Safe Nets Research Articles

Target-oriented Petri Net Synthesis

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

Compositional model checking of concurrent systems, with Petri nets

Compositionality and process equivalence are both standard concepts of process algebra. Compositionality means that the behaviour of a compound system relies only on the behaviour of its components, i.e. there is no emergent behaviour. Process equivalence means that the explicit statespace of a system takes a back seat to its interaction patterns: the information that an environment can obtain though interaction. Petri nets are a classical, yet widely used and understood, model of concurrency. Nevertheless, they have often been described as a non-compositional model, and tools tend to deal with monolithic, globally-specified models. This tutorial paper concentrates on Petri Nets with Boundaries (PNB): a compositional, graphical algebra of 1-safe nets, and its applications to reachability checking within the tool Penrose. The algorithms feature the use of compositionality and process equivalence, a powerful combination that can be harnessed to improve the performance of checking reachability and coverability in several common examples where Petri nets model realistic concurrent systems.

Persistent and Nonviolent Steps and the Design of GALS Systems

A concurrent system is persistent if throughout its operation no activity which became enabled can subsequently be prevented from being executed by any other activity. This is often a highly desirable or even necessary property; in particular, if the system is to be implemented in hardware. Over the past 40 years, persistence has been investigated and applied in practical implementations assuming that each activity is a single atomic action which can be represented, for example, by a single transition of a Petri net. In this paper we investigate the behaviour of GALS Globally Asynchronous Locally Synchronous systems in the context of VLSI circuits. The specification of a system is given in the form of a Petri net. Our aim is to re-design the system to optimise signal management, by grouping together concurrent events. Looking at the concurrent reachability graph of the given Petri net, we are interested in discovering events that appear in 'bundles', so that they all can be executed in a single clock tick. The best candidates for bundles are sets of events that appear and re-appear over and over again in the same configurations, forming 'persistent' sets of events. Persistence was considered so far only in the context of sequential semantics. In this paper, we move to the realm of step based execution and consider not only steps which are persistent and cannot be disabled by other steps, but also steps which are nonviolent and cannot disable other steps. We then introduce a formal definition of a bundle and propose an algorithm to prune the behaviour of a system, so that only bundled steps remain. The pruned reachability graph represents the behaviour of a re-engineered system, which in turn can be implemented in a new Petri net using the standard techniques of net synthesis. The proposed algorithm prunes reachability graphs of persistent and safe nets leaving bundles that represent maximally concurrent steps.

Branching Processes of General Petri Nets

We propose a new model of branching processes, suitable for describing the behavior of general Petri nets, without any finiteness or safeness assumption. In this framework, we define a new class of branching processes and unfoldings of a net N, which we call faithful. These coincide with the safe branching processes and unfoldings if N is safe, or weakly safe as in [Engelfriet 1991], but not in general. However, faithful branching processes and processes satisfy the good order-theoretic properties which make the safe processes of safe nets so useful in practice, and which are known not to hold for the safe processes of a general net. Faithful processes represent therefore good candidates to generalize the theory of safe nets to the general case.

Domain and event structure semantics for Petri nets with read and inhibitor arcs

We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets. Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW- IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures). As an intermediate semantic model, we introduce inhibitor event structures, an event-based model able to faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. Inhibitor event structures generalise several event structure models in the literature, like prime, asymmetric and bundle event structures.

Domain and event structure semantics for Petri nets with read and inhibitor arcs*1

We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets . Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW - IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures). As an intermediate semantic model, we introduce inhibitor event structures , an event-based model able to faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. Inhibitor event structures generalise several event structure models in the literature, like prime, asymmetric and bundle event structures.

Contextual Petri Nets, Asymmetric Event Structures, and Processes

We present an event structure semantics for contextual nets, an extension of P/T Petri nets where transitions can check for the presence of tokens without consuming them (read-only operations). A basic role is played by asymmetric event structures, a generalization of Winskel's prime event structures where symmetric conflict is replaced by a relation modelling asymmetric conflict or weak causality, used to represent a new kind of dependency between events arising in contextual nets. Extending Winskel's seminal work on safe nets, the truly concurrent event-based semantics of contextual nets is given at categorical level via a chain of coreflections leading from the category SW-CN of semi-weighted contextual nets to the category Dom of finitary prime algebraic domains. First an unfolding construction generates from a contextual net a corresponding occurrence contextual net, from where an asymmetric event structure is extracted. Then the configurations of the asymmetric event structure, endowed with a suitable order, are shown to form a finitary prime algebraic domain. We also investigate the relation between the proposed unfolding semantics and several deterministic process semantics for contextual nets in the literature. In particular, the domain obtained via the unfolding is characterized as the collection of the deterministic processes of the net endowed with a kind of prefix ordering.

Petri nets, traces, and local model checking

It has been observed that representing concurrent behaviour as sequences of interleaved events is not satisfactory — not all sequences should be considered as likely behaviours. Taking progress fairness assumptions into account one obtains a more realistic behavioural view of concurrent systems. In this paper we consider the problem of performing model-checking relative to this behavioural view. We present a CTL-like logic which is interpreted over labelled 1-safe nets. It turns out that Mazurkiewicz trace theory provides a natural setting in which progress fairness assumptions can be formalised. We provide the first, to our knowledge, set of sound and complete tableau rules for a CTL-like logic interpreted under progress fairness assumptions.

On the semantics of place/transition Petri nets

Place/transition (PT) Petri 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 terms 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 to relate multiple instances of the same place. The unfolding functor from PTNets to Occ reduces to Winskel's when restricted to safe nets. Moreover, 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.

