Linear and Branching Metrics for Quantitative Transition Systems

We extend the basic system relations of trace inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as real values in the interval [0,1]. Trace inclusion and equivalence give rise to asymmetrical and symmetrical linear distances, while simulation and bisimulation give rise to asymmetrical and symmetrical branching distances. We study the relationships among these distances, and we provide a full logical characterization of the distances in terms of quantitative versions of LTL and μ-calculus. We show that, while trace inclusion (resp. equivalence) coincides with simulation (resp. bisimulation) for deterministic boolean transition systems, linear and branching distances do not coincide for deterministic quantitative transition systems. Finally, we provide algorithms for computing the distances, together with matching lower and upper complexity bounds.