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**1 - 3**of**3**### Probabilistic Rely-guarantee Calculus ∗

"... Jones ’ rely-guarantee calculus [1] for shared variable concurrency is ex-tended to include probabilistic behaviours. We use an algebraic approach which combines and adapts probabilistic Kleene algebras with concurrent Kleene algebra. Soundness of the algebra is shown relative to a general probabili ..."

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Jones ’ rely-guarantee calculus [1] for shared variable concurrency is ex-tended to include probabilistic behaviours. We use an algebraic approach which combines and adapts probabilistic Kleene algebras with concurrent Kleene algebra. Soundness of the algebra is shown relative to a general probabilistic event structure semantics [21]. The main contribution of this paper is a collection of rely-guarantee rules built on top of that semantics. In particular, we show how to obtain bounds on probabilities by deriving rely-guarantee rules within the true-concurrent denotational semantics. The use of these rules is illustrated by a detailed verification of a simple probabilistic concurrent program: a faulty Eratosthenes sieve. 1

### Creative Commons Attribution License. Towards Verifying Safety Properties of Real-Time Probabilistic Systems

"... Using probabilities in the formal-methods-based development of safety-critical software has quick-ened interests in academia and industry. We address this area by our model-driven engineering method for reactive systems SPACE and its tool-set Reactive Blocks that provide an extension to support the ..."

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Using probabilities in the formal-methods-based development of safety-critical software has quick-ened interests in academia and industry. We address this area by our model-driven engineering method for reactive systems SPACE and its tool-set Reactive Blocks that provide an extension to support the modeling and verification of real-time behaviors. The approach facilitates the compo-sition of system models from reusable building blocks as well as the verification of functional and real-time properties and the automatic generation of Java code. In this paper, we describe the extension of the tool-set to enable the modeling and verification of probabilistic real-time system behavior with the focus on spatial properties that ensure system safety. In particular, we incorporate descriptions of probabilistic behavior into our Reactive Blocks models and integrate the model checker PRISM which allows to verify that a real-time system satis-fies certain safety properties with a given probability. Moreover, we consider the spatial implication of probabilistic system specifications by integrating the spatial verification tool BeSpaceD and give an automatic approach to translate system specifications to the input languages of PRISM and Be-SpaceD. The approach is highlighted by an example. 1