Elementary Net Systems Research Articles

Synthesising elementary net systems with localities

Elementary net systems with localities (enl-systems) is a class of Petri nets introduced to model globally asynchronous locally synchronous systems (gals), where some of the components can be considered as logically or physically close and acting synchronously, while others can be considered as loosely connected or residing at distant locations and communicating asynchronously with the rest of the system. The specification of the behaviour of a gals system comes very often in the form of a transition system. Automated synthesis based on the regions of transition systems is an approach that allows to construct Petri net models from their transition system specifications. In this paper we focus on developing algorithms and tool support for the synthesis of the enl-systems from transition systems, where transitions are labelled by steps (sets) of executed actions. We pay special attention to the subclass of enl-systems with localised conflicts where there is no conflict between events belonging to different localities. The algorithms are implemented within the workcraft framework.

Relational structures for concurrent behaviours

Relational structures based on acyclic relations can successfully model fundamental aspects of concurrent systems behaviour. Examples include Elementary Net systems and Mazurkiewicz traces. There are however cases where more general relational structures are needed. In this paper, we present a general model of relational structures which can be used for a broad class of concurrent behaviours. We demonstrate how this general set-up works for combined order structures which are based on two relations, viz. an acyclic ‘before’ relation and a possibly cyclic ‘not later than’ relation.

Petri nets based on Lawvere theories

AbstractWe give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory Q, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

Testing Equivalences of Time Petri Nets

In the paper, we study a family of testing equivalences in interleaving, partial-order semantics, and combined semantics in the context of safe time Petri nets (elementary net systems whose transitions are labeled with time firing intervals, can fire only if their lower time bounds are attained, and are forced to fire when their upper time bounds are reached). For this purpose, the following three representations of behavior of safe time Petri nets are developed: sequences of firings of net transitions, which represent interleaving semantics; time causal processes, from which partial orders are derived; and time causal tree, whose nodes are sequences of transition firings and arcs are labeled by information about partial orders. We establish relationships between these equivalences and show that semantics of time causal processes and time causal trees coincide.

The complexity of synthesizing elementary net systems relative to natural parameters

Elementary net systems (ENS) are the most fundamental class of Petri nets. Given a labeled transition system (TS) A, feasibility is the NP-complete decision problem whether A can be synthesized into an ENS. We analyze the impact of state degree, the number of allowed successors and predecessors of states in A, and event manifoldness, the amount of occurrences that an event can have in A, on the computational complexity of feasibility and the related event state separation property (ESSP) and state separation property (SSP). Feasibility, ESSP and SSP are NP-complete for TSs with state degree one and event manifoldness not less than three and for TSs with state degree and event manifoldness at least two. As we also show that SSP becomes tractable for TSs with state degree one and event manifoldness two, the only cases left open are ESSP and feasibility for the same input restriction.

Dynamic properties of fuzzy Petri net model and related analysis

Fuzzy Petri net (FPN) has been extensively applied in industrial fields for knowledge-based systems or systems with uncertainty. Although the applications of FPN are known to be successful, the theoretical research of FPN is still at an initial stage. To pave a way for further study, this work explores related dynamic properties of FPN including reachability, boundedness, safeness, liveness and fairness. The whole methodology is divided into two phases. In the first phase, a comparison between elementary net system (EN_system) and FPN is established to prove that the FPN is an extensive formalism of Petri nets using a backwards-compatible extension method. Next, current research results of dynamic properties are utilized to analyze FPN model. The results illustrate that FPN model is bounded, safe, weak live and fair, and can support theoretical evidences for designing related decomposition algorithm.

