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.

Abstract. In this paper we give an overview, how to apply region based methods for the synthesis of Petri nets from languages to process mining.The research domain of process mining aims at constructing a process model from an event log, such that the process model can reproduce the log, and does not allow for much more behaviour than shown in the log. We here consider Petri nets to represent process models. Event logs can be interpreted as finite languages. Region based synthesis methods can be used to construct a Petri net from a language generating the minimal net behaviour including the given language. Therefore, it seems natural to apply such methods in the process mining domain. There are several different region based methods in literature yielding different Petri nets. We adapt these methods to the process mining domain and compare them concerning efficiency and usefulness of the resulting Petri net.

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We introduce negotiations, a model of concurrency close to Petri nets, with multiparty negotiation as primitive. We study the problems of soundness of negotiations and of, given a negotiation with possibly many steps, computing a summary, i.e., an equivalent one-step negotiation. We provide a complete set of reduction rules for sound, acyclic, weakly deterministic negotiations and show that, for deterministic negotiations, the rules compute the summary in polynomial time.

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