# Learning Approximate and Exact Numeral Systems via Reinforcement Learning

arXiv: https://arxiv.org/abs/2105.13857 · HTML: https://mlsanity.com/paper/2105.13857/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 30 references that are in the map.
- First posted on arXiv 28 May 2021.

## Builds on (most cited first, 3 of 3)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Color naming reflects both perceptual structure and communicative need](https://mlsanity.com/paper/1805.06165/) (2018)

## Built on by (most cited first, 6 of 6)

- [Recursive numeral systems are highly regular and easy to process](https://mlsanity.com/paper/2510.27049/) (2025)
- [Learning Efficient Recursive Numeral Systems via Reinforcement Learning](https://mlsanity.com/paper/2409.07170/) (2024)
- [PACE: Procedural Abstractions for Communicating Efficiently](https://mlsanity.com/paper/2409.20120/) (2024)
- [Pragmatic Reasoning in Structured Signaling Games](https://mlsanity.com/paper/2305.10167/) (2023)
- [Evaluating the relationship between regularity and learnability in recursive numeral systems using Reinforcement Learning](https://mlsanity.com/paper/2602.21720/) (2026)
- [Agent-based imitation dynamics can yield efficiently compressed population-level vocabularies](https://mlsanity.com/paper/2603.15903/) (2026)

## Abstract (arXiv)

Recent work (Xu et al., 2020) has suggested that numeral systems in different languages are shaped by a functional need for efficient communication in an information-theoretic sense. Here we take a learning-theoretic approach and show how efficient communication emerges via reinforcement learning. In our framework, two artificial agents play a Lewis signaling game where the goal is to convey a numeral concept. The agents gradually learn to communicate using reinforcement learning and the resulting numeral systems are shown to be efficient in the information-theoretic framework of Regier et al. (2015); Gibson et al. (2017). They are also shown to be similar to human numeral systems of same type. Our results thus provide a mechanistic explanation via reinforcement learning of the recent results in Xu et al. (2020) and can potentially be generalized to other semantic domains.