Transition systems, link graphs and Petri nets - Inria
Math. Struct. in Comp. Science: page 1 of 59. doi:10.1017/S0960129506005664 c○ 2006 Cambridge University Press Transition systems, link graphs and Petri nets JAMES J. LEIFER † and ROBIN MILNER ‡ † INRIA, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France Email: James.Leifer@inria.fr ‡ University of Cambridge, Computer Laboratory, JJ Thomson Avenue, Cambridge CB3 0FD, U.K. Received 17 October 2004; revised 2 May 2006 A framework is defined within which reactive systems can be studied formally. The framework is based on s-categories, which are a new variety of categories within which reactive systems can be set up in such a way that labelled transition systems can be uniformlyextracted.Theseleadinturntobehaviouralpreordersandequivalences,suchas the failures preorder (treated elsewhere) and bisimilarity, which are guaranteed to be congruential. The theory rests on the notion of relative pushout, which was previously introduced by the authors. The framework is applied to a particular graphical model, known as link graphs, which encompasses a variety of calculi for mobile distributed processes. The specific theory of link graphs is developed. It is then applied to an established calculus, namely condition-event Petri nets. In particular, a labelled transition system is derived for condition-event nets, corresponding to a natural notion of observable actions in Petri-net theory. The transition system yields a congruential bisimilarity coinciding with one derived directly from the observable actions. This yields a calibration of the general theory of reactive systems and link graphs against known specific theories. 1. Introduction 1.1. Background Process calculi have made progress in modelling interactive concurrent systems (Brookes et al. 1984; Bergstra and Klop 1985; Hoare 1985; Milner 1980), systems with mobile connectivity (Milner et al. 1992; Fournet and Gonthier 1996), and systems with mobile locality (Berry and Boudol 1992; Cardelli and Gordon 2000). There is some agreement amongst all these approaches, both in their basic notions and in their theories; perhaps the strongest feature is a good understanding of behavioural specification and equivalence. At the same time, the space of possible calculi is large, we lack a uniform development of their theories, and, in particular, there is no settled way to combine the various kinds of mobility that they model. As shown by Castellani’s comprehensive survey, Castellani (2001), widely varying notions of locality have been explored, and this implies a similar variety in treating mobility.
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