# Contrastive Reinforcement Learning of Symbolic Reasoning Domains

arXiv: https://arxiv.org/abs/2106.09146 · HTML: https://mlsanity.com/paper/2106.09146/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 34 references that are in the map.
- First posted on arXiv 16 Jun 2021.

## Builds on (most cited first, 11 of 11)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [CURL: Contrastive Unsupervised Representations for Reinforcement Learning](https://mlsanity.com/paper/2004.04136/) (2020)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Learning to Prove Theorems via Interacting with Proof Assistants](https://mlsanity.com/paper/1905.09381/) (2019)
- [Deep Reinforcement Learning with a Natural Language Action Space](https://mlsanity.com/paper/1511.04636/) (2015)
- [DeepMath - Deep Sequence Models for Premise Selection](https://mlsanity.com/paper/1606.04442/) (2016)
- [Reinforcement Learning of Theorem Proving](https://mlsanity.com/paper/1805.07563/) (2018)
- [Conditional Negative Sampling for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2010.02037/) (2020)
- [Towards Effective Context for Meta-Reinforcement Learning: an Approach based on Contrastive Learning](https://mlsanity.com/paper/2009.13891/) (2020)
- [ATPboost: Learning Premise Selection in Binary Setting with ATP Feedback](https://mlsanity.com/paper/1802.03375/) (2018)

## Built on by (most cited first, 17 of 17)

- [Think or Not? Exploring Thinking Efficiency in Large Reasoning Models via an Information-Theoretic Lens](https://mlsanity.com/paper/2505.18237/) (2025)
- [Simplifying Polylogarithms with Machine Learning](https://mlsanity.com/paper/2206.04115/) (2022)
- [Peano: Learning Formal Mathematical Reasoning](https://mlsanity.com/paper/2211.15864/) (2022)
- [Assistive Teaching of Motor Control Tasks to Humans](https://mlsanity.com/paper/2211.14003/) (2022)
- [RLET: A Reinforcement Learning Based Approach for Explainable QA with Entailment Trees](https://mlsanity.com/paper/2210.17095/) (2022)
- [Contrastive Representations for Temporal Reasoning](https://mlsanity.com/paper/2508.13113/) (2025)
- [LEMMA: Bootstrapping High-Level Mathematical Reasoning with Learned Symbolic Abstractions](https://mlsanity.com/paper/2211.08671/) (2022)
- [Symbolic Equation Solving via Reinforcement Learning](https://mlsanity.com/paper/2401.13447/) (2024)
- [SEER: Facilitating Structured Reasoning and Explanation via Reinforcement Learning](https://mlsanity.com/paper/2401.13246/) (2024)
- [When Do Skills Help Reinforcement Learning? A Theoretical Analysis of Temporal Abstractions](https://mlsanity.com/paper/2406.07897/) (2024)
- [Application of AI to formal methods - an analysis of current trends](https://mlsanity.com/paper/2411.14870/) (2024)
- [Using Contrastive Samples for Identifying and Leveraging Possible Causal Relationships in Reinforcement Learning](https://mlsanity.com/paper/2210.17296/) (2022)
- [Gradient Boosting Reinforcement Learning](https://mlsanity.com/paper/2407.08250/) (2024)
- [MathDSL: A Domain-Specific Language for Concise Mathematical Solutions Via Program Synthesis](https://mlsanity.com/paper/2409.17490/) (2024)
- [Decomposed Inductive Procedure Learning: Learning Academic Tasks with Human-Like Data Efficiency](https://mlsanity.com/paper/2505.10422/) (2025)
- [Curiosity-driven RL for symbolic equation solving](https://mlsanity.com/paper/2510.17022/) (2025)
- [Reinforcement Learning for Symbolic Equation Solving](https://mlsanity.com/paper/2608.30162/) (2026)

## Abstract (arXiv)

Abstract symbolic reasoning, as required in domains such as mathematics and logic, is a key component of human intelligence. Solvers for these domains have important applications, especially to computer-assisted education. But learning to solve symbolic problems is challenging for machine learning algorithms. Existing models either learn from human solutions or use hand-engineered features, making them expensive to apply in new domains. In this paper, we instead consider symbolic domains as simple environments where states and actions are given as unstructured text, and binary rewards indicate whether a problem is solved. This flexible setup makes it easy to specify new domains, but search and planning become challenging. We introduce four environments inspired by the Mathematics Common Core Curriculum, and observe that existing Reinforcement Learning baselines perform poorly. We then present a novel learning algorithm, Contrastive Policy Learning (ConPoLe) that explicitly optimizes the InfoNCE loss, which lower bounds the mutual information between the current state and next states that continue on a path to the solution. ConPoLe successfully solves all four domains. Moreover, problem representations learned by ConPoLe enable accurate prediction of the categories of problems in a real mathematics curriculum. Our results suggest new directions for reinforcement learning in symbolic domains, as well as applications to mathematics education.