Theory of regions

The Theory of regions is an approach for synthesizing a Petri net from a transition system. As such, it aims at recovering concurrent, independent behavior from transitions between global states. Theory of regions handles elementary net systems as well as P/T nets and other kinds of nets. An important point is that the approach is aimed at the synthesis of unlabeled Petri nets only.

Definition

A region of a transition system ${\displaystyle (S,\Lambda ,\rightarrow )}$ is a mapping assigning to each state ${\displaystyle s\in S}$ a number ${\displaystyle \sigma (s)}$ (natural number for P/T nets, binary for ENS) and to each transition label a number ${\displaystyle \tau (\ell )}$ such that consistency conditions ${\displaystyle \sigma (s')=\sigma (s)+\tau (\ell )}$ holds whenever ${\displaystyle (s,\ell ,s')\in \rightarrow }$.^[1]

Intuitive explanation

Each region represents a potential place of a Petri net.

Mukund: event/state separation property, state separation property.^[2]