Complexity results for 1-safe nets

We study the complexity of several standard problems for 1-safe Petri nets and some of its subclasses. We prove that reachability, liveness, and deadlock are all PSPACE-complete for 1-safe nets. We also prove that deadlock is NP-complete for free-choice nets and for 1-safe free-choice nets. Finally, we prove that for arbitrary Petri nets, deadlock is equivalent to reachability and liveness.

Petri Nets, Traces, and Local Model Checking

<p>It has been observed that the behavioural view of concurrent<br />systems that all possible sequences of actions are relevant is too generous;<br />not all sequences should be considered as likely behaviours. Taking<br />progress fairness assumptions into account one obtains a more realistic<br />behavioural view of the systems. In this paper we consider the problem<br />of performing model checking relative to this behavioural view. We<br />present a CTL-like logic which is interpreted over the model of concurrent<br />systems labelled 1-safe nets. It turns out that Mazurkiewicz trace<br />theory provides a natural setting in which the progress fairness assumptions<br />can be formalized. We provide the first, to our knowledge, set of<br />sound and complete tableau rules for a CTL-like logic interpreted under<br />progress fairness assumptions.</p><p><br />keywords: fair progress, labelled 1-safe nets, local model checking, maximal traces, partial orders, inevitability</p>

Generalized OM-Bisimulation

History preserving bisimilarity and maximality preserving bisimilarity take full account of the interplay between causality and branching, and they are congruences w.r.t. action refinement for suitable classes of Petri nets. But even for finite safe Petri nets they are based on possibly infinite transition systems, and it is not obvious how to decide them. OM-bisimulation gives a characterization of history preserving bisimulation for safe nets without internal moves, and it allows a decidability result. In this paper we generalize OM-bisimulation and this decidability result to safe nets with internal moves, and also treat the case of maximality preserving bisimulation.

Complexity Results for 1-safe Nets

<p>We study the complexity of several standard problems for 1-safe Petri nets and some of its subclasses. We prove that reachability, liveness, and deadlock are all PSPACE-complete for 1-safe nets. We also prove that deadlock is NP-complete for free-choice nets and for 1-safe free-choice nets. Finally, we prove that for arbitrary Petri nets, deadlock is equivalent to reachability and liveness.</p><p> </p><p>This paper is to be presented at <em>FST & TCS, Foundations of Software Technology & Theoretical Computer Science</em>, to be held 15-17 December 1993, in Bombay, India.</p><p>A version of the paper with most proofs omitted is to appear in the proceedings.</p>

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>

Petri net semantics of priority systems

The specification of priorities provides a convenient way of resolving conflicts in the design of concurrent computing systems. Priorities have been widely used by operating systems to enforce the preferred order of the execution of jobs waiting for processing; while programming languages often provide primitives, such as prioritised choice operator, for expressing the intended preference of the execution of one enabled section of the program over another enabled section of the program. In this paper we consider priority systems ( Σ, ϱ), where Σ is a bounded Petri net, and ϱ is a priority relation on the transitions of the net. Our main goal is to give a formal semantics of ( Σ, ϱ) by constructing a Petri net Σ ϱ which would retain as much of the concurrency semantics of Σ as possible and at the same time not violate the priority constraints imposed by ϱ. In the construction provided by this paper, Σ ϱ is derived from Σ by adding additional places and arcs, and by splitting the transitions of the original net Σ if necessary. The way in which these new places are added generalises the standard complementation technique introduced for P/T-nets. For safe nets Σ he construction can be simplified and Σ ϱ built without splitting of any transitions. We then outline how the translation from ( Σ, ϱ) to Σ ϱ might be used to give a formal semantics of the prioritised choice operator.

Executions: a new partial-order semantics of Petri nets

Executions, a new partial-order semantics of P/T nets, are defined as a generalization of the processes of safe nets. Various relations between executions and processes are established; especially, it is shown that for each net N there is a safe net SN( N) such that the processes of SN( N) are isomorphic to the executions of N. Furthermore, executions are related to other partial-order semantics of nets in much the same way as processes of safe nets are. It is shown that nets are conflict-free (in some sense) if and only if they have just one maximal execution.

Symbolic execution of concurrent systems using Petri nets

Techniques for analyzing sequential programs in order to improve their reliability have been widely studied in the past. Among the most interesting analysis techniques, we consider symbolic execution. However, analysis techniques for concurrent programs, and in particular symbolic execution, are still an open research area. In this paper, we define a method for symbolic execution of concurrent systems, based on an extension of the Petri net formalism, called EF nets. EF nets are a powerful, highly expressive and general formalism. Depending on the level of abstraction of actions and predicates that one associates to the transitions of the net, EF nets can be used as a high-level specification formalism for concurrent systems, or as a lower level internal representation of concurrent programs. Thus, the model is not dependent on a particular concurrent programming language, but it is flexible enough to be the kernel model for the representation of a wide set of systems and programming languages. In the paper, in order to support the analysis of a concurrent system or program, at first a general algorithm for symbolically executing an EF net is defined. Then, a more efficient algorithm is given for the particular, though important, subclass of EF nets, defined as safe EF nets. Such algorithm is proved to significantly help in reducing the amount of information needed to characterize a symbolic execution. Both the modelling power of the EF nets and the usefulness of the concurrent symbolic execution algorithms defined are illustrated by means of a case study.

A fresh look at free choice nets

A subclass of Petri nets called live and safe free choice nets (LSFC nets) is studied. LSFC nets model distributed systems that can exhibit both nonsequential and nondeterministic behaviours. It is shown that the restricted combination of concurrency and choice as represented by LSFC nets leads to a number of attractive system properties. It is also shown, through examples, that a “less” restrictive combination of concurrency and choice destroys these properties.