Stochastic Reachability Based on Probabilistic Bisimulation

Abstract

This chapter provides a new approach for dealing with stochastic reachability. Instead of looking for new methods for computing reach set probabilities in the initial framework of stochastic hybrid processes, we propose to use the existing ones but in a changed framework. Precisely, the idea is to replace the complex stochastic processes that appear in the semantics of stochastic hybrid systems by simpler processes that are ‘probabilistically equivalent’. This means that the behaviour of the new simpler processes preserves only the probabilistic properties of interest for our analysis. The relation between a given stochastic hybrid process and its “equivalent version” is called bisimulation relation. This bisimulation relation can appear by relaxing the classical equivalence relations between stochastic processes that ask the exact equality between probabilities of trajectories at each moment of time. A process bisimilar to a given one should not be thought as a probabilistic version of it. The idea is to reduce the state space by lumping those states that behave in a similar way. Clearly, if we would be able to reduce the state space, the stochastic reachability analysis becomes more manageable. Usually, the main goal of the research in this field is to find suitable Markov chains (with a manageable state space) that are bisimilar with a given stochastic hybrid process.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2795-6_12