## Petri Nets And Step Transition Systems (1992)

Venue: | International Journal of Foundations of Computer Science |

Citations: | 43 - 1 self |

### BibTeX

@ARTICLE{Mukund92petrinets,

author = {Madhavan Mukund},

title = {Petri Nets And Step Transition Systems},

journal = {International Journal of Foundations of Computer Science},

year = {1992},

volume = {3},

pages = {443--478}

}

### Years of Citing Articles

### OpenURL

### Abstract

Labelled transition systems are a simple yet powerful formalism for describing the operational behaviour of computing systems. They can be extended to model concurrency faithfully by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step. Petri nets (or Place/Transition nets) give rise to such step transition systems in a natural way -- the marking diagram of a Petri net is the canonical transition system associated with it. In this paper, we characterize the class of PN-transition systems, which are precisely those step transition systems generated by Petri nets. We express the correspondence between PN-transition systems and Petri nets in terms of an adjunction between a category of PN-transition systems and a category of Petri nets in which the associated morphisms are behaviour-preserving in a strong and natural sense.

### Citations

3281 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...ons between dierent models of concurrency is that we can automatically translate results from one model to another. For instance, to obtain a non-interleaved model for a process calculus such as CCS [=-=9]-=-, it is intuitively easier to enrich the standard interleaved transition system semantics to obtain a more faithful representation of concurrency, rather than providing a semantics directly in terms o... |

938 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...have f 2��1 = f 2 �� f 1 and we are done. 2 8. The adjunction Having constructed the functor NT from PNet to PNts , we want to show that it has a left adjoint TN : PNts ! PNet . According to M=-=ac Lane [8], it-=- su��ces to construct a map TNO mapping objects in PNts to objects in PNet so that the diagram shown in Figure 7 commutes. The object map TNO can then be extended uniquely to a functor TN : PNts !... |

166 |
Petri nets are monoids
- Meseguer, Montanari
- 1990
(Show Context)
Citation Context ...s the same as that of the original transition system. We turn now to other categories of Petri nets that have been described in the literature|notably those of Winskel [18] and Meseguer and Montanari =-=[10-=-]. Our denition of net morphisms is a strengthened version of the one used by Winskel in [18]. Winskel permits the map on places to be an arbitrary relation (actually a multirelation) in the forward d... |

91 |
Catégories of Asynchronous Systems
- Bednarczyk
- 1988
(Show Context)
Citation Context ... corresponding to dierent models of concurrency. For instance, Winskel and Nielsen [20] use a version of regions to establish a core ection between a special class of asynchronous transition systems [=-=1, 17-=-] and 1-safe Petri nets. On a slightly dierent track, Hoogers, Kleijn and Thiagarajan [5] use regions to obtain a language-theoretic characterization of the non-sequential behaviour of Petri nets in t... |

76 | Event structure semantics in CCS and related languages - Winskel - 1982 |

73 |
Petri nets, algebras, morphisms, and compositionality
- Winskel
- 1987
(Show Context)
Citation Context ...orphisms we dene between nets are a smooth generalization of the morphisms dened between elementary net systems in [14]. These morphisms are strengthened versions of the morphisms dened by Winskel [18=-=]-=-. They preserve the dynamic behaviour of nets in a strong way. There is a natural functor NT : PNet ! PNts which maps each Petri net to the transition system associated with its marking diagram. Our m... |

48 |
A categorical linear framework for Petri nets, in
- Brown, Gum
- 1990
(Show Context)
Citation Context ... T 1 : If T (t 1 ) = t 2 then: { 1 S ( t 1 ) = t 2 and 1 S (t 1 ) = t 2 . { 8s 2 t 2 : W 1 (S (s); t 1 ) = W 2 (s; t 2 ). { 8s 2 t 2 : W 1 (t 1 ; S (s)) = W 2 (t 2 ; s). Following [2], we have directly dened the map on places as a partial function in the reverse direction, rather than as a relation in the forward direction whose inverse is a partial function (as in [14]). m m ... |

45 |
Concurrent machines
- Shields
- 1985
(Show Context)
Citation Context ... corresponding to dierent models of concurrency. For instance, Winskel and Nielsen [20] use a version of regions to establish a core ection between a special class of asynchronous transition systems [=-=1, 17-=-] and 1-safe Petri nets. On a slightly dierent track, Hoogers, Kleijn and Thiagarajan [5] use regions to obtain a language-theoretic characterization of the non-sequential behaviour of Petri nets in t... |

