Efficient model checking of the stochastic logic CSLTA

Abstract

CSLTA is a stochastic temporal logic for continuous-time Markov chains (CTMCs) which includes the well known CSL. CSLTA properties are defined using single-clock Deterministic Timed Automata (DTAs). The model checking of CSLTA amounts, in the worst-case, to the computation of the steady-state probability of a non-ergodic Markov Regenerative Process (MRgP) of size of |CTMC|×|DTA|. Various MRgP solution techniques are available in the literature, and we shall use the Component Method, which computes the steady state distribution of a non-ergodic MRgP by recognizing that, in an MRgP, certain components may actually be solved at a lower cost (same cost as that of a CTMC solution). Unfortunately the technique still requires the construction of the whole MRgP. This paper applies the Component Method to devise various CSLTA model checking algorithms, forward and backward. The Component Method can be applied to the MRgP constructed from the CTMC and the DTA, which is a rather straightforward application of the method, or to the MRgP constructed from the CTMC and the region graph of the DTA, a construction that takes into account the timed reachability in the DTA and that allows, in most cases, a significant reduction in the considered MRgP states. In both cases the whole MRgP is built. The primary result of this paper is instead to devise a model checking algorithm in which the component identification is based only on the region graph of the DTA. The MRgP components are generated “on-the-fly”, when needed, starting from the components of the region graph; they are then solved with the cheapest available solution method. Once a component has been solved it is discarded, therefore the whole MRgP is never constructed nor solved. The on-the-fly algorithm is “adaptive”: the time and space used depend on the formula, and, when the DTA actually expresses a CSL property, the algorithm complexity reduces, seamlessly, to that of standard CSL model checking algorithms.