Mutex Causality in Processes and Traces of General Elementary Nets

A concurrent history represented by a causality structure that captures the intrinsic, invariant dependencies between its actions, can be interpreted as defining a set of closely related observations e.g., step sequences. Depending on the relationships observed in the histories of a system, the concurrency paradigm to which it adheres may be identified, with different concurrency paradigms underpinned by different kinds of causality structures. Elementary net systems with inhibitor arcs and mutex arcs ENIM-systems are a system model that through its process semantics and associated causality structures fits the least restrictive concurrency paradigm. One can also investigate the abstract behaviour of an ENIM-system by grouping together step sequences in equivalence classes generalised comtraces using the structural relations between its transitions. The resulting concurrent histories of the ENIM-system are consistent with the generalised stratified order structures underlying its processes. The paper establishes a link between ENIM-systems and trace theory allowing one to discuss different observations of concurrent behaviour in a way that is consistent with the causality semantics defined by the operationally defined processes.

Betweenness Centrality of Event Graph Application to Metabolic Network Modelled by Elementary Net System

Betweenness Centrality of Event Graph Application to Metabolic Network Modelled by Elementary Net System

Minimal Regions of ENL-Transition Systems

One of the possible ways of constructing concurrent systems is their automated synthesis from behavioural specifications. In this paper, we look at a particular instance of this approach which aims at constructing GALS (globally asynchronous locally synchronous) systems from specifications given in terms of transition systems with arcs labelled by steps of executed actions. GALS systems are represented by Elementary Net Systems with Localities (ENL-systems), each locality defining a set of co-located actions. The synthesis procedure is based on the regions of transition systems and we provide a number of criteria aimed at generating a minimal set of regions (conditions) of an ENL-system generating a given transition system.

Structural non-interference in elementary and trace nets

Several notions of non-interference have been proposed in the literature for studying the problem of confidentiality in concurrent systems. The common feature of these non-interference properties is that they are all defined as extensional properties based on some notion of behavioural equivalence on systems. Here, instead, we address the problem of defining non-interference by looking at the structure of the systems under investigation. We use a simple class of Petri nets, namely, contact-free elementary net systems, as the system model and define structural non-interference properties based on the absence of particular places in the net: such places show that a suitable causality or conflict relation is present between a high-level transition and a low-level one. We characterise one structural property, called PBNI+, which we show to be equivalent to the well-known behavioural property SBNDC. It essentially captures all the positive information flows (that is, a low-level user can deduce that some high-level action has occurred). We start by providing a characterisation of PBNI+ on contact-free elementary net systems, then extend the definition to cope with the richer class of trace nets.

Nonautonomous Elementary Net Systems and Their Application to Programmable Logic Control

In this paper, a novel approach to Petri net modeling of programmable logic controller (PLC) programs is presented. The modeling approach is a simple extension of elementary net systems, and a graphical design tool that supports the use of this modeling approach is provided. A key characteristic of the model is that the binary sensory inputs and binary actuation outputs of the PLC are explicitly represented. This leads to the following two improvements: outputs are unambiguous, and interaction patterns are more clearly represented in the graphical form. The use of this modeling approach produces programs that are simple, lightweight, and portable. The approach is demonstrated by applying it to the development of a control module for a MonTech Positioning Station.

Undecidability of domino games and hhp-bisimilarity

History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is that they are preserved under action refinement, an operation important for the top-down design of concurrent systems. It is shown that, in contrast to hp-bisimilarity, checking hhp-bisimilarity for finite labelled asynchronous transition systems is undecidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable. It is also shown that the unlabelled domino bisimilarity problem is undecidable, which implies undecidability of hhp-bisimilarity for unlabelled finite asynchronous systems. Moreover, it is argued that the undecidability of hhp-bisimilarity holds for finite elementary net systems and 1-safe Petri nets.

A generic causal model for place latency

For a prototypical class of time-extended Petri nets it is shown that the extension does not increase their expressive power. The nets in this class have token latencies attributed to places. Place latency nets are formally defined together with their firing semantics. For any place latency system, an explicit construction of an elementary net system is given as an implementation, which is proven to be behaviourally equivalent. The adequacy problem of deciding which equivalence notion to apply is moderated by the fact that the implementation satisfies the B-condition, under which all better known equivalence notions coincide.

