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Issue RAIRO-Theor. Inf. Appl.
Volume 42, Number 2, April-June 2008
Page(s) 271 - 284
DOI 10.1051/ita:2007033
Published online 06 September 2007

RAIRO-Theor. Inf. Appl. 42, 271-284 (2008)
DOI: 10.1051/ita:2007033

Calculi of net structures and sets are similar

Ludwik Czaja

Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland; lczaja@mimuw.edu.pl


(Received August 3, 2006. Accepted March 2, 2007. Published online 6 September 2007.)

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.


Mathematics Subject Classification. 68Q85

Key words: Net structures -- synchronised operations -- distributive lattice -- Boolean algebra -- representation theorems


© EDP Sciences 2007


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