On the Semantics of Petri Nets

Petri Place/Transition (PT) nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the "token game"' is too intensional, even in its more abstract interpretations in term of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category {PTNets} of PT nets, we introduce a category {DecOcc} of decorated (nondeterministic) occurrence nets and we define adjunctions between {PTNets} and {DecOcc} and between {DecOcc} and {Occ}, the category of occurrence nets. The role of {DecOcc} is to provide natural unfoldings for PT nets, i.e. acyclic safe nets where a notion of family is used for relating multiple instances of the same place. The unfolding functor from {PTNets} to {Occ} reduces to Winskel's when restricted to safe nets, while the standard coreflection between {Occ} and {Dom}, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between {PTNets} and {Dom}

Extension of Petri Nets by Aspects to Apply the Model Driven Architecture Approach

Within MDA models are usually created in the UML. However, one may prefer to\ud use different notations such as Petri-nets, for example, for modelling concurrency\ud and synchronization properties of systems. This paper claims that techniques that\ud are adopted within the context of MDA can also be beneficial in modelling systems\ud by using notations other than the UML. Petri-Nets are widely used for modelling\ud of business and application logic of information systems with web services. For\ud certain kinds of applications, therefore, Petri Nets can be more suitable for building\ud Computation Independent, Platform Independent and Platform Specific Models\ud (CIM, PIM and PSM). Unfortunately, the well-known problems with separation of\ud concerns in Petri Nets and keeping track of changes may hinder achieving the aim of\ud MDA: building reusable, portable and interoperable models. In this paper we define\ud Aspect Petri Nets as a structure of several Petri Nets and quantification rules for\ud weaving of those Petri Nets. Aspect Petri Nets are suitable for application of MDA;\ud they support traceability of changes and reusability, portability and interoperability\ud of models. We illustrate advantages of modelling in Aspect Petri Nets for MDA\ud application and describe necessary tool support

Enhancing workflow-nets with data for trace completion

The growing adoption of IT-systems for modeling and executing (business) processes or services has thrust the scientific investigation towards techniques and tools which support more complex forms of process analysis. Many of them, such as conformance checking, process alignment, mining and enhancement, rely on complete observation of past (tracked and logged) executions. In many real cases, however, the lack of human or IT-support on all the steps of process execution, as well as information hiding and abstraction of model and data, result in incomplete log information of both data and activities. This paper tackles the issue of automatically repairing traces with missing information by notably considering not only activities but also data manipulated by them. Our technique recasts such a problem in a reachability problem and provides an encoding in an action language which allows to virtually use any state-of-the-art planning to return solutions

ACP Semantics for Petri Nets

The paper deals with algebraic semantics for Petri nets, based on process algebra ACP. The semantics is defined by assigning a special variable to every place of given Petri net, expressing the process initiated in the place. Algebraic semantics of the Petri net is then defined as a parallel composition of all the variables, where corresponding places hold tokens within the initial marking. Resulting algebraic specification preserves operational behavior of the original net-based specification

Finite petri nets as models for recursive causal behaviour

Goltz (1988) discussed whether or not there exist finite Petri nets (with unbounded capacities) modelling the causal behaviour of certain recursive CCS terms. As a representative example, the following term is considered: \ud \ud B=(a.nilb.B)+c.nil. \ud \ud We will show that the answer depends on the chosen notion of behaviour. It was already known that the interleaving behaviour and the branching structure of terms as B can be modelled as long as causality is not taken into account. We now show that also the causal behaviour of B can be modelled as long as the branching structure is not taken into account. However, it is not possible to represent both causal dependencies and the behaviour with respect to choices between alternatives in a finite net. We prove that there exists no finite Petri net modelling B with respect to both pomset trace equivalence and failure equivalence

The use of Petri nets for modeling pipelined processors

This paper discusses the use of Petri Nets for modeling and analyzing pipelined processors. Petri Nets are particularly well-suited to modeling the synchronization, buffering, resource contention and delicate timing so common in pipelined processors. Tools for simulating, animating and analyzing the behavior of the models are described. The usefulness of the tools and the analysis methods they support in evaluating the performance and analyzing the detailed timing of pipelined microprocessors is illustrated through an example

Formal Relationships Between Geometrical and Classical Models for Concurrency

A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any bunch of events, thus generalizing the principle of true concurrency. While they seem to be very promising - because they make possible the use of techniques from algebraic topology in order to study concurrent computations - they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel

Petri nets for systems and synthetic biology

We give a description of a Petri net-based framework for modelling and analysing biochemical pathways, which uni¯es the qualita- tive, stochastic and continuous paradigms. Each perspective adds its con- tribution to the understanding of the system, thus the three approaches do not compete, but complement each other. We illustrate our approach by applying it to an extended model of the three stage cascade, which forms the core of the ERK signal transduction pathway. Consequently our focus is on transient behaviour analysis. We demonstrate how quali- tative descriptions are abstractions over stochastic or continuous descrip- tions, and show that the stochastic and continuous models approximate each other. Although our framework is based on Petri nets, it can be applied more widely to other formalisms which are used to model and analyse biochemical networks

Forward Analysis and Model Checking for Trace Bounded WSTS

We investigate a subclass of well-structured transition systems (WSTS), the bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete deterministic ones, which we claim provide an adequate basis for the study of forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth. Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered previously for the termination of forward analysis, boundedness is decidable. Boundedness turns out to be a valuable restriction for WSTS verification, as we show that it further allows to decide all $\omega$-regular properties on the set of infinite traces of the system