1,823 research outputs found
A Congruence for Petri Nets
We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs
Compositional Semantics of Finite Petri Nets
Structure-preserving bisimilarity is a truly concurrent behavioral
equivalence for finite Petri nets, which relates markings (of the same size
only) generating the same causal nets, hence also the same partial orders of
events. The process algebra FNM truly represents all (and only) the finite
Petri nets, up to isomorphism. We prove that structure-preserving bisimilarity
is a congruence w.r.t. the FMN operators, In this way, we have defined a
compositional semantics, fully respecting causality and the branching structure
of systems, for the class of all the finite Petri nets. Moreover, we study some
algebraic properties of structure-preserving bisimilarity, that are at the base
of a sound (but incomplete) axiomatization over FNM process terms.Comment: arXiv admin note: substantial text overlap with arXiv:2301.0448
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An algebra of high level petri nets
PhD ThesisPetri nets were introduced by C.A. Petri as a theoretical model of concurrency in which the causal
relationship between actions, rather than just their temporal ordering, can be represented. As
a theoretical model of concurrency, Petri nets have been widely successful. Moreover, Petri nets
are popular with practitioners, providing practical tools for the designer and developer of real
concurrent and distributed systems.
However, it is from this second context that perhaps the most widely voiced criticism of Petri
nets comes. It is that Petri nets lack any algebraic structure or modularity, and this results in
large, unstructured models of real systems, which are consequently often intractable. Although
this is not a criticism of Petri nets per se, but rather of the uses to which Petri nets are put, the
criticism is well taken.
We attempt to answer this criticism in this work. To do this we return to the view of Petri nets
as a model of concurrency and consider how other models of concurrency counter this objection.
The foremost examples are then the synchronisation trees of Milner, and the traces of Hoare,
(against which such criticism is rarely, if ever, levelled). The difference between the models is
clear, and is to be found in the richness of the algebraic characterisations which have been made
for synchronisation trees in Milner's Calculus of Communicating Systems (CCS), and for traces
in Hoare's Communicating Sequential Processes (CSP).
With this in mind we define, in this thesis, a class of high level Petri nets, High Level Petri Boxes,
and provide for them a very general algebraic description language, the High Level Petri Box
Algebra, with novel ideas for synchronisation, and including both refinement and recursion among
its operators. We also begin on the (probably open-ended task of the) algebraic characterisation
of High Level Petri Boxes.
The major contribution of this thesis is a full behavioural characterisation of the High Level Petri
Boxes which form the semantic domain of the algebra. Other contributions are: a very general
method of describing communication protocols which extend the synchronisation algebras of
Winskel; a recursive operator that preserves finiteness of state (the best possible, given the
generality of the algebra); a refinement operator that is syntactic in nature, and for which the
recursive construct is a behavioural fix-point; and a notion of behavioural equivalence which is
a congruence with respect to a major part of the High Level Petri Box Algebra
Two polygraphic presentations of Petri nets
This document gives an algebraic and two polygraphic translations of Petri
nets, all three providing an easier way to describe reductions and to identify
some of them. The first one sees places as generators of a commutative monoid
and transitions as rewriting rules on it: this setting is totally equivalent to
Petri nets, but lacks any graphical intuition. The second one considers places
as 1-dimensional cells and transitions as 2-dimensional ones: this translation
recovers a graphical meaning but raises many difficulties since it uses
explicit permutations. Finally, the third translation sees places as
degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is
a setting equivalent to Petri nets, equipped with a graphical interpretation.Comment: 28 pages, 24 figure
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
Reactive Systems over Cospans
The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we offer a general construction of such bicolimits in a class of bicategories of cospans. The construction sheds light on as well as extends Ehrig and Konig’s rewriting via borrowed contexts and opens the way to a unified treatment of several applications
An Operational Petri Net Semantics for the Join-Calculus
We present a concurrent operational Petri net semantics for the
join-calculus, a process calculus for specifying concurrent and distributed
systems. There often is a gap between system specifications and the actual
implementations caused by synchrony assumptions on the specification side and
asynchronously interacting components in implementations. The join-calculus is
promising to reduce this gap by providing an abstract specification language
which is asynchronously distributable. Classical process semantics establish an
implicit order of actually independent actions, by means of an interleaving. So
does the semantics of the join-calculus. To capture such independent actions,
step-based semantics, e.g., as defined on Petri nets, are employed. Our Petri
net semantics for the join-calculus induces step-behavior in a natural way. We
prove our semantics behaviorally equivalent to the original join-calculus
semantics by means of a bisimulation. We discuss how join specific assumptions
influence an existing notion of distributability based on Petri nets.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244
Connector algebras for C/E and P/T nets interactions
A quite fourishing research thread in the recent literature on component based system is concerned with the algebraic properties of different classes of connectors. In a recent paper, an algebra of stateless connectors was presented that consists of five kinds of basic connectors, namely symmetry, synchronization, mutual exclusion, hiding and inaction, plus their duals and it was shown how they can be freely composed in series and in parallel to model sophisticated "glues". In this paper we explore the expressiveness of stateful connectors obtained by adding one-place buffers or unbounded buffers to the stateless connectors. The main results are: i) we show how different classes of connectors exactly correspond to suitable classes of Petri nets equipped with compositional interfaces, called nets with boundaries; ii) we show that the difference between strong and weak semantics in stateful connectors is reflected in the semantics of nets with boundaries by moving from the classic step semantics (strong case) to a novel banking semantics (weak case), where a step can be executed by taking some "debit" tokens to be given back during the same step; iii) we show that the corresponding bisimilarities are congruences (w.r.t. composition of connectors in series and in parallel); iv) we show that suitable monoidality laws, like those arising when representing stateful connectors in the tile model, can nicely capture concurrency aspects; and v) as a side result, we provide a basic algebra, with a finite set of symbols, out of which we can compose all P/T nets, fulfilling a long standing quest
A Process Calculus for Expressing Finite Place/Transition Petri Nets
We introduce the process calculus Multi-CCS, which extends conservatively CCS
with an operator of strong prefixing able to model atomic sequences of actions
as well as multiparty synchronization. Multi-CCS is equipped with a labeled
transition system semantics, which makes use of a minimal structural
congruence. Multi-CCS is also equipped with an unsafe P/T Petri net semantics
by means of a novel technique. This is the first rich process calculus,
including CCS as a subcalculus, which receives a semantics in terms of unsafe,
labeled P/T nets. The main result of the paper is that a class of Multi-CCS
processes, called finite-net processes, is able to represent all finite
(reduced) P/T nets.Comment: In Proceedings EXPRESS'10, arXiv:1011.601
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