Calculi of net structures and sets are similar

Ludwik Czaja

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 42, Issue: 2, page 271-284
  • ISSN: 0988-3754

Abstract

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Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and – if infinite pre/post sets of transitions are allowed – even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.

How to cite

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Czaja, Ludwik. "Calculi of net structures and sets are similar." RAIRO - Theoretical Informatics and Applications 42.2 (2007): 271-284. <http://eudml.org/doc/92871>.

@article{Czaja2007,
abstract = { Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and – if infinite pre/post sets of transitions are allowed – even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures. },
author = {Czaja, Ludwik},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Net structures; synchronised operations; distributive lattice; Boolean algebra; representation theorems; synchronised union; synchronised intersection; synchronised difference},
language = {eng},
month = {9},
number = {2},
pages = {271-284},
publisher = {EDP Sciences},
title = {Calculi of net structures and sets are similar},
url = {http://eudml.org/doc/92871},
volume = {42},
year = {2007},
}

TY - JOUR
AU - Czaja, Ludwik
TI - Calculi of net structures and sets are similar
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/9//
PB - EDP Sciences
VL - 42
IS - 2
SP - 271
EP - 284
AB - Three basic operations on labelled net structures are proposed: synchronised union, synchronised intersection and synchronised difference. The first of them is a version of known parallel composition with synchronised actions identically labelled. The operations work analogously to the ordinary union, intersection and difference on sets. It is shown that the universe of net structures with these operations is a distributive lattice and – if infinite pre/post sets of transitions are allowed – even a Boolean algebra. As a consequence, some representation theorems of this algebra are stated. The primitive objects are atomic net structures containing one transition with at most one pre-place or post-place (but not both). A simple example of a production system constructed by making use of the operations (and its transformations) is given. Some remarks on behavioural properties of compound nets are stated, in particular, how some constructing strategies may help to infer liveness. The latter issue is limited to semantics of place/transition nets without weights on arrows and with unbounded capacity of places and is not extensively investigated, since the main objective is focused on a calculus of net structures.
LA - eng
KW - Net structures; synchronised operations; distributive lattice; Boolean algebra; representation theorems; synchronised union; synchronised intersection; synchronised difference
UR - http://eudml.org/doc/92871
ER -

References

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