# Exact Learning of Arithmetic with Differentiable Agents

arXiv: https://arxiv.org/abs/2511.22751 · HTML: https://mlsanity.com/paper/2511.22751/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 44 references that are in the map.
- First posted on arXiv 27 Nov 2025.

## Builds on (most cited first, 30 of 34)

- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Neural Turing Machines](https://mlsanity.com/paper/1410.5401/) (2014)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [The Impact of Positional Encoding on Length Generalization in Transformers](https://mlsanity.com/paper/2305.19466/) (2023)
- [Theoretical Limitations of Self-Attention in Neural Sequence Models](https://mlsanity.com/paper/1906.06755/) (2019)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Neural Networks and the Chomsky Hierarchy](https://mlsanity.com/paper/2207.02098/) (2022)
- [Exploring Length Generalization in Large Language Models](https://mlsanity.com/paper/2207.04901/) (2022)
- [What Algorithms can Transformers Learn? A Study in Length Generalization](https://mlsanity.com/paper/2310.16028/) (2023)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [On the Turing Completeness of Modern Neural Network Architectures](https://mlsanity.com/paper/1901.03429/) (2019)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [Neural GPUs Learn Algorithms](https://mlsanity.com/paper/1511.08228/) (2015)
- [On the Practical Computational Power of Finite Precision RNNs for Language Recognition](https://mlsanity.com/paper/1805.04908/) (2018)
- [Neural Programmer-Interpreters](https://mlsanity.com/paper/1511.06279/) (2015)
- [Transformers Can Achieve Length Generalization But Not Robustly](https://mlsanity.com/paper/2402.09371/) (2024)
- [On the Computational Power of Transformers and its Implications in Sequence Modeling](https://mlsanity.com/paper/2006.09286/) (2020)
- [Inferring Algorithmic Patterns with Stack-Augmented Recurrent Nets](https://mlsanity.com/paper/1503.01007/) (2015)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)
- [Positional Description Matters for Transformers Arithmetic](https://mlsanity.com/paper/2311.14737/) (2023)
- [Learning to Transduce with Unbounded Memory](https://mlsanity.com/paper/1506.02516/) (2015)
- [Sequential Neural Networks as Automata](https://mlsanity.com/paper/1906.01615/) (2019)
- [Neural Random-Access Machines](https://mlsanity.com/paper/1511.06392/) (2015)
- [Universal Length Generalization with Turing Programs](https://mlsanity.com/paper/2407.03310/) (2024)
- [Position Coupling: Improving Length Generalization of Arithmetic Transformers Using Task Structure](https://mlsanity.com/paper/2405.20671/) (2024)
- [Arithmetic Transformers Can Length-Generalize in Both Operand Length and Count](https://mlsanity.com/paper/2410.15787/) (2024)
- [Autoregressive Large Language Models are Computationally Universal](https://mlsanity.com/paper/2410.03170/) (2024)
- [Learning Algorithms in the Limit](https://mlsanity.com/paper/2506.15543/) (2025)

## Built on by (most cited first, 1 of 1)

- [Certification from Examples is Hard for Circuits and Transformers under Minimal Overparametrization](https://mlsanity.com/paper/2605.22964/) (2026)

## Abstract (arXiv)

We explore the possibility of exact algorithmic learning with gradient-based methods and introduce a differentiable framework capable of strong length generalization on arithmetic tasks. Our approach centers on Differentiable Finite-State Transducers (DFSTs), a Turing-complete model family that avoids the pitfalls of prior architectures by enabling constant-precision, constant-time generation, and end-to-end log-parallel differentiable training. Leveraging policy-trajectory observations from expert agents, we train DFSTs to perform binary and decimal addition and multiplication. Remarkably, models trained on tiny datasets generalize without error to inputs thousands of times longer than the training examples. These results show that training differentiable agents on structured intermediate supervision could pave the way towards exact gradient-based learning of algorithmic skills. Code available at hrefhttps://github.com/dngfra/differentiable-exact-algorithmic-learner.githttps://github.com/dngfra/differentiable-exact-algorithmic-learner.git.