A Unifying Perspective for Safety of Stochastic Systems: From Barrier Functions to Finite Abstractions

Providing safety guarantees for stochastic dynamical systems is a central problem in various fields, including control theory, machine learning, and robotics. Existing methods either employ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Stochastic Barrier Functions</i> (SBFs) or rely on numerical approaches based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite abstractions</i>. SBFs, analogous to Lyapunov functions, are used to establish (probabilistic) set invariance, whereas abstraction-based approaches approximate the stochastic system with a finite model to compute safety probability bounds. This paper presents a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unifying perspective</i> on these seemingly different approaches. Specifically, we show that both methods can be interpreted as approximations of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic dynamic programming</i> problem. This perspective allows us to formally establish the correctness of both techniques, characterize their convergence and optimality properties, and analyze their respective assumptions, advantages, and limitations. Our analysis reveals that, unlike SBFs-based methods, abstraction-based approaches can provide asymptotically optimal safety certificates, albeit at the cost of increased computational effort.