## Process versus Unfolding Semantics for Place/Transition Petri Nets (1996)

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Venue: | Theoretical Computer Science |

Citations: | 26 - 15 self |

### BibTeX

@ARTICLE{Meseguer96processversus,

author = {José Meseguer and Ugo Montanari and Vladimiro Sassone},

title = {Process versus Unfolding Semantics for Place/Transition Petri Nets},

journal = {Theoretical Computer Science},

year = {1996},

volume = {153},

pages = {153--1}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In the last few years, the semantics of Petri nets has been investigated in several di#erent ways. Apart from the classical "token game," one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc, a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows...

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Citation Context ... [N ]. The following exposition assumes that the reader is acquainted with a few very basic notions of category theory, namely, category, functor and adjunction; an excellent introductory textbook is =-=[12]-=-. Some of the results presented here appear also in [27]. A short version of the paper appears as [17]. Notation. We denote indi#erently by juxtaposition (from right to left) and by # the composition ... |

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Citation Context ...for PT nets (see [8, 11] for related approaches). In Winskel's work---which in turn builds on the previous work [19]---the denotation of a safe net is a coherent finitary prime algebraic Scott domain =-=[31]-=-, or dI-domain [1]. Winskel shows that there exists a coreflection---a particularly nice form of adjunction---between the category Dom of (coherent) finitary prime algebraic domains and the category S... |

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Citation Context ...sential properties, and the global structure of the class of all nets. Providing, for example, useful combinators able to describe operations such as parallel and nondeterministic composition of nets =-=[33, 34, 14, 3, 4, 13, 16, 18]-=-. The formal framework which has proved superior for this kind of investigations is category theory. The discovery of categories, occurred in the context of algebraic topology in the early forties, em... |

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Citation Context ...net and preserving certain compositional properties in the assignment. This is exactly what the present authors---building on Winskel's work on safe nets [33]---have done in [15, 16] for PT nets (see =-=[8, 11]-=- for related approaches). In Winskel's work---which in turn builds on the previous work [19]---the denotation of a safe net is a coherent finitary prime algebraic Scott domain [31], or dI-domain [1]. ... |

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Citation Context ...es of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification. Introduction Petri nets, introduced by C.A. Petri in [22] (see also =-=[23, 25, 26]-=-), are a widely used model of concurrency. This model is attractive from a theoretical point of view because of its simplicity and because of its intrinsically concurrent nature, # Supported by O#ce o... |

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Citation Context ...many di#erent transitions and for the causal connections between the "events" which constitute them, the basic notion of computation of Petri nets has been formalized using various notions o=-=f process [24, 10, 2]-=-. The main criticism raised against process models is that they do not provide a semantics for a net as a whole, but specify only the meaning of single, deterministic computations, while the accurate ... |

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Citation Context ...CHBGCT920005. + Basic Research in Computer Science, Centre of the Danish National Research Foundation. and has often been used as a semantic basis on which to interpret concurrent languages (see e.g. =-=[32, 21, 9, 5]-=-). Concerning Petri nets themselves, several di#erent semantics have been proposed in the literature. Most of them can be coarsely classified as process-oriented semantics, unfolding semantics, or alg... |

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Citation Context ...CHBGCT920005. + Basic Research in Computer Science, Centre of the Danish National Research Foundation. and has often been used as a semantic basis on which to interpret concurrent languages (see e.g. =-=[32, 21, 9, 5]-=-). Concerning Petri nets themselves, several di#erent semantics have been proposed in the literature. Most of them can be coarsely classified as process-oriented semantics, unfolding semantics, or alg... |

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Citation Context ...es of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification. Introduction Petri nets, introduced by C.A. Petri in [22] (see also =-=[23, 25, 26]-=-), are a widely used model of concurrency. This model is attractive from a theoretical point of view because of its simplicity and because of its intrinsically concurrent nature, # Supported by O#ce o... |

