.com/paper/1710.05582/) (2017)
- [Two variable logic with ultimately periodic counting](https://mlsanity.com/paper/2006.01193/) (2020)
- [Towards Practical First-Order Model Counting](https://mlsanity.com/paper/2502.12278/) (2025)
- [Model Enumeration of Two-Variable Logic with Quadratic Delay Complexity](https://mlsanity.com/paper/2505.19648/) (2025)

## Abstract (arXiv)

Weighted first-order model counting (WFOMC) is a central task in lifted probabilistic inference: It asks for the weighted sum of all models of a first-order sentence over a finite domain. A long line of work has identified domain-liftable fragments of first-order logic, that is, syntactic classes for which WFOMC can be solved in time polynomial in the domain size. Among them, the two-variable fragment with counting quantifiers, C2, is one of the most expressive known liftable fragments. Existing algorithms for C2, however, establish tractability through multi-stage reductions that eliminate counting quantifiers via cardinality constraints, which introduces substantial practical overhead as the domain size grows. In this paper, we introduce IncrementalWFOMC3, a lifted algorithm for WFOMC on C2 and its modulo counting extension, C2mod. Instead of relying on reduction techniques, IncrementalWFOMC3 operates directly on a Scott normal form that retains counting quantifiers throughout inference. This direct treatment yields two main results. First, we derive a tighter data-complexity bound for WFOMC in C2, reducing the degree of the polynomial from quadratic to linear in the counting parameters. Second, we prove that C2mod is domain-liftable, extending tractability from C2 to a richer fragment with native modulo counting support. Finally, our empirical evaluation shows that IncrementalWFOMC3 delivers orders-of-magnitude runtime improvements and better scalability than both existing WFOMC algorithms and state-of-the-art propositional model counters.