Bayesian Hybrid Automata: A Formal Model of Justified Belief in Interacting Hybrid Systems Subject to Imprecise Observation

Hybrid discrete-continuous system dynamics arises when discrete actions, e.g. by a decision al-gorithm, meet continuous behaviour, e.g. due to physical processes and continuous control. A nat-ural domain of such systems are emerging smart technologies which add elements of intelligence, cooperation, and adaptivity to physical entities, enabling them to interact with each other and with humans as systems of (human-)cyber-physical systems or (H)CPSes. Various flavours of hybrid automata have been suggested as a means to formally analyse CPS dynamics. In a previous article, we demonstrated that all these variants of hybrid automata provide inaccurate, in the sense of either overly pessimistic or overly optimistic, verdicts for engineered systems operating under imprecise observation of their environment due to, e.g., measurement error. We suggested a revised formal model, called Bayesian hybrid automata, that is able to represent state tracking and estimation in hybrid systems and thereby enhances precision of verdicts obtained from the model in comparison to traditional model variants. In this article, we present an extended definition of Bayesian hybrid automata which incorporates a new class of guard and invariant functions that allow to evaluate traditional guards and invariants over probability distributions. The resulting framework allows to model observers with knowledge about the control strategy of an observed agent but with imprecise estimates of the data on which the control decisions are based.