Elementary Transition Systems Research Articles

Simple models for articulating complex work processes

Adaptability of workflow models can be achieved if the specification module of the Workflow Management System (WMS) is based on a theoretical kernel, allowing minimal effort for the creation of a workflow model capable of providing, when necessary, any view (and the related services) the users may want of it. The specification module of the Milano WMS has been developed on the basis of the above assumption [2, 3], so that it supports the articulation of complex work processes. Milano [1] is a system integrating in the user workspace a WMS with a multimedia conversation handler, and, finally, an object repository where an organizational handbook is stored.The workflow models of Milano constitute a small subcategory of ENS (Elementary Net Systems), namely Free-Choice Acyclic ENS. Free-choice Acyclic ENS have some nice, efficiently computable, properties: there is a synthesis algorithm from Elementary Transition Systems (ETS) to ENS so that the user can have different views of the workflow; it is possible to compute and classify the jumps recovering exceptions; morphisms between models allow an efficient correctness check for changes and support on-the-fly enactment of changes. The formal aspects of the Milano WMS are described in [2, 3]. Figure 1 presents the Workflow Net Model (WNM) (in the state {b4, b9}) of the first part of a Credit Procedure.

Deriving Petri nets from finite transition systems

This paper presents a novel method to derive a Petri net from any specification model that can be mapped into a state-based representation with arcs labeled with symbols from an alphabet of events (a Transition System, TS). The method is based on the theory of regions for Elementary Transition Systems (ETS). Previous work has shown that, for any ETS, there exists a Petri Net with minimum transition count (one transition for each label) with a reachability graph isomorphic to the original Transition System. Our method extends and implements that theory by using the following three mechanisms that provide a framework for synthesis of safe Petri nets from arbitrary TSs. First, the requirement of isomorphism is relaxed to bisimulation of TSs, thus extending the class of synthesizable TSs to a new class called Excitation-Closed Transition Systems (ECTS). Second, for the first time, we propose a method of PN synthesis for an arbitrary TS based on mapping a TS event into a set of transition labels in a PN. Third, the notion of irredundant region set is exploited, to minimize the number of places in the net without affecting its behavior. The synthesis method can derive different classes of place-irredundant Petri Nets (e.g., pure, free choice, unique choice) from the same TS, depending on the constraints imposed on the synthesis algorithm. This method has been implemented and applied in different frameworks. The results obtained from the experiments have demonstrated the wide applicability of the method.

The synthesis problem of Petri nets

The synthesis problem of concurrent systems is the problem of synthesizing a concurrent system model from sequential observations. The paper studies the synthesis problem for elementary Petri nets and transition systems. A characterization of the class of transition systems which correspond to elementary Petri nets is proven. It is shown how to generate all elementary Petri nets corresponding to a given transition system. If there is any such elementary Petri net, it is proven that there always exists a small one which has only polynomially many elements in the size of the transition system.

Transition-Systems, Event Structures, and Unfoldings

A subclass of transition systems called elementary transition systems can be identified with the help of axioms based on a structural notion called regions. Elementary transition systems have been shown to be the transition system model of a basic system model of net theory called elementary net systems. Here we show that by smoothly strengthening the regional axioms for elementary transition systems, one obtains a subclass called occurrence transition system. We then prove that occurrence transition systems are the transition system model of yet another basic model of concurrency, namely, prime event structures. We then propose an operation of unfolding elementary transition systems into occurrence transition systems, We prove that it is "correct" in a strong categorical sense.

Generating BDDs for symbolic model checking in CCS

Finite transition systems can easily be represented by binary decision diagrams (BDDs) through the characteristic function of the transition relation. Burch et al. have shown how model checking of a powerful version of the µ-calculus can be performed on such BDDs. In this paper we show how a BDD can be generated from elementary finite transition systems given as BDDs by applying the CCS operations of parallel composition, restriction, and relabelling. The resulting BDDs only grow linearly in the number of parallel components. This way bisimilarity checking can be performed for processes out of the reach of conventional process algebra tools.

Elementary transition systems and refinement

Elementary transition systems are-in a strong categorical sense-the transition system version of a basic system model of net theory called elementary net systems. The structural notion of a region associated with elementary transition systems captures the intuitive idea of a local state as modelled by the conditions of an elementary net system. In this paper we equip elementary transition systems with a refinement operation over the local states (regions). We then show our operation satisfies a number of interesting properties. In particular, this operation supports compositional reasoning. It is very hard if not impossible to define a corresponding operation at the level of nets which enjoys similar properties. This is due to the concrete choice of conditions used to enforce intended behaviour. Thus our results show that the more abstract-but essentially equivalent-model of elementary transition systems is the appropriate framework for theoretical studies concerning refinement operations for elementary net systems.

