## Axioms for Contextual Net Processes (1998)

Venue: | In Automata, Languages and Programming, volume 1443 of LNCS |

Citations: | 14 - 9 self |

### BibTeX

@INPROCEEDINGS{Gadducci98axiomsfor,

author = {F. Gadducci and U. Montanari},

title = {Axioms for Contextual Net Processes},

booktitle = {In Automata, Languages and Programming, volume 1443 of LNCS},

year = {1998},

pages = {296--308},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In the classical theory of Petri nets, a process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in [19, 11], processes of a P/T net are modeled as arrows of a suitable monoidal category: In this paper we lay the basis of a similar characterization for contextual P/T nets, that is, P/T nets extended with read arcs, which allows a transition to check for the presence of a token in a place, without consuming it. 1 Introduction Petri nets [24] are probably the best studied and most used model for concurrent systems: Their range of applications covers a wide spectrum, from their use as a specification tool to their analysis as a suitable semantical domain. A recent extension to the classical model concerns a class of nets where transitions are able to check for the presence of a token in a place without actually consuming it. While the possibility ...

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Citation Context ...ntextual C/E nets. 3 A Few Categorical Remarks We introduce now symmetric monoidal categories: Since our presentation is tailored over the needs of our representation theorems, we refer the reader to =-=[16]-=-. Definition 17 (symmetric monoidal categories). A monoidal category C is a triple C = hC 0 ;\Omega ; ei where C 0 is a category, e 2 C 0 is a distinguished object and\Omega : C 0 \Theta C 0 ! C 0 is ... |

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Citation Context ...rization for contextual P/T nets, that is, P/T nets extended with read arcs, which allows a transition to check for the presence of a token in a place, without consuming it. 1 Introduction Petri nets =-=[24]-=- are probably the best studied and most used model for concurrent systems: Their range of applications covers a wide spectrum, from their use as a specification tool to their analysis as a suitable se... |

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Citation Context ...process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in =-=[19, 11]-=-, processes of a P/T net are modeled as arrows of a suitable monoidal category: In this paper we lay the basis of a similar characterization for contextual P/T nets, that is, P/T nets extended with re... |

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Citation Context ..., SRI International, Menlo Park, USA, and visiting scholar at Stanford University The operational behaviour of Petri nets can be described either via firing sequences, or via non-sequential processes =-=[13]-=-. Even if tightly related, only the latter option fully exploits the ability of nets to describe concurrency. Processes are acyclic, deterministic safe nets whose transitions are occurrences of the tr... |

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Citation Context ...aic description and analysis of these nets and, as a consequence, of the concurrency paradigm based on shared memory they represent. To the best of our knowledge, the problem has been tackled only in =-=[18]. The-=- solution proposed there associates to a CP/T net N a monoidal category P 0 (N ), where the monoid of objects is not freely generated from the class of places. Our solution is instead "more in li... |

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Citation Context ...tions of dgs-monoidal categories surfaced in the literature along the years. In particular, a (bicategorical) presentation of them is used as a description of the (bi)category of relations already in =-=[6]-=-, and it forms the basis for a categorical description of circuits [15]. An equivalence induced by the dgs-monoidal axioms on the sequential composition of two arrows is shown in Fig. 2 (r 3 a denotes... |

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Citation Context ...richer theory and refer to several well-tailored applications. In fact, important constructions on ordinary nets can be extended to nets with read arcs, like those concerning non-sequential processes =-=[22, 28]-=- and event structures [1]. Moreover, these nets naturally model read-write access to shared memory, where readers are allowed to progress in parallel, with applications to transaction serializability ... |

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Citation Context ... presence of a token in a place without actually consuming it. While the possibility of sensing for both presence and absence of a token yields very expressive nets equipped also with inhibitory arcs =-=[14, 7, 4, 5]-=-, in the paper we focus our attention to nets extended with read arcs only, generically referred to as contextual nets, which have a richer theory and refer to several well-tailored applications. In f... |

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Citation Context ... presence of a token in a place without actually consuming it. While the possibility of sensing for both presence and absence of a token yields very expressive nets equipped also with inhibitory arcs =-=[14, 7, 4, 5]-=-, in the paper we focus our attention to nets extended with read arcs only, generically referred to as contextual nets, which have a richer theory and refer to several well-tailored applications. In f... |

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Citation Context ...years. In particular, a (bicategorical) presentation of them is used as a description of the (bi)category of relations already in [6], and it forms the basis for a categorical description of circuits =-=[15]-=-. An equivalence induced by the dgs-monoidal axioms on the sequential composition of two arrows is shown in Fig. 2 (r 3 a denotes the arrow r a ; (a\Omega r a ) = r; (r a\Omega a), and similarly for \... |

