Calculi of net structures and sets are similar

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.

Rough Net Structures - Example of Information System

“…It follows from Propositions 2.1 (ii) and 2.2 (ii) that LN S, ⊕, is a lattice. Let us quote from [3] some important result concerning this lattice:…”

“…Being sets, the net structures are combined by operations on sets. As shown in [3], if each transition gets a unique label in a net structure then the class of all such nets structures over a set of places, with operations similar to the set theoretical ones, becomes a distributive lattice. Its members may be taken as compound objects of information systems studied in the rough set theory see [10], [11], [13], [6].…”

“…The main message of this work is only a demonstration how to formulate the rough net structure concepts and problems (not rough nets on the whole!) in the calculus developed in [3] and to illustrate this by widely-known example. The running example for presentation of the idea, formalism and some problems of rough net structures, is the toy-task of dining philosophers [4].…”

“…This is an excerpt from [3] indispensable for understanding the main part of the present paper. The short Section 3 contains only an important consequence of Section 2, stating that the calculus is a distributive lattice.…”

“…The short Section 3 contains only an important consequence of Section 2, stating that the calculus is a distributive lattice. However the classical representation theorem originating from [17] and concisely proved in [3] for the lattice of net structures, as not essential for further considerations, has been mentioned only. Section 4 presents basic notions of the net version of information systems like lower/upper net-approximations of a net structure and example of their transformation under renaming operation.…”