We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational, and rational functions when some probabilities are left unspecified as parameters of the system. This allows for parametric model checking by evaluating the regular expression for different parameter values, for instance, to study the influence of a lossy channel in the overall reliability of a randomized protocol.

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Abstract. Practical real-time model checking suffers from the stateexplosion problem: the size of the state space grows exponentially with many system parameters: number of clocks, size of constants, number of system components. To cope with state explosion, we propose to use abstractions reducing the state-space while preserving teachability properties. Four exact, plus one safe abstractions are defined. In the main abstraction (simulation) a concrete state is mapped to a symbolic abstract state (a set of concrete states). The other four abstractions are defined on top of the simulation one. They can be computed on-the-fly in a completely orthogonal manner and thus can be combined to yield better reductions. A prototype implementation in the tool KRONOS has permitted to verify two benchmark examples with a significant scale-up in size.

is a software tool aiming at assisting designers of real-time systems to develop projects meeting the speci ed requirements. One major objective of Kronos is to provide a veri cation engine to be integrated into design environments for real-time systems in a wide range of application areas. Real-time communication protocols 8,10], timed asynchronous circuits 16,4], and hybrid systems 18,10] are some examples of application domains where Kronos has already been used. Kronos has been also used in analyzing real-time systems modeled in several other process description formalisms, such as Atp 17], Aorta 5], Et-lotos 8], and T-argos 15]. On the other direction, the tool itself provides an interface to untimed formalisms such as labeled-transition systems (LTS) which has been used to exploit untimed veri cation techniques 20]. Theoretical background The system-description language of Kronos is the model of timed automata 2], which are communicating nite-state machines extended with continuous realvalued variables (clocks) used to measure time delays. Usually a system is modeled as a network of automata. Communication is achieved by label synchronization a la CCS or CSP (binary or n-ary rendezvous), or shared variables (of bounded integer or enumeration type). System requirements can be speci ed in Kronos using a variety of formalisms, such as the real-time logic Tctl 1,14], timed B uchi automata, or ? Kronos is developed at Verimag, a joint laboratory of UJF, Ensimag and CNRS.

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