César Fernández

D-continuous causal nets: A model of non-sequential processes

We propose a model of non-sequential processes and study its properties. The model is called D-continuous causal nets (CCNs) and is based on a class of Petri nets. A causal net is essentially a partially ordered set of conditions and events which represents a fragment of the history of a concurrent system. D-continuity is a generalization of Dedekinds definition of the continuity (completeness) of the reals which is applicable to both totally and partially ordered sets as also to both dense and discrete structures. We develop a number of properties of our model and discuss why these might be useful properties for a non-sequential process to have.

A Note on D-Continuous Causal Nets

The aim of this paper is to augment the theory of D-continuous causal nets (CCN’s) initiated in [1]. A CCN is a model of non-sequential processes. Our motivation in formulating the model was to set up a formalism in which the properties of a non-sequential processes [2,3] could be exposed. Among these properties the one called generalized Dedekind continuity (D-continuity) is particularly intriguing in that it points to a way to bridge the gap between discrete and “continuous” modes of descriptions of the same phenomena. Consequently we decided to endow our process model with this property and study the resulting structure. Somewhat to our surprise, we have found that our model exhibits most of the properties that have been proposed by C.A.Petri. In other words, D-continuity not only has an important motivation and interpretation, it also has a variety of attractive mathematical consequences.

Connections Between Systems and Processes

Having defined the processes of a system net via the process definition 3.3.3, or (as we know: equivalently for system nets of finite synchronisation) via the construction given in Definition 3.4.1, it is now possible to translate properties from a system to the set of its processes and vice versa. It is possible to ask, for example, what it would mean for a system that all of its processes enjoy some interesting property, or conversely.

Some remarks on D-continuity

One afternoon in 1978 (or was it in 1977 ?), C.A. Petri called one of the authors of this paper to his room. He wanted to talk about some “axioms of concurrency”. By that time, we knew already that a non-sequential process -viewed as a record of conditions-holdings and events-occurrences of a system- could be modelled using a special type of net called occurrence net, [7], [6]. Since at the behavior level, cycles are unrolled, one may associate to each occurrence net a partially ordered set, poset for short, which is an appropriate mathematical machinery to study non-sequential processes.

Non-sequential Processes

This paper present a select1on of some propert1es of a non-sequentlal process. Three types of propert1es are stud1ed, namely: d1screteness propert1es, dens1ty properties and the o-continu1ty property. Relat1ons between these propert1es are establ1shed.