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Providing safety guarantees for stochastic dynamical systems has become acentral problem in many fields, including control theory, machine learning, androbotics. Existing methods either employ Stochastic Barrier Functions (SBFs) orrely on numerical approaches based on abstractions. While SBFs are analogous toLyapunov functions to prove (probabilistic) set invariance, abstraction-basedapproaches approximate the stochastic system into a finite model for thecomputation of safety probability bounds. In this paper, we offer a newperspective on these seemingly different methods. We show that both theseapproaches arise as approximations of a stochastic dynamic programming problem.Such a new and unifying perspective allows us to formally show the correctnessof both approaches, characterize their convergence and optimality properties,and examine advantages and disadvantages of each. Our analysis reveals thatabstraction-based methods can provide more accurate certificates of safety, butgenerally at a higher computational cost. We conclude the article with adiscussion that highlights possible future directions.

Unifying Safety Approaches for Stochastic Systems: From Barrier Functions to Uncertain Abstractions via Dynamic Programming