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Citation Context ...ruction provides a finite axiomatization of non-sequential processes. Moreover, the well-understood setting of monoidal categories allows for an easy comparison with related models, like linear logic =-=[17]-=-. The aim of this paper is to extend the above categorical approach to P/T nets with read arcs. Our results should enable a fully algebraic description and analysis of these nets and, as a consequence... |

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Citation Context ...process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in =-=[19, 11]-=-, processes of a P/T net are modeled as arrows of a suitable monoidal category: In this paper we lay the basis of a similar characterization for contextual P/T nets, that is, P/T nets extended with re... |

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Citation Context ... also [p 1 ] = [p 2 ] holds. The proof is carried out first by finding a suitable normal form for the arrows of the subcategory CN (CP(N )), and then by induction on the length of normal forms, as in =-=[11, 26, 9]-=-. 5 Conclusions In this paper we provided a categorical characterization of the behaviour of contextual P/T nets. We first defined the class of contextual processes of a CP/T net (see Definition 10); ... |

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Citation Context ...cartesian categories as pairings of two instances of an identity. However they do not form a natural transformation as duplicators in cartesian categories. Besides duplicators, gs-monoidal categories =-=[8]-=- are equipped with dischargers and they differ from cartesian categories just for missing the naturality axioms on duplicators and dischargers. The arrows of the gs-monoidal category freely generated ... |

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Citation Context ...ared memory, where readers are allowed to progress in parallel, with applications to transaction serializability in databases [25, 10], concurrent constraint programming [21, 3], asynchronous systems =-=[27]-=- and process algebras [20]. ? Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of Berlin and the University of... |

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Citation Context ...eral well-tailored applications. In fact, important constructions on ordinary nets can be extended to nets with read arcs, like those concerning non-sequential processes [22, 28] and event structures =-=[1]-=-. Moreover, these nets naturally model read-write access to shared memory, where readers are allowed to progress in parallel, with applications to transaction serializability in databases [25, 10], co... |

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Citation Context ... model read-write access to shared memory, where readers are allowed to progress in parallel, with applications to transaction serializability in databases [25, 10], concurrent constraint programming =-=[21, 3]-=-, asynchronous systems [27] and process algebras [20]. ? Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of B... |

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Citation Context ...uences corresponding to linearizations of the partial ordering are considered equivalent. Most semantic and logic notions specifying the concurrent behavior of nets are based on the notion of process =-=[23, 2]. Processe-=-s play an important role in the "Petri Nets are Monoids" approach to net theory [19, 11, 26]. In this approach, a net N is analogous to a signature \Sigma , and the symmetric monoidal catego... |

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Citation Context ...mantic and logic notions specifying the concurrent behavior of nets are based on the notion of process [23, 2]. Processes play an important role in the "Petri Nets are Monoids" approach to n=-=et theory [19, 11, 26]-=-. In this approach, a net N is analogous to a signature \Sigma , and the symmetric monoidal category P(N ) associated to N is analogous to the cartesian category L(\Sigma) of terms and substitutions f... |

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Citation Context ...ructures [1]. Moreover, these nets naturally model read-write access to shared memory, where readers are allowed to progress in parallel, with applications to transaction serializability in databases =-=[25, 10]-=-, concurrent constraint programming [21, 3], asynchronous systems [27] and process algebras [20]. ? Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Sys... |

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Citation Context ... model read-write access to shared memory, where readers are allowed to progress in parallel, with applications to transaction serializability in databases [25, 10], concurrent constraint programming =-=[21, 3]-=-, asynchronous systems [27] and process algebras [20]. ? Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of B... |

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Citation Context ... presence of a token in a place without actually consuming it. While the possibility of sensing for both presence and absence of a token yields very expressive nets equipped also with inhibitory arcs =-=[14, 7, 4, 5]-=-, in the paper we focus our attention to nets extended with read arcs only, generically referred to as contextual nets, which have a richer theory and refer to several well-tailored applications. In f... |

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Citation Context ...uences corresponding to linearizations of the partial ordering are considered equivalent. Most semantic and logic notions specifying the concurrent behavior of nets are based on the notion of process =-=[23, 2]. Processe-=-s play an important role in the "Petri Nets are Monoids" approach to net theory [19, 11, 26]. In this approach, a net N is analogous to a signature \Sigma , and the symmetric monoidal catego... |

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Citation Context ... are allowed to progress in parallel, with applications to transaction serializability in databases [25, 10], concurrent constraint programming [21, 3], asynchronous systems [27] and process algebras =-=[20]-=-. ? Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of Berlin and the University of Pisa; by the Office of Na... |