# LEMMA: Bootstrapping High-Level Mathematical Reasoning with Learned Symbolic Abstractions

arXiv: https://arxiv.org/abs/2211.08671 · HTML: https://mlsanity.com/paper/2211.08671/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 8 of its 17 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 8 of 8)

- [Thinking Fast and Slow with Deep Learning and Tree Search](https://mlsanity.com/paper/1705.08439/) (2017)
- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)
- [DeepCoder: Learning to Write Programs](https://mlsanity.com/paper/1611.01989/) (2016)
- [Formal Mathematics Statement Curriculum Learning](https://mlsanity.com/paper/2202.01344/) (2022)
- [Reinforcement Learning of Theorem Proving](https://mlsanity.com/paper/1805.07563/) (2018)
- [Leveraging Language to Learn Program Abstractions and Search Heuristics](https://mlsanity.com/paper/2106.11053/) (2021)
- [TacticZero: Learning to Prove Theorems from Scratch with Deep Reinforcement Learning](https://mlsanity.com/paper/2102.09756/) (2021)
- [Contrastive Reinforcement Learning of Symbolic Reasoning Domains](https://mlsanity.com/paper/2106.09146/) (2021)

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

- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [AI for Mathematics: A Cognitive Science Perspective](https://mlsanity.com/paper/2310.13021/) (2023)
- [When Do Skills Help Reinforcement Learning? A Theoretical Analysis of Temporal Abstractions](https://mlsanity.com/paper/2406.07897/) (2024)
- [MathDSL: A Domain-Specific Language for Concise Mathematical Solutions Via Program Synthesis](https://mlsanity.com/paper/2409.17490/) (2024)
- [Reinforcement Learning for Symbolic Equation Solving](https://mlsanity.com/paper/2608.30162/) (2026)

## Abstract (arXiv)

Humans tame the complexity of mathematical reasoning by developing hierarchies of abstractions. With proper abstractions, solutions to hard problems can be expressed concisely, thus making them more likely to be found. In this paper, we propose Learning Mathematical Abstractions (LEMMA): an algorithm that implements this idea for reinforcement learning agents in mathematical domains. LEMMA augments Expert Iteration with an abstraction step, where solutions found so far are revisited and rewritten in terms of new higher-level actions, which then become available to solve new problems. We evaluate LEMMA on two mathematical reasoning tasks--equation solving and fraction simplification--in a step-by-step fashion. In these two domains, LEMMA improves the ability of an existing agent, both solving more problems and generalizing more effectively to harder problems than those seen during training.