Formal Verification of Unknown Stochastic Systems via Non-parametric Estimation

TL;DR

This paper introduces a data-driven, model-free approach for the formal verification of stochastic systems using non-parametric estimation, providing probabilistic guarantees based solely on observation data.

Contribution

It develops a novel theoretical framework for estimating system Lipschitz constants non-parametrically and constructs finite abstractions for verification without prior model knowledge.

Findings

Estimates the Lipschitz constant with convergence rate $O(n^{-rac{1}{3+d}})$

Validates the approach through multiple case studies

Provides probabilistic guarantees for system satisfaction of specifications

Abstract

A novel data-driven method for formal verification is proposed to study complex systems operating in safety-critical domains. The proposed approach is able to formally verify discrete-time stochastic dynamical systems against temporal logic specifications only using observation samples and without the knowledge of the model, and provide a probabilistic guarantee on the satisfaction of the specification. We first propose the theoretical results for using non-parametric estimation to estimate an asymptotic upper bound for the \emph{Lipschitz constant} of the stochastic system, which can determine a finite abstraction of the system. Our results prove that the asymptotic convergence rate of the estimation is $O(n^{-\frac{1}{3+d}})$, where $d$ is the dimension of the system and $n$ is the data scale. We then construct interval Markov decision processes using two different data-driven methods, namely non-parametric estimation and empirical estimation of transition probabilities, to perform formal verification against a given temporal logic specification. Multiple case studies are presented to validate the effectiveness of the proposed methods.