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Coloured Petri Nets

Coloured Petri Nets is a graphical oriented language for design, specification, simulation and verification of systems. It is in particular well-suited for systems that consists of a number of processes which communicate and synchronise. Typical examples of application areas are communication protocols, distributed systems, automated production systems, work flow analysis and VLSI chips.

Brief Introduction to CP-nets

Coloured Petri Nets (CP-nets or CPNs) is a modelling language developed for systems in which communication, synchronisation and resource sharing play an important role. CP-nets combine the strengths of ordinary Petri nets with the strengths of a high-level programming language. Petri nets provide the primitives for process interaction, while the programming language provides the primitives for the definition of data types and the manipulations of data values.

CP-nets has an intuitive, graphical representation which is appealing to human beings. A CPN model consists of a set of modules (pages) which each contains a network of places, transitions and arcs. The modules interact with each other through a set of well-defined interfaces, in a similar way as known from many modern programming languages. The graphical representation makes it easy to see the basic structure of a complex CPN model, i.e., understand how the individual processes interact with each other.

CP-nets also has a formal, mathematical representation with a well-defined syntax and semantics. This representation is the foundation for the definition of the different behavioural properties and the analysis methods. Without the mathematical representation it would have been totally impossible to develop a sound and powerful CPN language. However, for the practical use of CP-nets and their tools, it suffices to have an intuitive understanding of the syntax and semantics. This is analogous to programming languages which are successfully applied by users who are not familiar with the formal, mathematical definitions of the languages.

CPN models can be made with or without explicit reference to time. Untimed CPN models are usually used to validate the functional/logical correctness of a system, while timed CPN models are used to evaluate the performance of the system. There are many other languages which can be used to investigate the functional/logical correctness of a system or the performance of it. However, it is rather seldom to find modelling languages that are well-suited for both kinds of analysis.

CP-nets can be simulated interactively or automatically. In an interactive simulation the user is in control. It is possible to see the effects of the individual steps directly on the graphical representation of the CP-net. This means that the user can investigate the different states and choose between the enabled transitions. An interactive simulation is similar to single-step debugging. It provides a way to "walk through" a CPN model, investigating different scenarios and checking whether the model works as expected. This is in contrast to many off-the-shelf simulation packages which often act as black boxes, where the user can define inputs and inspect the results, but otherwise have very little possibility to understand and validate the models on which the simulations build. It is our experience that the insight and detailed knowledge of a system, which the users gain during the development and validation of a simulation model, is often as important as the results that the users get from the actual simulation runs.

Automatic simulations are similar to program executions. Now the purpose is to be able to execute the CPN models as fast and efficient as possible, without detailed human interaction and inspection. However, the user still needs to interpret the simulation results. For this purpose it is often suitable to use animated, graphical representations providing an abstract, application-specific view of the current state and activities in the system.

CP-nets also offers more formal verification methods, known as state space analysis and invariant analysis. In this way it is possible to prove, in the mathematical sense of the word, that a system has a certain set of behavioural properties. However, industrial systems are often so complex that it is impossible or at least very expensive to make a full proof of system correctness. Hence, the formal verification methods should be seen as a complement to the more informal validation by means of simulation. The use of formal verification is often restricted to the most important subsystems or the most important aspects of a complex system.

CP-nets and their tools have been used in numerous practical projects within a large variety of different application areas. The CPN group at the University of Aarhus, Denmark, has developed two sets of computer tools, supporting the use of CP-nets:

  • Design/CPN was developed in the late 80'ies and early 90'ies. It has been used by 800 different organisations in 60 different countries.
  • CPN Tools was developed from 2000 to 2010 at which time it had more than 10,000 licenses in nearly 150 countries. From the autumn of 2010 CPN Tools is transferred to the AIS group, Eindhoven University of Technology, The Netherlands.

  • The CPN group maintains a list of published papers describing industrial CPN projects. This list contains more than 100 papers.

Why use CP-nets

CP-nets are used for three different - but closely related - purposes. First of all, a CP-net model is a description of the modelled system, and it can be used as a specification (of a system to be built) or as a presentation (of a system to be explained to other people, or ourselves). By creating a model we can investigate a new system before we construct it. This is an obvious advantage, in particular for systems where design errors may jeopardise security or be expensive to correct. Secondly, the behaviour of a CPN model can be analysed, either by means of simulation (which is equivalent to program execution and program debugging) or by means of more formal analysis methods (which are equivalent to program verification). Finally, it should be understood that the process of creating the description and performing the analysis usually gives the modeller a dramatically improved understanding of the modelled system - and it is often the case that this is more valid than the description and the analysis results themselves. Below, we give a brief description of some of the main qualities of CP-nets.

