A Lyapunov Approach for Time Bounded Reachability of CTMCs and CTMDPs

Abstract

Time bounded reachability is a fundamental problem in model checking continuous-time Markov chains (CTMCs) and Markov decision processes (CTMDPs) for specifications in continuous stochastic logics. It can be computed by numerically solving a characteristic linear dynamical system but the procedure is computationally expensive. We take a control-theoretic approach and propose a reduction technique that finds another dynamical system of lower dimension (number of variables), such that numerically solving the reduced dynamical system provides an approximation to the solution of the original system with guaranteed error bounds. Our technique generalises lumpability (or probabilistic bisimulation) to a quantitative setting. Our main result is a Lyapunov function characterisation of the difference in the trajectories of the two dynamics that depends on the initial mismatch and exponentially decreases over time. In particular, the Lyapunov function enables us to compute an error bound between the two dynamics as well as a convergence rate. Finally, we show that the search for the reduced dynamics can be computed in polynomial time using a Schur decomposition of the transition matrix. This enables us to efficiently solve the reduced dynamical system by computing the exponential of an upper-triangular matrix characterising the reduced dynamics. For CTMDPs, we generalise our approach using piecewise quadratic Lyapunov functions for switched affine dynamical systems. We synthesise a policy for the CTMDP via its reduced-order switched system that guarantees the time bounded reachability probability lies above a threshold. We provide error bounds that depend on the minimum dwell time of the policy. We demonstrate the technique on examples from queueing networks, for which lumpability does not produce any state space reduction but our technique synthesises policies using reduced version of the model.