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

### Intuitive explanation

Each region represents a potential place of a Petri net.

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

## References

1. ↑ "Madhavan Mukund". http://www.cmi.ac.in/~madhavan/papers/m-ijfocs92.php.
2. ↑ Mukund, Madhavan (1992-12-01). "Petri nets and step transition systems". International Journal of Foundations of Computer Science 03 (4): 443–478. doi:10.1142/S0129054192000231. ISSN 0129-0541. https://www.worldscientific.com/doi/abs/10.1142/S0129054192000231.

* Badouel, Eric; Darondeau, Philippe (1998), Reisig, Wolfgang; Rozenberg, Grzegorz, eds., "Theory of regions" (in en), Lectures on Petri Nets I: Basic Models: Advances in Petri Nets, Lecture Notes in Computer Science (Berlin, Heidelberg: Springer): pp. 529–586, doi:10.1007/3-540-65306-6_22, ISBN 978-3-540-49442-3

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