Reliable Computing 3: 209–217, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Diagrammatic Representation of Interval Space
in Proving Theorems about Interval Relations
Institute of Fundamental Technological Research, Polish Academy of Sciences, ul.
21, 00-049 Warsaw, Poland, e-mail: firstname.lastname@example.org, URL: http://www.ippt.gov.pl/˜zkulpa
(Received: 15 November 1996; accepted: 26 February 1997)
Abstract. The paper presents two-dimensional graphical representations for the space of intervals (the
IS-diagram) and arrangement interval relations (the W-diagram).Theusefulness of the representations
is illustrated with the example of proving equivalence of different characterizations of convex interval
As was stated by Simon : “
solving a problem simply means representing it
so as to make the solution transparent.” In fact, much of the progress in science
in general, and mathematics in particular, consisted of ﬁnding new representations
of various phenomena or formal constructs. Devising a new way of represent-
ing knowledge about some phenomenon, formal system, or problem class offers
new means of effective description of the domain objects and new possibilities of
reasoning about them and solving problems involving them.
Since some time, so-called diagrammatic representation and reasoning meth-
ods gain considerable interest, as they often provide more effective means than
other representations for storing, using and presenting complex information and
knowledge, see the survey paper .
In this paper a two-dimensional, diagrammatic representation of the space of
intervals, called an IS-diagram (introduced by Kulpa ), is elaborated. It consti-
tutes an extension and reﬁnement of the representation proposed by Rit  and
used to illustrate the concept of convex interval relations by N
okel . Anoth-
er diagrammatic notation based on it, called a W-diagram, is useful in depicting
interval relations and operations on them , . Two other auxiliary diagrammatic
notations, namely the conjunction diagram and lattice diagram, are also introduced.
Usefulness of these diagrams is illustrated with the example of proving equivalence
of different characterizations of convex interval relations , , .