# STL Robustness Risk over Discrete-Time Stochastic Processes

arXiv: https://arxiv.org/abs/2104.01503 · HTML: https://mlsanity.com/paper/2104.01503/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 59 references that are in the map.
- First posted on arXiv 3 Apr 2021.

## Builds on (most cited first, 20 of 20)

- [How Should a Robot Assess Risk? Towards an Axiomatic Theory of Risk in Robotics](https://mlsanity.com/paper/1710.11040/) (2017)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [On Robustness Metrics for Learning STL Tasks](https://mlsanity.com/paper/2003.06041/) (2020)
- [Arithmetic-Geometric Mean Robustness for Control from Signal Temporal Logic Specifications](https://mlsanity.com/paper/1903.05186/) (2019)
- [Control from Signal Temporal Logic Specifications with Smooth Cumulative Quantitative Semantics](https://mlsanity.com/paper/1904.11611/) (2019)
- [A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems](https://mlsanity.com/paper/1902.11277/) (2019)
- [PAC-Bayesian Bound for the Conditional Value at Risk](https://mlsanity.com/paper/2006.14763/) (2020)
- [Risk-Averse Planning Under Uncertainty](https://mlsanity.com/paper/1909.12499/) (2019)
- [Probabilistic Motion Planning under Temporal Tasks and Soft Constraints](https://mlsanity.com/paper/1706.05209/) (2017)
- [Metrics for Signal Temporal Logic Formulae](https://mlsanity.com/paper/1808.03315/) (2018)
- [Quantile Multi-Armed Bandits: Optimal Best-Arm Identification and a Differentially Private Scheme](https://mlsanity.com/paper/2006.06792/) (2020)
- [Time Robustness in MTL and Expressivity in Hybrid System Falsification (Extended Version)](https://mlsanity.com/paper/1505.06307/) (2015)
- [Safety-Aware Optimal Control of Stochastic Systems Using Conditional Value-at-Risk](https://mlsanity.com/paper/1802.07903/) (2018)
- [A Framework for Time-Consistent, Risk-Sensitive Model Predictive Control: Theory and Algorithms](https://mlsanity.com/paper/1703.01029/) (2017)
- [Temporal Logic as Filtering](https://mlsanity.com/paper/1510.08079/) (2015)
- [Shrinking Horizon Model Predictive Control with Signal Temporal Logic Constraints under Stochastic Disturbances](https://mlsanity.com/paper/1705.02152/) (2017)
- [On the Robustness of Temporal Properties for Stochastic Models](https://mlsanity.com/paper/1309.0866/) (2013)
- [Control Design for Risk-Based Signal Temporal Logic Specifications](https://mlsanity.com/paper/2006.00317/) (2020)
- [Synthesis of surveillance strategies via belief abstraction](https://mlsanity.com/paper/1709.05363/) (2017)
- [Learning-Based Distributionally Robust Model Predictive Control of Markovian Switching Systems with Guaranteed Stability and Recursive Feasibility](https://mlsanity.com/paper/2009.04422/) (2020)

## Built on by (most cited first, 4 of 4)

- [Probable Domain Generalization via Quantile Risk Minimization](https://mlsanity.com/paper/2207.09944/) (2022)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [Tolerance of Reinforcement Learning Controllers against Deviations in Cyber Physical Systems](https://mlsanity.com/paper/2406.17066/) (2024)

## Abstract (arXiv)

We present a framework to interpret signal temporal logic (STL) formulas over discrete-time stochastic processes in terms of the induced risk. Each realization of a stochastic process either satisfies or violates an STL formula. In fact, we can assign a robustness value to each realization that indicates how robustly this realization satisfies an STL formula. We then define the risk of a stochastic process not satisfying an STL formula robustly, referred to as the STL robustness risk. In our definition, we permit general classes of risk measures such as, but not limited to, the conditional value-at-risk. While in general hard to compute, we propose an approximation of the STL robustness risk. This approximation has the desirable property of being an upper bound of the STL robustness risk when the chosen risk measure is monotone, a property satisfied by most risk measures. Motivated by the interest in data-driven approaches, we present a sampling-based method for estimating the approximate STL robustness risk from data for the value-at-risk. While we consider the value-at-risk, we highlight that such sampling-based methods are viable for other risk measures.