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Citation Context ...many di#erent transitions and for the causal connections between the "events" which constitute them, the basic notion of computation of Petri nets has been formalized using various notions o=-=f process [24, 10, 2]-=-. The main criticism raised against process models is that they do not provide a semantics for a net as a whole, but specify only the meaning of single, deterministic computations, while the accurate ... |

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Citation Context ...sential properties, and the global structure of the class of all nets. Providing, for example, useful combinators able to describe operations such as parallel and nondeterministic composition of nets =-=[33, 34, 14, 3, 4, 13, 16, 18]-=-. The formal framework which has proved superior for this kind of investigations is category theory. The discovery of categories, occurred in the context of algebraic topology in the early forties, em... |

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41 | From Petri nets to linear logic through categories: a survey, Znternat - Marti-Oliet, Meseguer - 1991 |

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Citation Context ... Computations In this section we briefly recall some of the basic definitions about Petri nets [22, 25]. In particular, we remind their algebraic description as introduced in [14] and their processes =-=[24, 10, 2, 6, 7]. Given a set S -=-and a function �� from S to the set of natural numbers #, we write [[��]] to indicate the support of �� that is the subset of S consisting of those elements s such that ��(s) > 0. More... |

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Citation Context ...CHBGCT920005. + Basic Research in Computer Science, Centre of the Danish National Research Foundation. and has often been used as a semantic basis on which to interpret concurrent languages (see e.g. =-=[32, 21, 9, 5]-=-). Concerning Petri nets themselves, several di#erent semantics have been proposed in the literature. Most of them can be coarsely classified as process-oriented semantics, unfolding semantics, or alg... |

28 | A linear specification language for Petri nets
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Citation Context |

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Citation Context ..., 11] for related approaches). In Winskel's work---which in turn builds on the previous work [19]---the denotation of a safe net is a coherent finitary prime algebraic Scott domain [31], or dI-domain =-=[1]-=-. Winskel shows that there exists a coreflection---a particularly nice form of adjunction---between the category Dom of (coherent) finitary prime algebraic domains and the category Safe of safe Petri ... |

24 | On the semantics of Petri nets
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Citation Context ...g a denotation to each PT net and preserving certain compositional properties in the assignment. This is exactly what the present authors---building on Winskel's work on safe nets [33]---have done in =-=[15, 16]-=- for PT nets (see [8, 11] for related approaches). In Winskel's work---which in turn builds on the previous work [19]---the denotation of a safe net is a coherent finitary prime algebraic Scott domain... |

24 |
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Citation Context ...es of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification. Introduction Petri nets, introduced by C.A. Petri in [22] (see also =-=[23, 25, 26]-=-), are a widely used model of concurrency. This model is attractive from a theoretical point of view because of its simplicity and because of its intrinsically concurrent nature, # Supported by O#ce o... |

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Citation Context ...may call this use of categories "in the small," as opposed to their use "in the large" to study the global properties of the entire class of nets as illustrated above. This idea ha=-=s been exploited in [6]-=-, where it is shown that the commutative processes [2] of a net N are isomorphic to the arrows of a symmetric monoidal category T [N ]. Moreover, [6] introduced the concatenable processes of N ---a sl... |

22 | On the semantics of Place/Transition Petri nets
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Citation Context ....12 ([6, Theorem 20, pag. 184]) CP [N ] and P[N ] are isomorphic. 2 Unfolding Place/Transition Nets In this section we sketch the basic notions concerning the unfolding of PT Petri nets as defined in =-=[15, 16]-=-. In order to keep the exposition of the background material as short as possible, we limit ourselves to the definitions of the object components of the functors U [ ], F [ ], E [ ] and L[ ]. In parti... |

21 |
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Citation Context |

19 | An axiomatization of the algebra of Petri net concatenable processes. Theoret
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Citation Context ...mmetries. Therefore, we have that Sym P[N ] = Sym DP[N ] . Moreover, Proposition 1.12 shows that Sym P[N ] and Sym CP[N ] are isomorphic via a functor whose object component is the identity (see also =-=[7, 28]-=-). Now observe that, once the object component is fixed, there can be at most one symmetric strict monoidal functor F between two categories of symmetries. In fact, on the one hand we have that, by de... |

