Temporal Robustness of Stochastic Signals

Abstract

We study the temporal robustness of stochastic signals. This topic is of particular interest in interleaving processes such as multi-agent systems where communication and individual agents induce timing uncertainty. For a deterministic signal and a given specification, we first introduce the synchronous and the asynchronous temporal robustness to quantify the signal's robustness with respect to synchronous and asynchronous time shifts in its sub-signals. We then define the temporal robustness risk by investigating the temporal robustness of the realizations of a stochastic signal. This definition can be interpreted as the risk associated with a stochastic signal to not satisfy a specification robustly in time. In this definition, general forms of specifications such as signal temporal logic specifications are permitted. We show how the temporal robustness risk is estimated from data for the value-at-risk. The usefulness of the temporal robustness risk is underlined by both theoretical and empirical evidence. In particular, we provide various numerical case studies including a T-intersection scenario in autonomous driving.

MITL Verification Under Timing Uncertainty

“…It is also of interest to address Remark 6, by considering how information about the structure of the formula can be exploited in order to reduce conservativeness. Finally, the relationships between the gap Q + (ϕ) \ Q − (ϕ) and the notions of temporal robustness in [3,9] merit investigation. Recent works [12,8] consider the synthesis of controllers to maximise temporal robustness, so such problems may benefit from our interval queue-based analysis.…”

“…More generally, the timing uncertainty associated with satisfying the formula relates to the gap between the over-and under-approximations produced by the algorithm. Although its precise nature is deferred to future work, we anticipate a relationship between this gap and the notions of temporal robustness in [3,9]. Such is of relevance to recent works [8,12], which aim to synthesize controllers that maximise temporal robustness.…”