Automatic verification of real-time systems with discrete probability distributions

Abstract

We consider the timed automata model of [3], which allows the analysis of real-time systems expressed in terms of quantitative timing constraints. Traditional approaches to real-time system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for real-time systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, "with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2".

Safety Verification for Probabilistic Hybrid Systems

“…Abstraction approaches have also successfully been applied to probabilistic timed automata [24,25], a class of probabilistic hybrid automata, where only derivatives of constant value 1 occur. Their abstract analysis is based on difference-bound matrices (DBMs), and does not extend to the general setting considered here.…”

“…Because of this orthogonality, abstractions of probabilistic hybrid automata can be computed via abstractions for non-probabilistic hybrid automata and Markov decision processes. To show the applicability of this combination, we implemented a prototype tool, ProHVer, that builds an abstraction via a combination of existing techniques for classical hybrid automata [17] as well as methods for Markov decision processes [14,19,25]. Subsequently, a fixed-point engine computes the reachability probabilities on the abstraction, which provides a safe upper bound on the reachability probability in the model semantics.…”

Quantitative verification

“…Probabilistic timed automata (PTAs) [28] extend the timed automata formalism [2] with probabilistic choice over transitions. Similarly to timed automata, PTAs contain clocks, positive realvalued variables which increase uniformly with time, which can be referred to in its invariants and guards.…”

“…A probability measure over paths of a given adversary can be defined following [19] as for Markov decision processes. The logic for expressing properties of PTAs is PTCTL (Probabilistic Timed CTL) [28], derived from TCTL and PCTL, with which one can query the probability of reaching a state within a real-valued time bound. Model checking for this logic is obtained by adapting the techniques established for timed automata to the probabilistic case.…”

“…Observe that, since clocks are realvalued, the state space of a probabilistic timed automaton is infinite but can be partitioned into a finite set of symbolic states (regions or zones). The methods so far developed for PTAs include those based on the region graph construction [28], forward [28] and backward [29] zone graph exploration and the digital clocks approach [25]. The latter admits an extension with rewards, for example expected reachability properties, a generalisation of uniformly priced timed automata.…”