Reliable Computing 7: 77–111, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
A. SAINZ, LAMBERT JORBA,
REMEI CALM, ROSA ESTELA, HONORINO MIELGO, and
(members of the SIGLA/X group)
Departamento de Inform
atica y Matem
atica Aplicada, University of Girona, Campus Montilivi,
E-17071 Girona, Spain, e-mail: firstname.lastname@example.org
(Received: 5 May 1999; accepted: 19 October 1999)
Abstract. This paper summarizes the most important results and features of Modal Interval Analysis.
The ground idea of Interval mathematics is that ordinary set-theoretical intervals are the consistent
context for numerical computing. However, this interval context presents basic structural and semantic
rigidity arising from its set-theoretical foundation. To correct this situation, Modal Interval Analysis
deﬁnes intervals starting from the identiﬁcation of real numbers with the sets of predicates they
accept or reject. A modal interval is deﬁned as a pair formed by a classical interval (i.e. a set of
numbers) and a quantiﬁer, following a similar method to that in which real numbers are associated
in pairs having the same absolute value but opposite signs. Two different extensions for a continuous
function (called semantic extensions, since both will have a meaning thanks to the important semantic
theorems) are deﬁned and their properties are established. The deﬁnition of the rational extension, and
its relationships with the semantic extensions, it make possible to compute the semantic extensions
and to give a logical meaning to the interval results of a rational computations.
For some functions the semantic extensions are equal, for instance, for the arithmetic operators,
which can be obtained through computations with the intervals’ bounds, obtaining the deﬁnitions of
the well-known Kaucher’s completed interval arithmetic. It is important to remark that the process of
the construction of the Modal Interval Analysis is absolutely different from the process followed by
Similarly to the Kaucher’s completed interval arithmetic, with modal intervals it is possible to
solve the equations A + X = B or A ∗ X = B but with a very important difference. To ﬁnd the algebraic
solution for the equation A + X = B or A ∗ X = B when A, X and B are classical intervals is a
single problem: a) to ﬁnd and interval X which substituted in the corresponding equation, satisﬁes
the equality. Modal Interval Analysis not only solves that problem but also it gives a logical meaning
to the solution.
As a conclusion, the most important difference between Modal Interval Analysis versus Classical
Intervals + Kaucher’s Completed Arithmetic is the logical-semantic ground of the modal intervals
and the meaning for the interval results in the functional computations or in the solution of a linear
equation, provided by the semantic theorems.
The set-theoretical form of Interval analysis has stimulate a large body of work
from its inception in the late 1950’s , but with no fundamental departure from its
initial set-theoretical foundations in spite of some papers devoted to the structural
analysis of the method or to its completion (–, , , ).
In this paper a new interval analysis, the Modal Interval Analysis is presented.