There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The surface science of titanium dioxide

From the Publisher:
This book presents a unified theory of Generalized Stochastic Petri Nets (GSPNs) together with a set of illustrative examples from different application fields. The continuing success of GSPNs and the increasing interest in using them as a modelling paradigm for the quantitative analysis of distributed systems suggested the preparation of this volume with the intent of providing newcomers to the field with a useful tool for their first approach. Readers will find a clear and informal explanation of the concepts followed by formal definitions when necessary or helpful. The largest section of the book however is devoted to showing how this methodology can be applied in a range of domains.

Modelling with Generalized Stochastic Petri Nets

Coloured Petri Nets (CPNs) is a language for the modelling and validation of systems in which concurrency, communication, and synchronisation play a major role. Coloured Petri Nets is a discrete-event modelling language combining Petri nets with the functional programming language Standard ML. Petri nets provide the foundation of the graphical notation and the basic primitives for modelling concurrency, communication, and synchronisation. Standard ML provides the primitives for the definition of data types, describing data manipulation, and for creating compact and parameterisable models. A CPN model of a system is an executable model representing the states of the system and the events (transitions) that can cause the system to change state. The CPN language makes it possible to organise a model as a set of modules, and it includes a time concept for representing the time taken to execute events in the modelled system. CPN Tools is an industrial-strength computer tool for constructing and analysing CPN models. Using CPN Tools, it is possible to investigate the behaviour of the modelled system using simulation, to verify properties by means of state space methods and model checking, and to conduct simulation-based performance analysis. User interaction with CPN Tools is based on direct manipulation of the graphical representation of the CPN model using interaction techniques, such as tool palettes and marking menus. A license for CPN Tools can be obtained free of charge, also for commercial use.

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Coloured Petri Nets and CPN Tools for modelling and validation of concurrent systems

Under the umbrella of buzzwords such as “Business Activity Monitoring” (BAM) and “Business Process Intelligence” (BPI) both academic (e.g., EMiT, Little Thumb, InWoLvE, Process Miner, and MinSoN) and commercial tools (e.g., ARIS PPM, HP BPI, and ILOG JViews) have been developed. The goal of these tools is to extract knowledge from event logs (e.g., transaction logs in an ERP system or audit trails in a WFM system), i.e., to do process mining. Unfortunately, tools use different formats for reading/storing log files and present their results in different ways. This makes it difficult to use different tools on the same data set and to compare the mining results. Furthermore, some of these tools implement concepts that can be very useful in the other tools but it is often difficult to combine tools. As a result, researchers working on new process mining techniques are forced to build a mining infrastructure from scratch or test their techniques in an isolated way, disconnected from any practical applications. To overcome these kind of problems, we have developed the ProM framework, i.e., an “pluggable” environment for process mining. The framework is flexible with respect to the input and output format, and is also open enough to allow for the easy reuse of code during the implementation of new process mining ideas. This paper introduces the ProM framework and gives an overview of the plug-ins that have been developed.

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The prom framework: a new era in process mining tool support

The Petri Net Markup Language (PNML) is an XML-based interchange format for Petri nets. In order to support different versions of Petri nets and, in particular, future versions of Petri nets, PNML allows the definition of Petri net types. Due to this flexibility, PNML is a starting point for a standard interchange format for Petri nets. This paper discusses the design principles, the basic concepts, and the underlying XML technology of PNML. The main purpose of this paper is to disseminate the ideas of PNML and to stimulate discussion on and contributions to a standard Petri net interchange format.

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The Petri net markup language: concepts, technology, and tools

CPN Tools is a tool for editing, simulating and analysing Coloured Petri Nets. The GUI is based on advanced interaction techniques, such as toolglasses, marking menus, and bi-manual interaction. Feedback facilities provide contextual error messages and indicate dependency relationships between net elements. The tool features incremental syntax checking and code generation which take place while a net is being constructed. A fast simulator efficiently handles both untimed and timed nets. Full and partial state spaces can be generated and analysed, and a standard state space report contains information such as boundedness properties and liveness properties. The functionality of the simulation engine and state space facilities are similar to the corresponding components in Design/CPN, which is a widespread tool for Coloured Petri Nets.

CPN tools for editing, simulating, and analysing coloured Petri nets

Coloured Petri nets (CP-nets or CPNs) provide a framework for the design, specification, validation, and verification of systems. CP-nets have a wide range of application areas and many CPN projects have been carried out in industry, e.g., in the areas of communication protocols, operating systems, hardware designs, embedded systems, software system designs, and business process re-engineering. Design/CPN is a graphical computer tool supporting the practical use of CP-nets. The tool supports the construction, simulation, and functional and performance analysis of CPN models. The tool is used by more than four hundred organisations in forty different countries – including one hundred commercial companies. It is available free of charge, also for commercial use. This paper provides a comprehensive road map to the practical use of CP-nets and the Design/CPN tool. We give an informal introduction to the basic concepts and ideas underlying CP-nets. The key components and facilities of the Design/CPN tool are presented and their use illustrated. The paper is self-contained and does not assume any prior knowledge of Petri nets and CP-nets nor any experience with the Design/CPN tool.

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The practitioner's guide to coloured Petri nets

In this paper we show how to extend Coloured Petri Nets (CP-nets), with three new modelling primitives—place capacities, test arcs and inhibitor arcs. The new modelling primitives are introduced to improve the possibilities of creating models that are on the one hand compact and comprehensive and on the other hand easy to develop, understand and analyse. A number of different place capacity and inhibitor concepts have been suggested earlier, e.g., integer and multi-set capacities and zero-testing and threshold inhibitors. These concepts can all be described as special cases of the more general place capacity and inhibitor concepts defined in this paper.

Coloured Petri Nets Extended with Place Capacities, Test Arcs and Inhibitor Arcs

This paper shows how two of the most important analysis methods for Petri nets can be performed in a modular way. We illustrate our techniques by means of modular Place/Transitions nets (modular PT-nets) in which the individual modules interact via shared places and shared transitions. For place invariants we show that it is possible to construct invariants of the total modular PT-net from invariants of the individual modules. For state spaces, we show that it is possible to decide behavioural properties of the modular PT-net from state spaces of the individual modules plus a synchronization graph, without unfolding to the ordinary state space. The generalization of our techniques to high-level Petri nets is rather straightforward.

Søren Christensen

Papers

The Petri net markup language: concepts, technology, and tools

CPN tools for editing, simulating, and analysing coloured Petri nets

The practitioner's guide to coloured Petri nets

Coloured Petri Nets Extended with Place Capacities, Test Arcs and Inhibitor Arcs

Modular Analysis of Petri Nets