# Deep Learning with Logical Constraints

arXiv: https://arxiv.org/abs/2205.00523 · HTML: https://mlsanity.com/paper/2205.00523/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 67 references that are in the map.
- First posted on arXiv 1 May 2022.

## Builds on (most cited first, 21 of 21)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Analyzing Differentiable Fuzzy Logic Operators](https://mlsanity.com/paper/2002.06100/) (2020)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [Incorporating Domain Knowledge into Deep Neural Networks](https://mlsanity.com/paper/2103.00180/) (2021)
- [Embedding Symbolic Knowledge into Deep Networks](https://mlsanity.com/paper/1909.01161/) (2019)
- [Adversarially Regularising Neural NLI Models to Integrate Logical Background Knowledge](https://mlsanity.com/paper/1808.08609/) (2018)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [On Tractable Representations of Binary Neural Networks](https://mlsanity.com/paper/2004.02082/) (2020)
- [Label Relation Graphs Enhanced Hierarchical Residual Network for Hierarchical Multi-Granularity Classification](https://mlsanity.com/paper/2201.03194/) (2022)
- [DeepGO: Predicting protein functions from sequence and interactions using a deep ontology-aware classifier](https://mlsanity.com/paper/1705.05919/) (2017)
- [Analyzing Differentiable Fuzzy Implications](https://mlsanity.com/paper/2006.03472/) (2020)
- [Neuro-Symbolic Constraint Programming for Structured Prediction](https://mlsanity.com/paper/2103.17232/) (2021)

## Built on by (most cited first, 30 of 44)

- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts](https://mlsanity.com/paper/2305.19951/) (2023)
- [A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference](https://mlsanity.com/paper/2212.12393/) (2022)
- [BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts](https://mlsanity.com/paper/2402.12240/) (2024)
- [Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing](https://mlsanity.com/paper/2210.15889/) (2022)
- [ROAD-R: The Autonomous Driving Dataset with Logical Requirements](https://mlsanity.com/paper/2210.01597/) (2022)
- [A Neuro-Symbolic Benchmark Suite for Concept Quality and Reasoning Shortcuts](https://mlsanity.com/paper/2406.10368/) (2024)
- [On the Independence Assumption in Neurosymbolic Learning](https://mlsanity.com/paper/2404.08458/) (2024)
- [LOGICSEG: Parsing Visual Semantics with Neural Logic Learning and Reasoning](https://mlsanity.com/paper/2309.13556/) (2023)
- [Predicated Diffusion: Predicate Logic-Based Attention Guidance for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2311.16117/) (2023)
- [Logic of Differentiable Logics: Towards a Uniform Semantics of DL](https://mlsanity.com/paper/2303.10650/) (2023)
- [A Survey on Deep Learning Approaches for Tabular Data Generation: Utility, Alignment, Fidelity, Privacy, Diversity, and Beyond](https://mlsanity.com/paper/2503.05954/) (2025)
- [Vehicle: Bridging the Embedding Gap in the Verification of Neuro-Symbolic Programs](https://mlsanity.com/paper/2401.06379/) (2024)
- [Neural Network Verification is a Programming Language Challenge](https://mlsanity.com/paper/2501.05867/) (2025)
- [Refining neural network predictions using background knowledge](https://mlsanity.com/paper/2206.04976/) (2022)
- [Learning with Logical Constraints but without Shortcut Satisfaction](https://mlsanity.com/paper/2403.00329/) (2024)
- [ULLER: A Unified Language for Learning and Reasoning](https://mlsanity.com/paper/2405.00532/) (2024)
- [Hierarchical Interdisciplinary Topic Detection Model for Research Proposal Classification](https://mlsanity.com/paper/2209.13519/) (2022)
- [Naturally Supervised 3D Visual Grounding with Language-Regularized Concept Learners](https://mlsanity.com/paper/2404.19696/) (2024)
- [Concept-Centric Transformers: Enhancing Model Interpretability through Object-Centric Concept Learning within a Shared Global Workspace](https://mlsanity.com/paper/2305.15775/) (2023)
- [EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic](https://mlsanity.com/paper/2408.08133/) (2024)
- [Exploiting T-norms for Deep Learning in Autonomous Driving](https://mlsanity.com/paper/2402.11362/) (2024)
- [Improving deep learning with prior knowledge and cognitive models: A survey on enhancing explainability, adversarial robustness and zero-shot learning](https://mlsanity.com/paper/2403.07078/) (2024)
- [Semantic Objective Functions: A distribution-aware method for adding logical constraints in deep learning](https://mlsanity.com/paper/2405.15789/) (2024)
- [stl2vec: Semantic and Interpretable Vector Representation of Temporal Logic](https://mlsanity.com/paper/2405.14389/) (2024)
- [ABS: Enforcing Constraint Satisfaction On Generated Sequences Via Automata-Guided Beam Search](https://mlsanity.com/paper/2506.09701/) (2025)
- [Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI](https://mlsanity.com/paper/2510.25497/) (2025)
- [To Neuro-Symbolic Classification and Beyond by Compiling Description Logic Ontologies to Probabilistic Circuits](https://mlsanity.com/paper/2601.14894/) (2026)
- [PiShield: A PyTorch Package for Learning with Requirements](https://mlsanity.com/paper/2402.18285/) (2024)
- [Informed deep hierarchical classification: a non-standard analysis inspired approach](https://mlsanity.com/paper/2409.16956/) (2024)

## Abstract (arXiv)

In recent years, there has been an increasing interest in exploiting logically specified background knowledge in order to obtain neural models (i) with a better performance, (ii) able to learn from less data, and/or (iii) guaranteed to be compliant with the background knowledge itself, e.g., for safety-critical applications. In this survey, we retrace such works and categorize them based on (i) the logical language that they use to express the background knowledge and (ii) the goals that they achieve.