pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems

• URL: http://arxiv.org/abs/2606.19561v1
• Date: Wed, 17 Jun 2026 20:01:56 GMT
• Title: pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems
• Abstract summary: We introduce pdSTL (probabilistic differentiable Signal Temporal Logic), a framework that unifies probabilistic semantics with differentiable robustness over belief trajectories.<n>Results demonstrate that pdSTL efficient optimization with formal probabilistic guarantees, significantly outperforming deterministic differentiable STL in maintaining safety margins under real-world uncertainty.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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