Safety in MDPs by measuring the probability of reaching dangerous states

Rafal Wisniewski, Luminita-Manuela Bujorianu, Safety of stochastic systems: An analytic and computational approach, . Automatica, Volume 133, 2021 DOI: 10.1016/j.automatica.2021.109839.

We refine the concept of stochastic reach avoidance for a general class of Markov processes introducing a threshold of p for the reaching probability. This new problem is called p-safety, and it aims to ensure that the given process reaches a forbidden set before leaving its ‘working’ state space with a probability of less than p. In the situation when an initial probability measure characterizes the initial states, a variant of p-safety is put forward. We call this form of safety weak p-safety. In this work, we characterize both p-safety and weak p-safety and show how to compute them. We employ semi-definite programming to compute p-safety and linear programming to compute weak p-safety. To get to this point, we use certificates of positivity of polynomials translated into the sum of squares and the Bernstein forms.

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