Volume 14, 2002

University of Bialystok

Copyright (c) 2002 Association of Mizar Users

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

- 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.

- Preliminaries
- Markings on Petri Nets
- Transitions and Firing
- Firing Sequences of Transitions
- Sequential Composition
- Concurrent Composition
- Neutral Process

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek and Andrzej Trybulec.
Miscellaneous facts about functions.
*Journal of Formalized Mathematics*, 8, 1996. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
Patricia L. Carlson and Grzegorz Bancerek.
Context-free grammar --- part I.
*Journal of Formalized Mathematics*, 4, 1992. - [9]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Beata Madras.
Product of family of universal algebras.
*Journal of Formalized Mathematics*, 5, 1993. - [11]
Andrzej Trybulec.
Binary operations applied to functions.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Andrzej Trybulec.
Domains and their Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [14]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [16]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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