  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.
