We survey the work on both discrete and continuous-space probabilistic systems as coalgebras, starting with how probabilistic systems are modeled as coalgebras and followed by a discussion of their bisimilarity and behavioral equivalence, mentioning results that follow from the coalgebraic treatment of probabilistic systems. The syntax and semantics of χ are discussed, and it is explained how equational reasoning can simplify, among others, tool implementations for simulation and verification. Abstraction from internal actions is the most important tool for system verification. Central to theories of concurrency is the notion of abstraction. Finally, a bottle filling line example is introduced to illustrate system anal ysis by means of equational reasoning. .
We briefly look at recursive specifications in this theory, and discuss the relations with Milner's silent step. A widely accepted method to specify possibly infinite behaviour is to define it as the solution, in some process algebra, of a recursive specification, i. As the vehicle of our exposition, we use the hybrid process algebra χ Chi. It is interesting to note that, for different reasons, for both discrete and continuous probabilistic systems it may be more convenient to work with behavioral equivalence than with bisimilarity. A hybrid process algebra can be used for the specification, simulation, control and verification of embedded systems in combination with their environment, and for any dynamic system in general.
Process algebra is the study of distributed or parallel syst ems by algebraic means. In this paper we investigate to what extent guardedness is also a necessary requirement to ensure unique solutions. The method only works if the recursive specification has a unique solution in the process algebra; it is well-known that guardedness is a sufficient requirement on a recursive specification to guarantee a unique solution in any of the standard process algebras. Originating in computer science, process algebra has been extended in recent years to encompass not just discrete event, reactive systems, but also continuously evolving phenomena, resulting in so-called hybrid process algebras. . .
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