Transition System Models for Concurrency

<p>Labelled transition systems can be extended to faithfully model concurrency by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step, We can characterize a subclass of these <em>step transition systems</em>, called PN-transition systems, which describe the behaviour of Petri nets.</p><p>This correspondence is formally described in terms of a coreflection between a category of <em>PN</em>-transition systems and a category of Petri nets.</p><p>In this paper, we show that we can define subcategories of <em>PN</em>-transition systems whose objects are <em> safe PN-transition systems and elementary PN-transition systems</em> such that there is a coreflection between these subcategories and subcategories of our category of Petri nets corresponding to safe nets and elementary net systems.</p><p>We also prove that our category of elementary <em>PN</em>-transition systems is equivalent to the category of (sequential) <em> elementary transition systems</em> defined by Nielsen, Rozenberg and Thiagarajan, thereby establishing that the concurrent behaviour of an elementary net system can be completely recovered from a description of its sequential behaviour. Finally, we establish a coreflection between our category of safe <em>PN</em>-transition system and a subcategory of <em>asynchronous transition systems</em> which has been shown by Winskel and Nielsen to be closely linked to safe nets.</p>

Elementary transition systems

Transition systems are a simple and powerful formalism for explaining the operational behaviour of models of concurrency. They provide a common framework for investigating the interrelationships between different approaches to the study of distributed systems. Hence an important question to be answered is: which subclass of transition systems corresponds to a particular model of distribted systems? In this paper we provide an answer to this question for elementary net systems. Within net theory, which is one well-established theory of distributed systems, elementary net systems constitute a basic systems model. Using this model, fundamental concepts such as causality, concurrency, conflict and confusion can be clearly defined and separated from each other (see [ 151). Much is known about the behavioural aspects of elementary net systems in terms of trace theory, nonsequential processes and event structures as shown in [lo]. Trace theory was initiated by Mazurkiewicz [7] (see also [l]). The theory of nonsequential processes originates from the work of Petri [12]; see also [2]. Event structures arose out of the work of Nielsen, Plotkin and Winskel [9] and they now possess a rich theory mainly due to the efforts of Winskel [18]. Elementary net systems also have a strong relationship to transition systems. More precisely, there is a natural way of associating a transition system with each elementary net system in order to explain the operational behaviour of elementary net systems in purely sequential terms. Hence the question arises as to which transition systems correspond to elementary net systems.

Transition Systems, Event Structures and Unfoldings

<p>Elementary transition systems were introduced by the authors in DAIMI PB-310. They were proved to be, in a strong categorical sense, the transition system version of elementray net systems. The question arises whether the notion of a region and the axioms (mostly based on regions) imposed on ordinary transition systems to obtain elementray net systems. Stated differently, one colud ask whether elementray transition systems could also play a role in characterizing other models of concurrency.</p><p> </p><p>We show here that by smoothly stengthening the axioms of elementary transition systems one obtains a subclass called occurrence transitions systems which turn out to be categorically equivalent to the well-known model of concurrency called prime event structures.</p><p> </p><p>Next we show that occurrence transition systems are to elementry transition systems what occurrence nets are to elementary nets systems. We define an ''unfold'' operation on elementry transition systems which yields occurrence transistion systems. We then prove that this operation uniquely extends to a functor which is the right adjoint to the inclusion functor from (the full subcategory of) occurrence transition systems to (the category of) elementary transition systems. Thus the results of this paper also show that the semantic theory of elementray net systems has a nice counterpart in the more abstract world of transition systems.</p>

Elementary Transition Systems and Refinement

The model of Elementary Transition Systems has been introduced by the authors as an abstraction of Elementary Net Systems - with a formal embedding in terms of a categorical coreflection, keeping behavioural information like causality, concurrency and conflict, but forgetting the concrete programming of a particular behaviour over an event set using conditions. In this paper we give one example of the advantages of ETS over ENS, - the definition of local state refinement. We show that the well known problems in understanding within nets the simple notion of syntactic substitution of conditions by (sub) nets behaviourally, - these problems seem to disappear when moving to the more abstract level of ETS. Formally, we show that the ETS-version of condition-substitution does satisfy nice and natural properties, e.g., projection and compositionality results w.r.t. a standard notion of transition system morphisms.

Elementary transition systems

<p>Transition systems are a simple and powerful formalism for explaining the operational behaviour of models of concurrency. They provide a common framework for investigating the interrelationships between different approaches to the study of distributed systems. Hence an important question to be answered is: which subclass of transition systems corresponds to a particular model of distributed systems? In this paper we provide an answer to this question for elementary net systems.</p>