Reduction Semantics in Markovian Process Algebra

Markovian process algebras allow for performance analysis by automatic generation of Continuous Time Markov Chains. The inclusion of exponential distribution rate information in process algebra terms, however, causes non trivial issues to arise in the definition of their semantics. As a consequence, technical settings previously considered do not make it possible to base Markovian semantics on directly computing reductions between communicating processes: this would require the ability to readjust processes, i.e. a commutative and associative parallel operator and a congruence relation on terms enacting such properties. Semantics in reduction style is, however, often used for complex languages, due to its simplicity and conciseness. In this paper we introduce a new technique based on stochastic binders that allows us to define Markovian semantics in the presence of such a structural congruence. As application examples, we define the reduction semantics of Markovian versions of the pi-calculus, considering both the cases of Markovian durations: being additional standalone prefixes (as in Interactive Markov Chains) and being, instead, associated to standard synchronizable actions, giving them a duration (as in Stochastic pi-calculus). Notably, in the latter case, we obtain a ``natural'' Markovian semantics for pi-calculus (CCS) parallel that preserves, for the first time, its associativity. In both cases we show our technique for defining reduction semantics to be correct with respect to the standard Markovian one (in labeled style) by providing Markovian extensions of the classical pi-calculus Harmony theorem.