  1. CP-nets have a graphical representation. The graphical form is intuitively very appealing. It is extremely easy to understand and grasp - even for people who are not very familiar with the details of CP-nets. This is due to the fact that CPN diagrams resemble many of the informal drawings which designers and engineers make while they construct and analyse a system.
  2. CP-nets have a well-defined semantics which unambiguously defines the behaviour of each CP-net. The presence of the semantics makes it possible to implement simulators for CP-nets, and it also forms the foundation for the formal analysis methods.
  3. CP-nets are very general and can be used to describe a large variety of different systems. The applications of CP-nets range from informal systems (such as the description of work processes) to formal systems (such as communication protocols). They also range from software systems (such as distributed algorithms) to hardware systems (such as VLSI chips).
  4. CP-nets have very few, but powerful, primitives. The definition of CP-nets is rather short and it builds upon standard concepts which many system modellers already know from mathematics and programming languages. This means that it is relatively easy to learn to use CP-nets. However, the small number of primitives also means that it is easy to develop strong analysis methods.
  5. CP-nets have an explicit description of both states and actions. This is in contrast to most system description languages which describe either the states or the actions - but not both. Using CP-nets, the reader may easily change the point of focus - from states to actions or vice versa.
  6. CP-nets have a semantics which builds upon true concurrency, instead of interleaving. In an interleaving semantics it is impossible to have two actions in the same step, and thus concurrency only means that the actions can occur after each other, in any order. A true concurrency semantics is easier to work with, because it is closer to the way human beings think about concurrent actions.
  7. CP-nets offer hierarchical descriptions. This means that we can construct a large CP-net by relating smaller CP-nets to each other, in a well-defined way. The hierarchy constructs of CP-nets play a role similar to that of subroutines, procedures and modules of programming languages. The existence of hierarchical CP-nets makes it possible to model large systems in a manageable and modular way.
  8. CP-nets integrate the description of control and synchronisation with the description of data manipulation. This means that on a single sheet of paper it can be seen what the environment, enabling conditions and effects of an action are. Many other graphical description languages work with graphs which only describe the environment of an action - while the detailed behaviour is specified separately (often by means of unstructured prose).
  9. CP-nets can be extended with a time concept. This means that it is possible to use the same modelling language for the specification/ validation of functional/logical properties (such as absence of deadlocks) and performance properties (such as average waiting times). The basic idea behind the time extension is to introduce a global clock and to allow each token to carry a time stamp - indicating when it is ready to be consumed by a transition.
  10. CP-nets are stable towards minor changes of the modelled system. This is proved by many practical experiences and it means that small modifications of the modelled system do not completely change the structure of the CP-net. In particular, it is possible to add a new sequential process without changing the net structure representing existing processes.
  11. CP-nets offer interactive simulations where the results are presented directly on the CPN diagram. The simulation makes it possible to debug a large model while it is being constructed - analogously to a good programmer debugging the individual parts of a program as as they are finished The data values of the moving tokens can be inspected.
  12. CP-nets have a number of formal analysis methods by which properties of CP-nets can be proved. The two most important analysis methods are known as occurrence graphs and place invariants.
  13. CP-nets have computer tools supporting their drawing, simulation and formal analysis. This makes it possible to handle even large nets without drowning in details and without making trivial calculation errors. The existence of such computer tools is extremely important for the practical use of CP-nets.

Analysis of CP-nets

CP-nets can be analysed in four different ways:

  1. The first analysis method is interactive simulation. It is very similar to debugging and prototyping. This means that we can execute a CP-net model, to make a detailed investigation of the behaviour of the modelled system. It is possible to set breakpoints and to visualise the simulation results by means of different kinds of graphics, e.g. Message Sequence Charts (also know as event traces).
  2. The second analysis method is automatic simulation which is similar program execution. It allows a fast execution of thousands or millions of transitions. The purpose is to investigate the functional correctness of the system or to investigate the performance of the system, e.g. to identify bottlenecks, to predict the use of buffer space or the mean/maximal service time.
  3. The third analysis method is state spaces (also called occurrence graphs or reachability graphs). The basic idea behind occurrence graphs is to construct a directed graph which has a node for each reachable system state and an arc for each possible state change. Obviously, such a graph may become very large, even for small CP-nets. However, it can be constructed and analysed totally automatically, and there exist techniques which makes it possible to work with condensed occurrence graphs without losing analytic power. These techniques build upon equivalence classes.
  4. The fourth analysis method is place invariants. This method is very similar to the use of invariants in ordinary program verification. The user constructs a set of equations which is proved to be satisfied for all reachable system states. The equations are used to prove properties of the modelled system, e.g., absence of deadlock.

History of Petri Nets

Petri nets were originally developed in the 60'ies and the 70'ies, and they were soon recognised as being one of the most adequate and sound languages for description and analysis of synchronisation, communication and resource sharing between concurrent processes. However, attempts to use Petri nets in practice revealed two serious drawbacks. First of all, there were no data concepts and hence the models often became excessively large, because all data manipulation had to be represented directly into the net structure (i.e., by means of places and transitions). Secondly, there were no hierarchy concepts, and thus it was not possible to build a large model via a set of separate submodels with well-defined interfaces.

The development of high-level Petri nets in the late 70'ies and hierarchical Petri nets in the late 80'ies removed these two serious problems. Coloured Petri Nets (also called CP-nets or CPN) is the most well-known kind of high-level Petri nets. CP-nets incorporate both data structuring and hierarchical decomposition - without compromising the qualities of the original Petri nets.