Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

## Processes in Petri nets

Grzegorz Bancerek
Bialystok Technical University
Mitsuru Aoki
Shinshu University, Nagano
Akio Matsumoto
Shinshu University, Nagano
Yasunari Shidama
Shinshu University, Nagano

### Summary.

Sequential and concurrent compositions of processes in Petri nets are introduced. A process is formalized as a set of (possible), so called, firing sequences. In the definition of the sequential composition the standard concatenation is used $$R_1 \mathop{\rm before} R_2 = \{p_1\mathop{^\frown}p_2: p_1\in R_1\ \land\ p_2\in R_2\}$$ The definition of the concurrent composition is more complicated $$R_1 \mathop{\rm concur} R_2 = \{ q_1\cup q_2: q_1\ {\rm misses}\ q_2\ \land\ \mathop{\rm Seq} q_1\in R_1\ \land\ \mathop{\rm Seq} q_2\in R_2\}$$ For example, $$\{\langle t_0\rangle\} \mathop{\rm concur} \{\langle t_1,t_2\rangle\} = \{\langle t_0,t_1,t_2\rangle , \langle t_1,t_0,t_2\rangle , \langle t_1,t_2,t_0\rangle\}$$ The basic properties of the compositions are shown.

#### MML Identifier: PNPROC_1

The terminology and notation used in this paper have been introduced in the following articles [13] [12] [18] [5] [17] [9] [1] [3] [6] [11] [15] [2] [14] [7] [16] [8] [10] [4]

#### Contents (PDF format)

1. Preliminaries
2. Markings on Petri Nets
3. Transitions and Firing
4. Firing Sequences of Transitions
5. Sequential Composition
6. Concurrent Composition
7. Neutral Process

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