Qualitative probabilistic modelling in event-B
IFM'07 Proceedings of the 6th international conference on Integrated formal methods
Automating refinement checking in probabilistic system design
ICFEM'07 Proceedings of the formal engineering methods 9th international conference on Formal methods and software engineering
Quantitative µ-calculus analysis of power management in wireless networks
ICTAC'06 Proceedings of the Third international conference on Theoretical Aspects of Computing
Programming-logic analysis of fault tolerance: expected performance of self-stabilisation
Rigorous Development of Complex Fault-Tolerant Systems
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Formal notations like B or action systems support a notion of refinement. Refinement relates an abstract specification A to a concrete specification C that is as least as deterministic. Knowing A and C one proves that C refines, or implements, specification A. In this study we consider specification A as given and concern ourselves with a way to find a good candidate for implementation C. To this end we classify all implementations of an abstract specification according to their performance. We distinguish performance from correctness. Concrete systems that do not meet the abstract specification correctly are excluded. Only the remaining correct implementations C are considered with respect to their performance. A good implementation of a specification is identified by having some optimal behaviour in common with it. In other words, a good refinement corresponds to a reduction of non-optimal behaviour. This also means that the abstract specification sets a boundary for the performance of any implementation. We introduce the probabilistic action system formalism which combines refinement with performance. In our current study we measure performance in terms of long-run expected average-cost. Performance is expressed by means of probability and expected costs. Probability is needed to express uncertainty present in physical environments. Expected costs express physical or abstract quantities that describe a system. They encode the performance objective. The behaviour of probabilistic action systems is described by traces of expected costs. A corresponding notion of refinement and simulation-based proof rules are introduced. Probabilistic action systems are based on discrete-time Markov decision processes. Numerical methods solving the optimisation problems posed by Markov decision processes are well-known, and used in a software tool that we have developed. The tool computes an optimal behaviour of a specification A thus assisting in the search for a good implementation C.