Free Choice Petri Nets

Abstract

Library of Congress cataloguing in publication data available British Library cataloguing in publication data available ISBN 0 521 46519 2 hardback 3.2 T-systems 4 Liveness in free-choice systems 4.1 Free-choice systems 4.2 Stable predicates: siphons and traps 4.3 Commoner's Theorem 4.4 The non-liveness problem is NP-complete 4.5 Minimal siphons 4.6 Liveness and deadlock-freedom 5 The Coverability Theorems 5.1 The S-coverability Theorem 5.2 Derived results 5.3 The T-coverability Theorem 5.4 Derived results vi Contents 6 The Rank Theorem 111 6.1 Characterizations of well-formedness Ill 6.2 The non-well-formed case 6.3 The well-formed case 6.4 Derived results 7 Reduction and synthesis 7.1 Basic notions 7.2 The reduction rules 7.3 An example of reduction 7.4 Completeness 7.5 Synthesis rules 8 Home markings 8.1 Existence of home markings 8.2 A characterization of the home markings 8.3 Derived results 9 Reachability and shortest sequences 9.1 The Reachability Theorem 9.2 The Shortest Sequence Theorem 10 Generalizations 10.1 Asymmetric-choice nets 10.2 A necessary condition for well-formedness 10.3 A sufficient condition for well-formedness Index 235 List of symbols List of main results Preface Free-choice Petri nets have been around for more than twenty years, and are a successful branch of net theory. Nearly all the introductory texts on Petri nets devote some pages to them. This book is intended for those who wish to go further. It brings together the classical theorems of free-choice theory obtained by Commoner and Hack in the seventies, and a selection of new results, like the Rank Theorem, which were so far scattered among papers, reports and theses, some of them difficult to access. Much of the recent research which found its way into the book was funded by the ESPRIT II BRA Action DEMON, and the ESPRIT III Working Group CALIBAN. The book is self-contained, in the sense that no previous knowledge of Petri nets is required. We assume that the reader is familiar with naive set theory and with some elementary notions of graph theory (e.g. path, circuit, strong connectedness) and linear algebra (e.g. linear independence, rank of a matrix). One result of Chapter 4 requires some knowledge of the theory of NP-completeness. The book can be the subject of an undergraduate course of one semester if the proofs of the most difficult theorems are omitted. If they are included, we suggest the course be restricted to Chapters 1 through 5, which contain most of the classical results on Sand T-systems and free-choice Petri nets. A postgraduate course could cover the whole book. All chapters are accompanied by a list of exercises. Difficult exercises are marked with asterisks. We would like to express our warmest thanks to the many people who have helped us to write the book. Eike Best encouraged us, offered advice and criticism, and was a good friend. Raymond Devillers flooded us with helpful comments, and corrected many mistakes.

Workflow Verification: Finding Control-Flow Errors Using Petri-Net-Based Techniques

“…Best [7], Desel and Esparza [10,9,12], Hack [15]) because they seem to be a good compromise between expressive power and analyzability. It is a class of Petri nets for which strong theoretical results and efficient analysis techniques exist.…”

“…It is a class of Petri nets for which strong theoretical results and efficient analysis techniques exist. For example, the well-known Rank Theorem (Desel and Esparza [10]) enables us to formulate the following corollary. Finding Control-Flow Errors Using Petri-Net-Based Techniques …”

“…Therefore, the problem of deciding whether´ÈAE µ is live and bounded can be solved in polynomial time (Rank Theorem [10]). By Theorem 1, this corresponds to soundness.…”

“…ÈAE is free-choice and well-formed. Hence, ÈAE is Scoverable [10], i.e., each place is part of an embedded strongly connected state-machine component. Since initially there is just one token´ÈAE µ is safe and so is´ÈAE µ. Ù Ø Safeness is a desirable property, because it makes no sense to have multiple tokens in a place representing a condition.…”

Structural Methods to Improve the Symbolic Analysis of Petri Nets

“…Petri nets (PNs) are a graph-based mathematical formalism that allows to describe systems modeling causality, concurrency and con ict relations among its events 16,7]. In particular, PNs play an important role in the synthesis and veri cation of concurrent systems.…”