Petri Nets and Discrete Events Systems

Introduction

The interaction between perception and/or memorization with information exposing time are consistent with a theory which relates processing time to perceived duration, see Thomas (1975). It is mainly due because such temporal information is obtained from both a sort of timer processing and a visual information processing, so attention effectiveness is a function of these two processings, in which the range of durations become so important that it has been proved that there is a minimum quantum of time (discrete measurements) below which neither work. These facts also open the field of considering timed formalisms (Pelayo, 2005) for properly describing and analyzing these issues.

In Wang (2002), a seed study of the Cognitive Models of the brain can be found. Among others, a description of the memorizing process is there shown, moreover Wang has given an implementation of this cognitive process in RTPA, which has been taken by the authors of this paper as reference in Barquilla (2008) to formally describe this cognitive process

Petri Nets (Petri, 1966) and Dynamical Systems (Arrowsmith, 1990) have increased their popularity during the last decades. Both mathematical tools have many analysis methods available and have become classical for modeling discrete-event processes.

In fact, many papers (David, 1991, 1994; Silva, 1990) show how discrete-event processes can be modeled by Petri Nets. There are also papers that present how to model these processes by discrete-event dynamical systems (Foursov, 2006).

Thus, it motivates to analyze if there exists any relation between these tools and what advantages are taking each one. Mainly, we wonder if a given Petri net corresponds directly to dynamical systems and the Petri Net properties have counterparts in the corresponding dynamical system.

In Foursov (2006), the authors make a partial attempt to do that, i.e., based on formal power series, they obtain an algorithm that allows checking whether a given weighted Petri net corresponds to a continuous polynomial dynamical system. But in this context, different initial states of the Petri net (different initial markings) could correspond to different dynamical systems.

Therefore, the work of giving a correspondence between Petri nets and dynamical systems was incomplete and we proposed (Guirao, 2011) different ways that try to solve not only the problem of reachability, but also the one of describing the periodicity of markings and their similarity.

The structure of the paper is as follows: We show how Petri nets can be used to model discrete-event processes. It is explained how to define the state space and the evolution operator in order to encode Petri nets as discrete dynamical systems. The next section is dedicated to introduce a metric for the states space; it is based on the Bayre metric, usually employed when modeling computer processes. We deal with giving a proper metric for non-bounded Petri nets. Finally, some conclusions and further research directions are presented.