# Differentiable Logic Machines

arXiv: https://arxiv.org/abs/2102.11529 · HTML: https://mlsanity.com/paper/2102.11529/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 47 references that are in the map.
- First posted on arXiv 23 Feb 2021.

## Builds on (most cited first, 25 of 25)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Hindsight Experience Replay](https://mlsanity.com/paper/1707.01495/) (2017)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding](https://mlsanity.com/paper/1810.02338/) (2018)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Neural Logic Machines](https://mlsanity.com/paper/1904.11694/) (2019)
- [SDRL: Interpretable and Data-efficient Deep Reinforcement Learning Leveraging Symbolic Planning](https://mlsanity.com/paper/1811.00090/) (2018)
- [From Statistical Relational to Neuro-Symbolic Artificial Intelligence](https://mlsanity.com/paper/2003.08316/) (2020)
- [Neural Logic Reinforcement Learning](https://mlsanity.com/paper/1904.10729/) (2019)
- [Turning 30: New Ideas in Inductive Logic Programming](https://mlsanity.com/paper/2002.11002/) (2020)
- [Learn to Explain Efficiently via Neural Logic Inductive Learning](https://mlsanity.com/paper/1910.02481/) (2019)
- [Inductive Logic Programming via Differentiable Deep Neural Logic Networks](https://mlsanity.com/paper/1906.03523/) (2019)
- [Inductive Learning of Answer Set Programs from Noisy Examples](https://mlsanity.com/paper/1808.08441/) (2018)
- [Learning Algorithms via Neural Logic Networks](https://mlsanity.com/paper/1904.01554/) (2019)
- [Incorporating Relational Background Knowledge into Reinforcement Learning via Differentiable Inductive Logic Programming](https://mlsanity.com/paper/2003.10386/) (2020)

## Built on by (most cited first, 18 of 18)

- [A Survey on Interpretable Reinforcement Learning](https://mlsanity.com/paper/2112.13112/) (2021)
- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [Neuro-Symbolic Hierarchical Rule Induction](https://mlsanity.com/paper/2112.13418/) (2021)
- [Learning Interpretable Rules for Scalable Data Representation and Classification](https://mlsanity.com/paper/2310.14336/) (2023)
- [DeiSAM: Segment Anything with Deictic Prompting](https://mlsanity.com/paper/2402.14123/) (2024)
- [Comprehension Without Competence: Architectural Limits of LLMs in Symbolic Computation and Reasoning](https://mlsanity.com/paper/2507.10624/) (2025)
- [R5: Rule Discovery with Reinforced and Recurrent Relational Reasoning](https://mlsanity.com/paper/2205.06454/) (2022)
- [Neuro-symbolic Rule Learning in Real-world Classification Tasks](https://mlsanity.com/paper/2303.16674/) (2023)
- [From Data to Commonsense Reasoning: The Use of Large Language Models for Explainable AI](https://mlsanity.com/paper/2407.03778/) (2024)
- [LogicCBMs: Logic-Enhanced Concept-Based Learning](https://mlsanity.com/paper/2512.07383/) (2025)
- [Neural DNF-MT: A Neuro-symbolic Approach for Learning Interpretable and Editable Policies](https://mlsanity.com/paper/2501.03888/) (2025)
- [A Survey on Explainable Reinforcement Learning: Concepts, Algorithms, Challenges](https://mlsanity.com/paper/2211.06665/) (2022)
- [ShortCircuit: AlphaZero-Driven Circuit Design](https://mlsanity.com/paper/2408.09858/) (2024)
- [Are Large Language Models Capable of Deep Relational Reasoning? Insights from DeepSeek-R1 and Benchmark Comparisons](https://mlsanity.com/paper/2506.23128/) (2025)
- [PRIMA: Planner-Reasoner Inside a Multi-task Reasoning Agent](https://mlsanity.com/paper/2202.00531/) (2022)
- [ESC-Rules: Explainable, Semantically Constrained Rule Sets](https://mlsanity.com/paper/2208.12523/) (2022)
- [Symbolic Imitation Learning: From Black-Box to Explainable Driving Policies](https://mlsanity.com/paper/2309.16025/) (2023)
- [From Chain to Tree: Refining Chain-like Rules into Tree-like Rules on Knowledge Graphs](https://mlsanity.com/paper/2403.05130/) (2024)

## Abstract (arXiv)

The integration of reasoning, learning, and decision-making is key to build more general artificial intelligence systems. As a step in this direction, we propose a novel neural-logic architecture, called differentiable logic machine (DLM), that can solve both inductive logic programming (ILP) and reinforcement learning (RL) problems, where the solution can be interpreted as a first-order logic program. Our proposition includes several innovations. Firstly, our architecture defines a restricted but expressive continuous relaxation of the space of first-order logic programs by assigning weights to predicates instead of rules, in contrast to most previous neural-logic approaches. Secondly, with this differentiable architecture, we propose several (supervised and RL) training procedures, based on gradient descent, which can recover a fully-interpretable solution (i.e., logic formula). Thirdly, to accelerate RL training, we also design a novel critic architecture that enables actor-critic algorithms. Fourthly, to solve hard problems, we propose an incremental training procedure that can learn a logic program progressively. Compared to state-of-the-art (SOTA) differentiable ILP methods, DLM successfully solves all the considered ILP problems with a higher percentage of successful seeds (up to 3.5×). On RL problems, without requiring an interpretable solution, DLM outperforms other non-interpretable neural-logic RL approaches in terms of rewards (up to 3.9%). When enforcing interpretability, DLM can solve harder RL problems (e.g., Sorting, Path) Moreover, we show that deep logic programs can be learned via incremental supervised training. In addition to this excellent performance, DLM can scale well in terms of memory and computational time, especially during the testing phase where it can deal with much more constants (>2×) than SOTA.