Algorithmic Minimization of Uncertain Continuous-Time Markov Chains

The assumption of perfect knowledge of rate parameters in continuous-time Markov chains (CTMCs) is undermined when confronted with reality, where they may be uncertain due to lack of information or because of measurement noise. Here, we consider uncertain CTMCs (UCTMCs), where rates are assumed to vary nondeterministically with time from bounded continuous intervals. A UCTMC can be, therefore, seen as a specific type of Markov decision process for which the analysis is computationally difficult. To tackle this, we develop a theory of minimization, which generalizes the notion of lumpability for CTMCs. Our first result is a quantitative and logical characterization of minimization. Specifically, we show that the reduced UCTMC model has a macrostate for each block of a partition of the state space, which preserves value functions and logical formulae whenever rewards are equal within each block. The second result is an efficient minimization algorithm for UCTMCs by means of partition refinement. As an application, we show that reductions in a number of CTMC benchmark models are robust with respect to uncertainties in original rates.