Dr. Carl Adam Petri, former honorary professor

Dr. Mark-Oliver Stehr, former research assistant

Dr. Olaf Kummer, former research assistant

Stefan Haar

Uwe Fenske

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Duration:
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since
04/1994

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Keywords:
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Concurrency-Theorie, Axiome, Elementare Netz-Systeme, Zyklische Ordnungen, Markierte Graphen

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Objectives:
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The aim of this project is the investigation and coherent presentation of axiomatic concurrency theory. Axiomatic concurrency theory was originally proposed by C.A. Petri as a foundation for net theory based on the notions of causality and concurrency. Mathematically these two concepts are formalized as symmetric binary relations on the domain of (local) states and events. Physical considerations play a major role for the motivation of axioms and interpretation of results. The emphasis of this project is put on the analysis of interactions between different sets of axioms from a mathematical point of view.

An introduction to concurrency theory and in particular to the
historical, philosophical and physical background is given by Hartmut
Müller in *Geschichte und Entwicklung der Concurrency
Theorie*. Also most of the original axiomatic systems and ideas of
C.A.Petri are presented in this work.

In *Physically Motivated Axiomatic Concurrency Theory - A Posetless
Approach* (and in the revised version *Concurrency Theory of
Cyclic and Acyclic Processes*) Mark-Oliver Stehr treats concurrency
theory without the assumption of an underlying partial order from
which the causality relation is derived. In addition to (infinite)
acyclic models this leads to consideration of cyclic but finite models
leading to several difficulties. Furthermore this work explores the
possibility of associating a net and even an elementary net system
with the concurrency structure in a natural way. Some sufficient
conditions for a succesfull connection between these worlds are
proposed.

Olaf Kummer gave a coherent formal treatment of the subject in
*Axiomensysteme für die Theorie der Nebenläufigkeit*.
This work focuses on the difficulties (which are illustrated by
strange models) arising in concurrency theory and presents possible
solutions. For this task serveral new properties are formulated and
their connections are explored. The work concludes with a comparison
of different axiomatic systems. One of these shows a strong relation
to the notion of D-continuity, which was developed by C.A.Petri as a
(partial order) generalization of Dedekind continuity.
In the meantime this work was also published as a book.

The work *Zyklische Ordnungen - Axiome und einfache Eigenschaften*
by Mark-Oliver Stehr describes a mathematical theory of
(partial) cyclic orders. This theory was developed by the author
in order to cope with the problems arising when the underlying
partial order structure of causality is given up and cyclic models
are considered. Although this was the initial motivation, it seems
that application for cyclic orders arise in several field where
cyclic dependencies are relevant.

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Publications:
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- Hartmut Müller:
*Geschichte und Entwicklung der Concurrency Theorie.*Diplomarbeit, Universität Hamburg, Fachbereich Informatik, 1993. - Mark-Oliver Stehr:
*Physically Motivated Axiomatic Concurrency Theory - A Posetless Approach*, Studienarbeit, Fachbereich Informatik, Universität Hamburg, December 1993. -
Axiomensysteme für die Theorie der Nebenläufigkeit.
- Diplomarbeit, Universität Hamburg, Fachbereich Informatik, 1996.
- Logos Verlag, Berlin, 1996. ISBN 3-931216-28-4.

- Mark-Oliver Stehr:
*Zyklische Ordnungen - Axiome und einfache Eigenschaften*, Diplomarbeit, Fachbereich Informatik, Universität Hamburg, 1996. - Mark-Oliver Stehr:
*Concurrency Theory of Cyclic and Acyclic Processes*, Fachbereichsbericht FBI-HH-B-190/96, Universität Hamburg, Fachbereich Informatik, September 1996. - Olaf Kummer, Mark-Oliver Stehr:
*Petri's Axioms of Concurrency - A Selection of Recent Results*, Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997