# Exploiting T-norms for Deep Learning in Autonomous Driving

arXiv: https://arxiv.org/abs/2402.11362 · HTML: https://mlsanity.com/paper/2402.11362/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 33 references that are in the map.
- First posted on arXiv 17 Feb 2024.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [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)
- [ROAD: The ROad event Awareness Dataset for Autonomous Driving](https://mlsanity.com/paper/2102.11585/) (2021)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Deep Learning with Logical Constraints](https://mlsanity.com/paper/2205.00523/) (2022)
- [Autonomous Driving with Deep Learning: A Survey of State-of-Art Technologies](https://mlsanity.com/paper/2006.06091/) (2020)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [ROAD-R: The Autonomous Driving Dataset with Logical Requirements](https://mlsanity.com/paper/2210.01597/) (2022)
- [Neuro-Symbolic Entropy Regularization](https://mlsanity.com/paper/2201.11250/) (2022)
- [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)
- [Refining neural network predictions using background knowledge](https://mlsanity.com/paper/2206.04976/) (2022)
- [Traffic Prediction Based on Random Connectivity in Deep Learning with Long Short-Term Memory](https://mlsanity.com/paper/1711.02833/) (2017)

## Built on by (most cited first, 4 of 4)

- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [PiShield: A PyTorch Package for Learning with Requirements](https://mlsanity.com/paper/2402.18285/) (2024)
- [Deep Generative Models with Hard Linear Equality Constraints](https://mlsanity.com/paper/2502.05416/) (2025)
- [Integrating Legal and Logical Specifications in Perception, Prediction, and Planning for Automated Driving: A Survey of Methods](https://mlsanity.com/paper/2510.25386/) (2025)

## Abstract (arXiv)

Deep learning has been at the core of the autonomous driving field development, due to the neural networks' success in finding patterns in raw data and turning them into accurate predictions. Moreover, recent neuro-symbolic works have shown that incorporating the available background knowledge about the problem at hand in the loss function via t-norms can further improve the deep learning models' performance. However, t-norm-based losses may have very high memory requirements and, thus, they may be impossible to apply in complex application domains like autonomous driving. In this paper, we show how it is possible to define memory-efficient t-norm-based losses, allowing for exploiting t-norms for the task of event detection in autonomous driving. We conduct an extensive experimental analysis on the ROAD-R dataset and show (i) that our proposal can be implemented and run on GPUs with less than 25 GiB of available memory, while standard t-norm-based losses are estimated to require more than 100 GiB, far exceeding the amount of memory normally available, (ii) that t-norm-based losses improve performance, especially when limited labelled data are available, and (iii) that t-norm-based losses can further improve performance when exploited on both labelled and unlabelled data.