Algebraic Verification of Probabilistic and Concurrent Systems

This thesis provides an algebraic modelling and verification of probabilistic concurrent systems in the style of Kleene algebra. Without concurrency, it is shown that the equational theory of continuous probabilistic Kleene algebra is complete with respect to an automata model under standard simulation equivalence. This yields a minimisation-based decision procedure for the algebra. Without probability, an event structure model of Hoare et al.'s concurrent Kleene algebra is constructed. These two algebras are then ``merged" to provide probabilistic concurrent Kleene algebra which is used to discover and prove development rules for probabilistic concurrent systems (e.g. rely/guarantee calculus). Soundness of the new algebra is ensured by models based on probabilistic automata (interleaving) and probabilistic bundle event structures (true concurrency) quotiented with the respective simulation equivalences. Lastly, event structures with implicit probabilities are constructed to provide a state based model for the soundness of the probabilistic rely/guarantee rules.