# Probabilistic Model Counting with Short XORs

arXiv: https://arxiv.org/abs/1707.09467 · HTML: https://mlsanity.com/paper/1707.09467/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2017.

## Built on by (most cited first, 5 of 5)

- [Projected Model Counting: Beyond Independent Support](https://mlsanity.com/paper/2110.09171/) (2021)
- [Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration With Provable Guarantees](https://mlsanity.com/paper/2309.08883/) (2023)
- [Learning Markov Random Fields for Combinatorial Structures via Sampling through Lov'asz Local Lemma](https://mlsanity.com/paper/2212.00296/) (2022)
- [Solving Satisfiability Modulo Counting Exactly with Probabilistic Circuits](https://mlsanity.com/paper/2503.01009/) (2025)
- [Approximate SMT Counting Beyond Discrete Domains](https://mlsanity.com/paper/2507.18612/) (2025)

## Abstract (arXiv)

The idea of counting the number of satisfying truth assignments (models) of a formula by adding random parity constraints can be traced back to the seminal work of Valiant and Vazirani, showing that NP is as easy as detecting unique solutions. While theoretically sound, the random parity constraints in that construction have the following drawback: each constraint, on average, involves half of all variables. As a result, the branching factor associated with searching for models that also satisfy the parity constraints quickly gets out of hand. In this work we prove that one can work with much shorter parity constraints and still get rigorous mathematical guarantees, especially when the number of models is large so that many constraints need to be added. Our work is based on the realization that the essential feature for random systems of parity constraints to be useful in probabilistic model counting is that the geometry of their set of solutions resembles an error-correcting code.