A short note on Simulation and Abstraction

interpretation setting that we use, a bounded linear operator and its Moore-Penrose Pseudo Inverse are the analogue of the adjoined pair of monotonic functions in a Galois insertion. The abstract systems are obtained by lumping states, i.e. by identifying each concrete state si with a class C j of states which are all behavioural equivalent to each other. Concretely, we compute this via n×m matrices K (where n is the number of concrete states and m the number of abstract classes) with Kij = 1 iff si ∈C j and 0 otherwise. We refer to such matrices which have exactly one entry 1 in each row while all other entries are 0 as classification matrices, and denote the set of all classification matrices by K . The abstract systems are then given by Mi = K † i MiKi with Ki some classification matrix and † constructing the so called Moore-Penrose pseudo-inverse – in the case of classification matrices K† can be constructed as the row-normalised transpose of K. The problem of showing that two systems M1 and M2 are behaviourally equivalent, i.e. are (probabilistically) bisimilar, is now translated into finding two classification matrices Ki ∈ K such that M1 = K † 1M1K1 = K † 2M2K2 = M # 2 In case that two systems are not bisimilar we can still define a quantity ε which describes how (non-)bisimilar the two systems are. This ε is formally defined in terms of the norm of a linear operator representing the partition induced by the ‘minimal’ bisimulation on the set of the states of a given system, i.e. the one minimising the observational difference between the system’s components (see again [10] for further details, in particular regarding labeled probabilistic transition systems): Definition 2 Let M1 and M2 be the matrix representations of two probabilistic transition systems. We say that M1 and M2 are ε-bisimilar, denoted by M1 ∼b M2, iff inf K1,K2∈K ‖K1M1K1 −K † 2M2K2‖= ε where ‖.‖ denotes an appropriate norm, e.g. the supremum norm ‖.‖∞. In [10] we show that, when ε = 0 this gives the standard notion of probabilistic bisimulation.