Multi-layered simulation relations for linear stochastic systems

The design of provably correct controllers for continuous-state stochastic systems crucially depends on approximate finite-state abstractions and their accuracy quantification. For this quantification, one generally uses approximate stochastic simulation relations, whose constant precision limits the achievable guarantees on the control design. This limitation especially affects higher dimensional stochastic systems and complex formal specifications. This work allows for variable precision by defining a simulation relation that contains multiple precision layers. For bi-layered simulation relations, we develop a robust dynamic programming approach yielding a lower bound on the satisfaction probability of temporal logic specifications. We illustrate the benefit of bi-layered simulation relations for linear stochastic systems in an example.