13 |
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Citation Context ...corated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification. Introduction Petri nets, introduced by C.A. Petri in =-=[22]-=- (see also [23, 25, 26]), are a widely used model of concurrency. This model is attractive from a theoretical point of view because of its simplicity and because of its intrinsically concurrent nature... |

12 |
Non-Sequential Processes. Interner Bericht ISF--77--5, Gesellschaft fur Mathematik und Datenverarbeitung
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Citation Context ...many di#erent transitions and for the causal connections between the "events" which constitute them, the basic notion of computation of Petri nets has been formalized using various notions o=-=f process [24, 10, 2]-=-. The main criticism raised against process models is that they do not provide a semantics for a net as a whole, but specify only the meaning of single, deterministic computations, while the accurate ... |

9 |
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Citation Context ...category functor (c, C) ## #c#C# (see Definition 3.15), and LF returns the finite configurations of prime event structures ordered by inclusion. We remark that a similar approach has been followed in =-=[20]-=- in the case of elementary net systems---a particular class of safe nets without self-looping transitions---for unfoldings and non-sequential processes. It should be stressed that our concern here is ... |

9 | On the category of Petri net computations
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Citation Context ...nto the subject, whilst, on the other hand, it would still leave unresolved the key issue of functoriality for DP [ ] (and P[ ]). (Further research is currently ongoing on these open questions, e.g., =-=[29, 27]-=-.) Although DP [ ] is defined only at the object level, we think that the paper presents interesting results, providing a natural and unified account of the algebraic, the process-oriented, and the de... |

8 |
Local event structures and Petri nets
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Citation Context ...net and preserving certain compositional properties in the assignment. This is exactly what the present authors---building on Winskel's work on safe nets [33]---have done in [15, 16] for PT nets (see =-=[8, 11]-=- for related approaches). In Winskel's work---which in turn builds on the previous work [19]---the denotation of a safe net is a coherent finitary prime algebraic Scott domain [31], or dI-domain [1]. ... |

8 |
On the Semantics of Petri Nets: Processes, Unfolding and Infinite Computations
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Citation Context ...nto the subject, whilst, on the other hand, it would still leave unresolved the key issue of functoriality for DP [ ] (and P[ ]). (Further research is currently ongoing on these open questions, e.g., =-=[29, 27]-=-.) Although DP [ ] is defined only at the object level, we think that the paper presents interesting results, providing a natural and unified account of the algebraic, the process-oriented, and the de... |

7 | On the model of computation of Place/Transition Petri nets
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(Show Context)
Citation Context ... category theory, namely, category, functor and adjunction; an excellent introductory textbook is [12]. Some of the results presented here appear also in [27]. A short version of the paper appears as =-=[17]-=-. Notation. We denote indi#erently by juxtaposition (from right to left) and by # the composition of functors, while the composition of arrows is always written as # , except in the categories---such ... |

6 | Some remarks on concatenable processes - Sassone - 1994 |

5 |
Inductive completion of monoidal categories and infinite net computations
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Citation Context ...appears to be of some interest on its own. Concluding this discussion, we would like to mention that the correspondence of semantics presented here can be lifted smoothly to infinite computations. In =-=[30]-=-, the present authors show that the symmetric monoidal category P[N ] # obtained as the completion of P[N ] by colimits of #-diagrams can be understood as the category of possibly infinite concatenabl... |

4 | Strong Concatenable Processes: An Approach to the Category of Petri Net Computations - Sassone - 1994 |

2 | Stable Models of Typed A-calculi - Berry - 1978 |

1 | On the semantics of place/transition Petri nets, Technical Report TR 27/92 Dipartimento Informatica, Universim di Pisa - Meseguer, Montanari, et al. - 1992 |

1 | Non-sequential processes, Intemer Bericht ISF-77-5, Gesellschafi liir Mathematik und Datenverarbeitung - Petri - 1977 |

1 | o-inductive completion of monoidal categories and infinite Petri net computations, presentation at - Montanari - 1993 |