---
title: "Theory of regions"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Theory_of_regions"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:39:47.623283+00:00"
instance: "kb-cron"
---

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


=== Intuitive explanation ===
Each region represents a potential place of a Petri net.
Mukund: event/state separation property, state separation property.


== References ==

Badouel, Eric; Darondeau, Philippe (1998), Reisig, Wolfgang; Rozenberg, Grzegorz (eds.), "Theory of regions", 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{{citation}}:  CS1 maint: work parameter with ISBN (link)