Models and axioms for a fragment of real time process algebra

Process algebra is the study of concurrent communicating processes in an algebraic framework. It has been initiated by Milner, who has developed the process algebra CCS [Mil80],[Mil89]. Bergstra and Klop presented in [BK84b] the process algebra ACP, fora textbook we refer to [BW90]. Another process algebra that we mention is CSP that has been developed by Hoare [Hoa85]. In the first part of this thesis we give a brief introduetion to ACP, that is basedon [BW90]. Over the last few years there have been several attemps to extend process algebra, which has resulted in higher order process algebra, process algebra with value passing and process algebra with time. Baeten and Bergstra have extended ACP with time by decorating the atomie actions with time stamps [BB91]. These time stamps are taken from some time domain, which may be the set of real numbers. For example, a(t) is the process which executes the atomie action a at timet. They also introduced the process fvES p( v), where v is a so-called time variable, S is a subset of the time domain, and p is a process expression in which the free occurrences of the time variabie v become bound. This integrated process expression denotes the alternative composition over the timestampsin S. In this thesis we study this approach in more detail. We restriet ourselves to prefix integrated process terms, that is, we will allow only processes of the form fvEV a(v) and fvEv(a(v) · p), where V is an interval over the time domain. This restrietion is necessary to obtain a traetabie calculus for which in principle a complete axiomatization can be found. In the second part of this thesis we study the bisimulation semantics and axiom systems for real time ACP with prefix integration without recursion. The problem with process terms with prefixed integration, is to reason with terms containing free occurrences of time variables. To tackle this problem, we generalize our syntax. We allow termsof the form fa(a(v) · p), where a is a condition, that is a boolean expression over time variables. We introduce a finite axiom system for this generalized class of terms, and we prove completeness and decidability of bisimulation equivalence. The material of this part originates from [BB91], [Klu91b] and [FK92], it has partly been written together with Willem Jan Fokkink. In the literature several equivalences, and their characterizing laws, are known which deal with abstraction in process algebra without time. In the third part of this thesis we define branching, delay and weak bisimulation equivalence in the context of time, and we study the axiomatizations of these equivalences. Earlier versions of this work can be found in [Klu91a] and [Klu92]. Process algebra can be used in the specification and verification of protocols. In part four of this thesis we study guarded recursion in real time ACP in more detail, as it is an essential feature in the specification of protocols. In this part we show as well how a real time protocol can be verified, using the axioms for abstraction. The verification originates from [Klu91a]. In real time ACP consecutive actions can not happen at the same point in time, although they can occur arbitrary close to each other. In part five of this thesis