Parametric Model Checking Continuous-Time Markov Chains

CSL is a well-known temporal logic for specifying properties of real-time stochastic systems, such as continuous-time Markov chains. We introduce PCSL, an extension of CSL that allows using existentially quantified parameters in timing constraints, and investigate its expressiveness and decidability over properties of continuous-time Markov chains. Assuming Schanuel’s Conjecture, we prove the decidability of model checking the one-parameter fragment of PCSL on continuous-time Markov chains. Technically, the central problem we solve (relying on Schanuel’s Conjecture) is to decide positivity of real-valued exponential polynomial functions on bounded intervals. A second contribution is to give a reduction of the Positivity Problem for matrix exponentials to the PCSL model checking problem, suggesting that it will be difficult to give an unconditional proof of the decidability of model checking PCSL. 2012 ACM Subject Classification Theory of computation → Logic and verification; Theory of computation → Verification by model checking; Theory of computation → Random walks and Markov chains