Safety Certification is Classification

TL;DR

This paper introduces a kernel embedding classification framework for safety certification of uncertain dynamical systems, avoiding recursive errors and handling non-Markovian dynamics effectively.

Contribution

It presents a novel non-recursive approach that unifies existing methods and extends safety certification to complex, non-Markovian systems.

Findings

The proposed method remains stable regardless of the certification horizon.

It successfully certifies safety for systems with non-Markovian dynamics.

Simulation confirms DP-based methods become unsound over long horizons.

Abstract

The goal of this paper is certifying safety of dynamical systems subject to uncertainty. Existing approaches use trajectory data to estimate transition probabilities, and compute safety probabilities recursively via dynamic programming (DP). This recursion may lead to compounding errors in the certified safety probability, thus collapsing to a vacuous lower bound for growing horizons $T$. We propose a kernel embedding framework that treats safety certification as a classification problem on trajectory data, directly estimating the $T$-step safety probability without recursion. We show that the framework subsumes well-established approaches from the literature (e.g., barrier certificates, robust Markov models) as special cases, and allows us to go beyond their limitations. As the main consequence, it bypasses compounding error across the horizon and enables certification for systems with non-Markovian dynamics. We demonstrate that direct estimators remain stable independent of the certification horizon and in the non-Markovian setting, whilst DP-based certificates silently go unsound -- confirmed in simulation on a neural-controlled quadrotor.