Event structures for Petri nets with persistence

Event structures are a well-accepted model of concurrency. In a seminal paper by Nielsen, Plotkin and Winskel, they are used to establish a bridge between the theory of domains and the approach to concurrency proposed by Petri. A basic role is played by an unfolding construction that maps (safe) Petri nets into a subclass of event structures, called prime event structures, where each event has a uniquely determined set of causes. Prime event structures, in turn, can be identified with their domain of configurations. At a categorical level, this is nicely formalised by Winskel as a chain of coreflections. Contrary to prime event structures, general event structures allow for the presence of disjunctive causes, i.e., events can be enabled by distinct minimal sets of events. In this paper, we extend the connection between Petri nets and event structures in order to include disjunctive causes. In particular, we show that, at the level of nets, disjunctive causes are well accounted for by persistent places. These are places where tokens, once generated, can be used several times without being consumed and where multiple tokens are interpreted collectively, i.e., their histories are inessential. Generalising the work on ordinary nets, Petri nets with persistence are related to a new subclass of general event structures, called locally connected, by means of a chain of coreflections relying on an unfolding construction

Bisimilarity and Behaviour-Preserving Reconfigurations of Open Petri Nets

We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open systems which might interact with the surrounding environment and endowed with a colimit-based composition operation. We show that natural notions of bisimilarity over open nets are congruences with respect to the composition operation. The considered behavioural equivalences differ for the choice of the observations, which can be single firings or parallel steps. Additionally, we consider weak forms of such equivalences, arising in the presence of unobservable actions. We also provide an up-to technique for facilitating bisimilarity proofs. The theory is used to identify suitable classes of reconfiguration rules (in the double-pushout approach to rewriting) whose application preserves the observational semantics of the net.Comment: To appear in "Logical Methods in Computer Science", 41 page

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

PP-persistent homology of finite topological spaces

Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$

A regular viewpoint on processes and algebra

While different algebraic structures have been proposed for the treatment of concurrency, finding solutions for equations over these structures needs to be worked on further. This article is a survey of process algebra from a very narrow viewpoint, that of finite automata and regular languages. What have automata theorists learnt from process algebra about finite state concurrency? The title is stolen from [31]. There is a recent survey article [7] on finite state processes which deals extensively with rational expressions. The aim of the present article is different. How do standard notions such as Petri nets, Mazurkiewicz trace languages and Zielonka automata fare in the world of process algebra? This article has no original results, and the attempt is to raise questions rather than answer them

Theory of traces

AbstractThe theory of traces, originated by A. Mazurkiewicz in 1977, is an attempt to provide a mathematical description of the behavior of concurrent systems. Its aim is to reconcile the sequential nature of observations of the system behavior on the one hand and the nonsequential nature of causality between the actions of the system on the other hand.One can see the theory of traces to be rooted in formal string language theory with the notion of partial commutativity playing the central role. Alternatively one can see the theory of traces to be rooted in the theory of labeled acyclic directed graphs (or even in the theory of labeled partial orders).This paper attempts to present a major portion of the theory of traces in a unified way. However, it is not a survey in the sense that a number of new notions are introduced and a number of new results are proved. Although traditionally most of the development in the theory of traces follows the string-language-theoretic line, we try to demonstrate to the reader that the graph-theoretic point of view may be more appropriate.The paper essentially consists of two parts. The first one (Sections 1 through 4) is concerned with the basic theory of traces. The second one (Section 5) presents applications of the theory of traces to the theory of the behavior of concurrent systems, where the basic system model we have chosen is the condition/event system introduced by C.A. Petri