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Modelling Concurrency with Partial Orders
, 1986
"... Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a n ..."
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Cited by 263 (18 self)
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Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. Th...
On the semantics of Petri nets
 Proceedings Third International Conference on Concurrency Theory, CONCUR'92, Stony Brook, NY, USA, LNCS 630
, 1992
"... Petri Place/Transition (PT) 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 term of nonsequential processes and monoidal cat ..."
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Cited by 26 (11 self)
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Petri Place/Transition (PT) 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 term 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 for relating multiple instances of the same place. The unfolding functor from PTNets to Occ reduces to Winskel’s when restricted to safe nets, while 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.
An axiomatization of the category of Petri net computations
 Math. Struct. in Comput. Sci
, 1998
"... Abstract. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a ..."
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Cited by 13 (5 self)
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Abstract. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a
On the Category of Petri Net Computations
, 1995
"... . We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each ne ..."
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Cited by 10 (6 self)
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. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a functor Q[ ] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net N , the strongly concatenableprocesses of N are isomorphic to the arrows of Q[N ]. In addition, we identify a coreflection right adjoint to Q[ ] and characterize its replete image, thus yielding an axiomatization of the category of net computations. Introduction Petri nets, introduced by C.A. Petri [8] (see also [10]), are unanimously considered among the most representative models for concurrency, since they are a fairly simple and natural model of concurrent and distributed computations. However, Petri nets are, in our opinion, not yet completely understood. Among the semantics proposed for Petri nets, a relevant role is played by the various notions of process [9, 4, 1], whose merit is...
The unfolding of general Petri nets ∗
"... ABSTRACT. The unfolding of (1)safe Petri nets to occurrence nets is well understood. There is a universal characterization of the unfolding of a safe net which is part and parcel of a coreflection ..."
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Cited by 9 (3 self)
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ABSTRACT. The unfolding of (1)safe Petri nets to occurrence nets is well understood. There is a universal characterization of the unfolding of a safe net which is part and parcel of a coreflection
Events, Causality and Symmetry
, 2008
"... The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications, and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences ..."
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Cited by 6 (2 self)
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The article discusses causal models, such as Petri nets and event structures, how they have been rediscovered in a wide variety of recent applications, and why they are fundamental to computer science. A discussion of their present limitations leads to their extension with symmetry. The consequences, actual and potential, are discussed.
Strong Concatenable Processes: An Approach to the Category of Petri Net Computations
 BRICSComputer Science Department, University of Aarhus
, 1994
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."
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Cited by 4 (1 self)
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
On the Compositionality and Analysis of Algebraic HighLevel Nets
 RESEARCH REPORT A16, DIGITAL SYSTEMS LABORATORY
, 1991
"... This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In cate ..."
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Cited by 3 (1 self)
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This work discusses three aspects of net theory: compositionality of nets, analysis of nets and highlevel nets. Net theory has often been criticised for the difficulty of giving a compositional semantics to a net. In this work we discuss this problem form a category theoretic point of view. In category theory compositionality is represented by colimits. We show how a highlevel net can be mapped into a lowlevel net that represents its behaviour. This construction is functorial and preserves colimits, giving a compositional semantics for these highlevel nets. Using this semantics we propose some proof rules for compositional reasoning with highlevel nets. Linear logic is a recently discovered logic that has many interesting properties. From a net theoretic point of view its interest lies in the fact that it is able to express resources in an analogous manner to nets. Several interpretations of Petri nets in terms of linear logic exist. In this work we study a model theoretic inter...
Symmetry in Petri nets
, 2008
"... An algebraic treatment of symmetry in Petri nets is proposed. The standard definition of Petri net is that it has precisely one initial marking. Motivated by work on defining symmetry across models for concurrency, we extend the definitions of forms of net to allow them to have multiple initial mark ..."
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Cited by 1 (0 self)
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An algebraic treatment of symmetry in Petri nets is proposed. The standard definition of Petri net is that it has precisely one initial marking. Motivated by work on defining symmetry across models for concurrency, we extend the definitions of forms of net to allow them to have multiple initial markings. Existing coreflections between event structures and occurrence nets and between occurrence nets and P/T nets are generalized, and from them coreflections between categories of nets with symmetry are obtained.
IOS Press Modular Construction of Finite and Complete Prefixes of Petri net
"... Abstract. This paper considers distributed systems, defined as a collection of components interacting through interfaces. Components, interfaces and distributed systems are modeled as Petri nets. It is well known that the unfolding of such a distributed system factorises, in the sense that it can be ..."
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Abstract. This paper considers distributed systems, defined as a collection of components interacting through interfaces. Components, interfaces and distributed systems are modeled as Petri nets. It is well known that the unfolding of such a distributed system factorises, in the sense that it can be expressed as the composition of unfoldings of its components. This factorised form of the unfolding generally provides a more compact representation of the system runs, because each component does not need to represent the possible choices (conflicts) appearing in the other components. Moreover, the unfolding factorisation makes it possible to analyse the system by parts. The paper focuses on the derivation of a finite and complete prefix (FCP) in the unfolding of a distributed system. Specifically, one would like to directly obtain such a FCP in factorised form, without computing first a FCP of the global distributed system and then factorising it. The construction of such a “modular FCP ” is based on deriving summaries of component behaviours w.r.t. their interfaces, that are then communicated to the neighbouring components. The latter combine