Oliver Schön

Oliver Schön, Shammakh Naseer, Ben Wooding et al.

To advance formal verification of stochastic systems against temporal logic requirements for handling unknown dynamics, researchers have been designing data-driven approaches inspired by breakthroughs in the underlying machine learning techniques. As one promising research direction, abstraction-based solutions based on Gaussian process (GP) regression have become popular for their ability to learn a representation of the latent system from data with a quantified error. Results obtained based on this model are then translated to the true system via various methods. In a recent publication, GPs using a so-called binary-tree kernel have demonstrated a polynomial speedup w.r.t. the size of the data compared to their vanilla version, outcompeting all existing sparse GP approximations. Incidentally, the resulting binary-tree Gaussian process (BTGP) is characteristic for its piecewise-constant posterior mean and covariance functions, naturally abstracting the input space into discrete partitions. In this paper, we leverage this natural abstraction of the BTGP for formal verification, eliminating the need for cumbersome abstraction and error quantification procedures. We show that the BTGP allows us to construct an interval Markov chain model of the unknown system with a speedup that is polynomial w.r.t. the size of the abstraction compared to alternative approaches. We provide a delocalized error quantification via a unified formula even when the true dynamics do not live in the function space of the BTGP. This allows us to compute upper and lower bounds on the probability of satisfying reachability specifications that are robust to both aleatoric and epistemic uncertainties.

Bardh Hoxha, Oliver Schön, Hideki Okamoto et al.

We study certified runtime monitoring of past-time signal temporal logic (ptSTL) from visual observations under partial observability. The monitor must infer safety-relevant quantities from images and provide finite-sample guarantees, while being \emph{reusable}: once trained and calibrated, it should certify any formula in a target fragment without per-formula retraining. For fragments induced by a finite dictionary of temporal atoms, we prove that the \emph{semantic basis}, the vector of atom robustness scores, is the minimum prediction target within the class of monotone, 1-Lipschitz reusable interfaces: any formula is evaluated by a deterministic decoder derived from the parse tree, and a single conformal calibration pass certifies the entire fragment with no union bound. We also introduce a \emph{rolling prediction monitor} that predicts only current predicate values and reconstructs temporal history online; this is easier to learn but grows conservative at long horizons. On a pedestrian-crossroad benchmark, rolling achieves tighter certified bounds at short horizons while the semantic-basis monitor is up to 4-times tighter at long horizons. We validate the presented monitors on real-world Waymo driving data, where both monitors satisfy the conformal coverage guarantee empirically.

Oliver Schön, Licio Romao, Sadegh Soudjani

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.

Oliver Schön, Zhengang Zhong, Sadegh Soudjani

Algorithmic verification of realistic systems to satisfy safety and other temporal requirements has suffered from poor scalability of the employed formal approaches. To design systems with rigorous guarantees, many approaches still rely on exact models of the underlying systems. Since this assumption can rarely be met in practice, models have to be inferred from measurement data or are bypassed completely. Whilst former usually requires the model structure to be known a-priori and immense amounts of data to be available, latter gives rise to a plethora of restrictive mathematical assumptions about the unknown dynamics. In a pursuit of developing scalable formal verification algorithms without shifting the problem to unrealistic assumptions, we employ the concept of barrier certificates, which can guarantee safety of the system, and learn the certificate directly from a compact set of system trajectories. We use conditional mean embeddings to embed data from the system into a reproducing kernel Hilbert space (RKHS) and construct an RKHS ambiguity set that can be inflated to robustify the result w.r.t. a set of plausible transition kernels. We show how to solve the resulting program efficiently using sum-of-squares optimization and a Gaussian process envelope. Our approach lifts the need for restrictive assumptions on the system dynamics and uncertainty, and suggests an improvement in the sample complexity of verifying the safety of a system on a tested case study compared to a state-of-the-art approach.