Model Checking Markov Chains with Actions and State Labels

Abstract

Abstract-In the past, logics of several kinds have been proposed for reasoning about discrete-time or continuous-time Markov chains. Most of these logics rely on either state labels (atomic propositions) or on transition labels (actions). However, in several applications it is useful to reason about both state properties and action sequences. For this purpose, we introduce the logic asCSL which provides a powerful means to characterize execution paths of Markov chains with actions and state labels. asCSL can be regarded as an extension of the purely state-based logic CSL (continuous stochastic logic). In asCSL, path properties are characterized by regular expressions over actions and state formulas. Thus, the truth value of path formulas depends not only on the available actions in a given time interval, but also on the validity of certain state formulas in intermediate states. We compare the expressive power of CSL and asCSL and show that even the state-based fragment of asCSL is strictly more expressive than CSL if time intervals starting at zero are employed. Using an automaton-based technique, an asCSL formula and a Markov chain with actions and state labels are combined into a product Markov chain. For time intervals starting at zero, we establish a reduction of the model checking problem for asCSL to CSL model checking on this product Markov chain. The usefulness of our approach is illustrated with an elaborate model of a scalable cellular communication system, for which several properties are formalized by means of asCSL formulas and checked using the new procedure.

Time-Bounded Verification of CTMCs against Real-Time Specifications

“…These include properties of the form, "what is the probability to reach a given target state within the deadline, while avoiding unsafe states and not staying too long in any of the dangerous states on the way?". Such properties can neither be expressed in CSL nor in its dialects [5,17]. Model checking DTA properties can be done by a reduction to computing the reachability probability in a piecewise deterministic Markov process, based on the product construction between the CTMC and DTA [15,9].…”

“…Model checking CTMCs against linear real-time specifications has received scant attention so far. To our knowledge, this issue has only been (partially) addressed in [14,5,17]. Baier et al [5] define the logic asCSL where path properties are characterized by (time-bounded) regular expressions over actions and state formulas.…”

“…To our knowledge, this issue has only been (partially) addressed in [14,5,17]. Baier et al [5] define the logic asCSL where path properties are characterized by (time-bounded) regular expressions over actions and state formulas. The truth value of path formulas depends not only on the available actions in a given time interval, but also on the validity of certain state formulas in intermediate states.…”

“…The truth value of path formulas depends not only on the available actions in a given time interval, but also on the validity of certain state formulas in intermediate states. asCSL is strictly more expressive than CSL [5]. Model checking asCSL is performed by representing the regular expressions as finite-state automata, followed by computing time-bounded reachability probabilities in the product of CTMC C and this automaton.…”

Quantitative Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications

“…Related work is model checking of asCSL [6] and CSL TA [13]. asCSL allows to impose a time constraint on action sequences described by regular expressions; its model-checking algorithm is based on a determin-istic Rabin automaton construction.…”