# Backpropagation through Signal Temporal Logic Specifications: Infusing Logical Structure into Gradient-Based Methods

arXiv: https://arxiv.org/abs/2008.00097 · HTML: https://mlsanity.com/paper/2008.00097/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jul 2020.

## Built on by (most cited first, 30 of 54)

- [Guided Conditional Diffusion for Controllable Traffic Simulation](https://mlsanity.com/paper/2210.17366/) (2022)
- [Toward General-Purpose Robots via Foundation Models: A Survey and Meta-Analysis](https://mlsanity.com/paper/2312.08782/) (2023)
- [LTLDoG: Satisfying Temporally-Extended Symbolic Constraints for Safe Diffusion-based Planning](https://mlsanity.com/paper/2405.04235/) (2024)
- [Learning Signal Temporal Logic through Neural Network for Interpretable Classification](https://mlsanity.com/paper/2210.01910/) (2022)
- [TLINet: Differentiable Neural Network Temporal Logic Inference](https://mlsanity.com/paper/2405.06670/) (2024)
- [TeLoGraF: Temporal Logic Planning via Graph-encoded Flow Matching](https://mlsanity.com/paper/2505.00562/) (2025)
- [SafeVLA-Bench: A Benchmark for the Success-Safety Gap in Vision-Language-Action Models](https://mlsanity.com/paper/2606.00773/) (2026)
- [STLCG++: A Masking Approach for Differentiable Signal Temporal Logic Specification](https://mlsanity.com/paper/2501.04194/) (2025)
- [Diverse Controllable Diffusion Policy with Signal Temporal Logic](https://mlsanity.com/paper/2503.02924/) (2025)
- [Follow The Rules: Online Signal Temporal Logic Tree Search for Guided Imitation Learning in Stochastic Domains](https://mlsanity.com/paper/2209.13737/) (2022)
- [Receding Horizon Planning with Rule Hierarchies for Autonomous Vehicles](https://mlsanity.com/paper/2212.03323/) (2022)
- [Learning to Boost the Performance of Stable Nonlinear Systems](https://mlsanity.com/paper/2405.00871/) (2024)
- [Learning Robust and Correct Controllers from Signal Temporal Logic Specifications Using BarrierNet](https://mlsanity.com/paper/2304.06160/) (2023)
- [Zero-Shot Trajectory Planning for Signal Temporal Logic Tasks](https://mlsanity.com/paper/2501.13457/) (2025)
- [TGPO: Temporal Grounded Policy Optimization for Signal Temporal Logic Tasks](https://mlsanity.com/paper/2510.00225/) (2025)
- [Differentiable Inference of Temporal Logic Formulas](https://mlsanity.com/paper/2208.05440/) (2022)
- [Risk-Driven Design of Perception Systems](https://mlsanity.com/paper/2205.10677/) (2022)
- [Logically Constrained Robotics Transformers for Enhanced Perception-Action Planning](https://mlsanity.com/paper/2408.05336/) (2024)
- [Safe Model-based Control from Signal Temporal Logic Specifications Using Recurrent Neural Networks](https://mlsanity.com/paper/2103.15938/) (2021)
- [Multi-Predictor Fusion: Combining Learning-based and Rule-based Trajectory Predictors](https://mlsanity.com/paper/2307.01408/) (2023)
- [Learning Temporal Logic Predicates from Data with Statistical Guarantees](https://mlsanity.com/paper/2406.10449/) (2024)
- [Traffic-Rule-Compliant Trajectory Repair via Satisfiability Modulo Theories and Reachability Analysis](https://mlsanity.com/paper/2412.15837/) (2024)
- [Pretrained Embeddings as a Behavior Specification Mechanism](https://mlsanity.com/paper/2503.02012/) (2025)
- [Modal Logic Neural Networks](https://mlsanity.com/paper/2512.03491/) (2025)
- [Co-learning Planning and Control Policies Constrained by Differentiable Logic Specifications](https://mlsanity.com/paper/2303.01346/) (2023)
- [RuleFuser: An Evidential Bayes Approach for Rule Injection in Imitation Learned Planners and Predictors for Robustness under Distribution Shifts](https://mlsanity.com/paper/2405.11139/) (2024)
- [STeP: Signal Temporal Logic for Precise Specifications for Action Generation with Vision Language Models](https://mlsanity.com/paper/2607.18580/) (2026)
- [Learning Provably Robust Motion Planners Using Funnel Libraries](https://mlsanity.com/paper/2111.08733/) (2021)
- [Neural Controller Synthesis for Signal Temporal Logic Specifications Using Encoder-Decoder Structured Networks](https://mlsanity.com/paper/2212.05200/) (2022)
- [A Safe Preference Learning Approach for Personalization with Applications to Autonomous Vehicles](https://mlsanity.com/paper/2311.02099/) (2023)

## Abstract (arXiv)

This paper presents a technique, named STLCG, to compute the quantitative semantics of Signal Temporal Logic (STL) formulas using computation graphs. STLCG provides a platform which enables the incorporation of logical specifications into robotics problems that benefit from gradient-based solutions. Specifically, STL is a powerful and expressive formal language that can specify spatial and temporal properties of signals generated by both continuous and hybrid systems. The quantitative semantics of STL provide a robustness metric, i.e., how much a signal satisfies or violates an STL specification. In this work, we devise a systematic methodology for translating STL robustness formulas into computation graphs. With this representation, and by leveraging off-the-shelf automatic differentiation tools, we are able to efficiently backpropagate through STL robustness formulas and hence enable a natural and easy-to-use integration of STL specifications with many gradient-based approaches used in robotics. Through a number of examples stemming from various robotics applications, we demonstrate that STLCG is versatile, computationally efficient, and capable of incorporating human-domain knowledge into the problem formulation.