Reliable Computing 9: 1–20, 2003.
2003 Kluwer Academic Publishers. Printed in the Netherlands.
Diagrammatic Analysis of Interval Linear
Part I: Basic Notions and the One-Dimensional Case
Institute of Fundamental Technological Research, Polish Academy of Sciences,
Swie¸tokrzyska 21, 00–049 Warsaw, Poland, e-mail: firstname.lastname@example.org
(Received: 8 September 2001; accepted: 11 June 2002)
Abstract. Different solution sets for theintervallinear system A
x = b are characterized and classiﬁed
using diagrammatic tools for interval analysis developed recently. In Part I, a thorough analysis of
the basic, one-dimensional expression a
x ♦ b is conducted, with the help of an appropriate interval
space diagram, in which all the needed relations ♦
⊃⊂, ⊇, ⊆,
=} are directly representable. The
solution sets are obtained with simple diagrammatic constructions, and are characterized by quotient
sequences of a and b. A complete classiﬁcation of all possible solution types is developed in this way,
with various ways of visualizing the structure of the set of these types.
This paper continues, following –, , , , the development of dia-
grammatic tools for interval research. Here we will introduce a uniﬁed, diagram-
matic approach that allows for determination of types and study of properties of
basic solution sets of the interval linear equation Σ
= b in an integrated way.
First, the correctness of the phrase “linear equation” is discussed. In consequence,
the above formula is properly renamed as an interval relational expression, and its
four basic solution sets Σ, Σ
Next, the diagrammatic tools needed are introduced—for additional details, see
, , . Diagrammatic methods of knowledge representation and reasoning
recently became a subject of intensive research, including both basic theoretical
work concerning their cognitive sources and mathematical foundations, and their
numerous applications in various ﬁelds, especially in mathematics , , .
These tools are then used to analyze in detail the simplest, one-dimensional
interval relational expressions a
x ♦ b,where♦
⊃⊂, ⊇, ⊆,
=}. The analysis also
uses the quotient sequence concept, and leads to the full classiﬁcation of possible
structural cases and solution types. These results, besides their own value, are
needed to conduct further analysis reported in the second part of the paper .
The second part contains the analysis of the two-dimensional interval relational
♦ b and their solution sets. The generalization of the