A pragmatic behavior subtyping relation based on both states and actions

A behavior preserving relation between Petri-net systems is introduced in this paper, based on the observability of both places and transitions, which is important in modeling the dynamic behavior of concurrent object-oriented systems with Petri nets. Each group of closely related attributes of a concurrent object is modeled by the state of a collection of observable places, and each of its methods by a group of observable transitions. The grouping distinguishes the definition in this paper from others, which makes it easy to work together with the static object models, to reuse the models and to dispel the interference among groups, thus relieving the problem of inheritance anomaly by the possibility of dividing the synchronization code into independent parts. For a formal definition of this behavior subtyping relation, Elementary Net systems, with both S-elements and T-elements labeled, are used. Then it is extended informally to the state based coloured Petri net systems. Finally, the background of the definitions and the future work are presented.

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.

The Synthesis Problem for Elementary Net Systems with Inhibitor Arcs

We investigate the synthesis problem for the Elementary Net Systems with Inhibitor Arcs (ENI-systems) executed according to the a-priori semantics. We characterise transition systems generated by ENI-systems, called TSENI transition systems, by adapting the notion of a step transition system whose arcs are labelled by sets of concurrently executed events. The relationship between the ENI-systems and TSENI transition systems is established via the notion of a region. We define, and show consistency of, two behaviour preserving translations between nets and transition systems. We also discuss how to optimise the synthesis procedure by using only minimal regions and selected inhibitor arcs.

The synthesis problem for elementary net systems is NP-complete

Given a labelled graph representing the sequential behaviour of some system, the synthesis problem consists in deciding whether it is the behaviour of a Petri net. This problem was solved by Ehrenfeucht and Rozenberg for the class of elementary net systems, relying on regions in graphs introduced as sets of nodes liable to represent extensions of places of a net. The solution was extended later on to pure place/transition nets using a variant notion of generalized regions. The naive method of synthesis which relies on this principle leads to exponential algorithms for an arbitrary class of nets. In an earlier study, we gave an algorithm that solves the synthesis problem in polynomial time for the class of pure place/transition nets. We show here that in contrast the synthesis problem is indeed NP-complete for the class of elementary nets.

Semantics of Inhibitor Nets

We discuss an abstract semantics of concurrent systems generalising causal partial orders. The new semantics employs relational structures-called stratified order structures-which comprise causal partial orders and weak causal partial orders. Stratified order structures can be represented by certain equivalence classes of step sequences-comtraces-directly generalising Mazurkiewicz traces. We use Elementary Net Systems with inhibitor arcs as a system model and show that stratified order structures can provide an abstract semantics which is consistent with their operational semantics expressed in terms of step sequences. Two different types of operational rules are considered. We also construct occurrence nets to enable the generation of stratified order structures for a given run of the net.

Expansion nets and expansion processes of elementary net systems

Occurrence nets are insufficient to precisely describe executions of elementary net systems with contacts. Traditionally, S-complementation is used for removal of contacts from the systems. Although the main behavior and properties of the original elementary net systems are preserved during S-complementation, their topologies may be changed greatly. This paper introduces a new kind of nets—expansion nets—for representing behavior of elementary net systems. As shown in the paper, expansion nets are natural as well as sufficient for describing the precise behavior of elementary net systems with or without contacts.

A Logical Study of Distributed Transition Systems

We extend labelled transition systems to distributed transition systems by labelling the transition relation with a finite set of actions, representing the fact that the actions occur as a concurrent step . We design an addition-based temporal logic in which one can explicity talk about steps. The logic is studied to establish a variety of positive and negative results in terms of axiomatizability and decidability. Our positive results show that the step notion is amenable to logical treatment via standard techniques. They also help us to obtain a logical characterization of two well known models for distributed systems: labelled elementary net systems and labelled prime event structures. Our negative results show that demanding deterministic structures when dealing with a "non-interleaved" notion of transitions is, from a logical standpoint, very expressive. They also show that another well known model of distributed systems called asynchronous transition systems exhibits a surprising amount of expressive power in a natural logical setting.