# Predictive Monitoring with Logic-Calibrated Uncertainty for Cyber-Physical Systems

arXiv: https://arxiv.org/abs/2011.00384 · HTML: https://mlsanity.com/paper/2011.00384/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2020.

## Built on by (most cited first, 6 of 6)

- [Formalizing and Evaluating Requirements of Perception Systems for Automated Vehicles using Spatio-Temporal Perception Logic](https://mlsanity.com/paper/2206.14372/) (2022)
- [LogiDebrief: A Signal-Temporal Logic based Automated Debriefing Approach with Large Language Models Integration](https://mlsanity.com/paper/2505.03985/) (2025)
- [Neural Network-based Control for Multi-Agent Systems from Spatio-Temporal Specifications](https://mlsanity.com/paper/2104.02737/) (2021)
- [Learning Spatio-Temporal Specifications for Dynamical Systems](https://mlsanity.com/paper/2112.10714/) (2021)
- [Inference of Human-derived Specifications of Object Placement via Demonstration](https://mlsanity.com/paper/2508.19367/) (2025)
- [Multi-Agent Planning with Spatio-Temporal and Topological Constraints using STL-GO](https://mlsanity.com/paper/2607.28679/) (2026)

## Abstract (arXiv)

Predictive monitoring -- making predictions about future states and monitoring if the predicted states satisfy requirements -- offers a promising paradigm in supporting the decision making of Cyber-Physical Systems (CPS). Existing works of predictive monitoring mostly focus on monitoring individual predictions rather than sequential predictions. We develop a novel approach for monitoring sequential predictions generated from Bayesian Recurrent Neural Networks (RNNs) that can capture the inherent uncertainty in CPS, drawing on insights from our study of real-world CPS datasets. We propose a new logic named Signal Temporal Logic with Uncertainty (STL-U) to monitor a flowpipe containing an infinite set of uncertain sequences predicted by Bayesian RNNs. We define STL-U strong and weak satisfaction semantics based on if all or some sequences contained in a flowpipe satisfy the requirement. We also develop methods to compute the range of confidence levels under which a flowpipe is guaranteed to strongly (weakly) satisfy an STL-U formula. Furthermore, we develop novel criteria that leverage STL-U monitoring results to calibrate the uncertainty estimation in Bayesian RNNs. Finally, we evaluate the proposed approach via experiments with real-world datasets and a simulated smart city case study, which show very encouraging results of STL-U based predictive monitoring approach outperforming baselines.