36 |
A distributed operational semantics for CCS based on condition/event systems
- Degano, Nicola, et al.
- 1988
(Show Context)
Citation Context ... intuitively easier to enrich the standard interleaved transition system semantics to obtain a more faithful representation of concurrency, rather than providing a semantics directly in terms of nets =-=[3, 15]-=- or event structures [19]. So, for example, we can use a very simple extension of the standard operational semantics for CCS to provide a non-interleaved semantics for a rich subclass of the language ... |

32 |
Nets, terms, and formulas
- Olderog
- 1991
(Show Context)
Citation Context ... intuitively easier to enrich the standard interleaved transition system semantics to obtain a more faithful representation of concurrency, rather than providing a semantics directly in terms of nets =-=[3, 15]-=- or event structures [19]. So, for example, we can use a very simple extension of the standard operational semantics for CCS to provide a non-interleaved semantics for a rich subclass of the language ... |

19 |
and P.S.Thiagarajan: A Trace Semantics for Petri Nets
- Hoogers
- 1995
(Show Context)
Citation Context ...a version of regions to establish a core ection between a special class of asynchronous transition systems [1, 17] and 1-safe Petri nets. On a slightly dierent track, Hoogers, Kleijn and Thiagarajan [=-=5]-=- use regions to obtain a language-theoretic characterization of the non-sequential behaviour of Petri nets in terms of a generalized version of Mazurkiewicz trace languages. We shall discuss the relat... |

15 |
locations and asynchronous transition systems
- Mukund, Nielsen, et al.
- 1992
(Show Context)
Citation Context ...e can use a very simple extension of the standard operational semantics for CCS to provide a non-interleaved semantics for a rich subclass of the language in terms of asynchronous transtition systems =-=[13]. The-=-se asynchronous transition systems belong to the special subcategory shown to correspond closely to safe nets in [20]. This implies that we obtain \for free" a net semantics for this language. An... |

3 |
Partial 2-structures; Part II, State spaces of concurrent systems
- Ehrenfeucht, Rozenberg
- 1990
(Show Context)
Citation Context ...ystems To describe PN-transition systems, we need to introduce the notion of a region. Regions have originally been dened in the context of sequential transition systems by Ehrenfeucht and Rozenberg [=-=4]-=- as a transition system counterpart of the notion of a condition in an elementary net system. Using regions, they characterize the class of elementary transition systems which represent the behaviour ... |

3 |
A logic for distributed transition systems
- Lodaya, Ramanujam, et al.
- 1988
(Show Context)
Citation Context ...m (A3), q 0 = q 00 . 2 As we had mentioned earlier, we have to ensure that the steps in a step transition system actually represent concurrent actions. For instance, Lodaya, Ramanujam and Thiagarajan =-=[7] den-=-e distributed transition systems , which are basically step transition systems with a \step axiom" which insists that for every step, all substeps must be present and combine together in a consis... |

3 | A transition system characterization of Petri nets - Mukund - 1990 |

1 |
Diamond Properties of Elementary Net Systems
- Hoogeboom, Rozenberg
- 1991
(Show Context)
Citation Context ...14]. Since elementary transition systems are a subclass of conventional sequential transition systems, this categorical equivalence oers an alternative proof of the result of Hoogeboom and Rozenberg [=-=6-=-] Elementary net systems Safe nets ,! Petri nets * * Asynchronous transition systems (Winskel, Nielsen) * * Elementary PN-transition systems ,! Safe PN-transition systems ,! PN-transition systems = E... |

1 |
Transition System Models for Concurrency
- Mukund
- 1992
(Show Context)
Citation Context ...shed a similar result relating a subclass of asynchronous transition systems to 1-safe Petri nets. By tuning our regions appropriately, we canst the results of [14] and [20] neatly into our framework =-=[12]-=-. To begin with, for a region r, we can restrict the range of r Q to f0; 1g and the range of r E to f(0; 0); (0; 1); (1; 0); (1; 1)g. If we enforce the regional axioms (A3) and (A4) for step transitio... |

1 |
Models for Concurrency", (to appear
- Winskel, Nielsen
(Show Context)
Citation Context ... m m m m A A AU A A AU ? ? 6 6 a b m J J J^ J J J^ b b a a Figure 1: of transition systems corresponding to dierent models of concurrency. For instance, Winskel and Nielsen [20] use a version of regions to establish a core ection between a special class of asynchronous transition systems [1, 17] and 1-safe Petri nets. On a slightly dierent track, Hoogers, Kleijn and Thiagar... |