pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems

2026-06-17

AI Summary

This work addresses the challenge of enabling autonomous robots to simultaneously satisfy complex temporal tasks and safety requirements in uncertain environments, where existing approaches are either non-differentiable or neglect uncertainty in belief space. The paper introduces a differentiable probabilistic Signal Temporal Logic (pdSTL) framework that unifies probabilistic semantics with differentiable robustness for the first time. It computes conservative satisfaction bounds via interval-valued probability semantics and unfolds STL operators through an LSTM-like recursive structure, achieving linear-time, end-to-end differentiable monitoring and optimization. This approach enables safe policy optimization over belief trajectories with formal probabilistic guarantees. Evaluations in simulated obstacle avoidance and lane-changing scenarios, as well as real-world experiments on a Crazyflie quadrotor, demonstrate that pdSTL significantly improves both safety margins and optimization efficiency compared to deterministic differentiable STL.

Abstract

Autonomous robots operating in uncertain environments must satisfy complex temporal and safety specifications despite stochastic dynamics and sensing noise. While Signal Temporal Logic (STL) offers robustness measures for gradient-based optimization, existing extensions either lack differentiability or ignore belief-space uncertainty. We introduce pdSTL (probabilistic differentiable Signal Temporal Logic), a framework that unifies probabilistic semantics with differentiable robustness over belief trajectories. pdSTL employs interval-valued probabilistic semantics to compute conservative satisfaction bounds, propagated compositionally through the STL syntax tree. We formulate the temporal robustness evaluation as a recurrent, LSTM-style unfolding of STL operators, enabling linear-time, differentiable monitoring suitable for end-to-end trajectory optimization. We validate pdSTL on simulated obstacle avoidance, lane-change maneuvers, and real-world Crazyflie quadcopter flight experiments under aerodynamic disturbances. Results demonstrate that pdSTL achieves efficient optimization with formal probabilistic guarantees, significantly outperforming deterministic differentiable STL in maintaining safety margins under real-world uncertainty.

Problem

• stochastic systems
• safety specifications
• temporal logic
• uncertainty
• autonomous robots

Innovation

• probabilistic differentiable STL
• interval-valued semantics
• belief-space robustness
• differentiable monitoring
• temporal logic optimization