# Differentiable Inference of Temporal Logic Formulas

arXiv: https://arxiv.org/abs/2208.05440 · HTML: https://mlsanity.com/paper/2208.05440/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 10 Aug 2022.

## Builds on (most cited first, 7 of 7)

- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [One Thousand and One Hours: Self-driving Motion Prediction Dataset](https://mlsanity.com/paper/2006.14480/) (2020)
- [Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective](https://mlsanity.com/paper/2003.00330/) (2020)
- [Backpropagation through Signal Temporal Logic Specifications: Infusing Logical Structure into Gradient-Based Methods](https://mlsanity.com/paper/2008.00097/) (2020)
- [Interpretable Classification of Time-Series Data using Efficient Enumerative Techniques](https://mlsanity.com/paper/1907.10265/) (2019)
- [Learning Linear Temporal Properties from Noisy Data: A MaxSAT Approach](https://mlsanity.com/paper/2104.15083/) (2021)
- [Assume/Guarantee Contracts for Dynamical Systems: Theory and Computational Tools](https://mlsanity.com/paper/2012.12657/) (2020)

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

- [Conformal Prediction for Signal Temporal Logic Inference](https://mlsanity.com/paper/2509.25473/) (2025)
- [Learning Probabilistic Temporal Logic Specifications for Stochastic Systems](https://mlsanity.com/paper/2505.12107/) (2025)
- [Joint Learning of Policy with Unknown Temporal Constraints for Safe Reinforcement Learning](https://mlsanity.com/paper/2305.00576/) (2023)
- [A Safe Preference Learning Approach for Personalization with Applications to Autonomous Vehicles](https://mlsanity.com/paper/2311.02099/) (2023)
- [On Synthesis of Metric Interval Temporal Logics](https://mlsanity.com/paper/2609.01032/) (2026)

## Abstract (arXiv)

We demonstrate the first Recurrent Neural Network architecture for learning Signal Temporal Logic formulas, and present the first systematic comparison of formula inference methods. Legacy systems embed much expert knowledge which is not explicitly formalized. There is great interest in learning formal specifications that characterize the ideal behavior of such systems -- that is, formulas in temporal logic that are satisfied by the system's output signals. Such specifications can be used to better understand the system's behavior and improve design of its next iteration. Previous inference methods either assumed certain formula templates, or did a heuristic enumeration of all possible templates. This work proposes a neural network architecture that infers the formula structure via gradient descent, eliminating the need for imposing any specific templates. It combines learning of formula structure and parameters in one optimization. Through systematic comparison, we demonstrate that this method achieves similar or better mis-classification rates (MCR) than enumerative and lattice methods. We also observe that different formulas can achieve similar MCR, empirically demonstrating the under-determinism of the problem of temporal logic inference.