Reliable Computing 8: 213–227, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
In Case of Interval (or More General)
Uncertainty, No Algorithm Can Choose the
at GH Wuppertal, Fachbereich Mathematik, Gaußstraße 20, 42097 Wuppertal,
Germany, e-mail: firstname.lastname@example.org
Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA,
LIP6, Pole IA, Universit
e Pierre et Marie Curie, Case 169, 4 place Jussieu, 75252 Paris C
France, e-mail: Maria.Rifqi@lip6.fr
(Received: 23 July 1996; accepted: 3 August 2001)
Abstract. When we only know the interval of possible values of a certain quantity (or a more general
set of possible values), it is desirable to characterize this interval by supplying the user with the
“simplest” element from this interval, and by characterizing how different from this value we can get.
For example, if, for some unknown physical quantity x, measurements result in the interval [1
of possible values, then, most probably, the physicist will publish this result as y
2. Similarly, a
natural representation of the measurement result x
141593] is x
In this paper, we show that the problem of choosing the simplest element from a given interval
(or from a given set) is, in general, not algorithmically solvable.
1. In Case of Interval (or More General Set) Uncertainty, a User Would Like
to Have a Representative Value from This Interval (Set)
The valueof a physical quantity y is usually obtained either by a direct measurement,
or by an indirect measurement, i.e., by processing the results of some related
. Since measurements are normally not 100% precise, their
results may differ from the actual values of the measured quantities. As a result,
after the measurement (direct or indirect), we do not get the exact value of the
desired quantity, we only get a set Y of its possible values.
In many cases, this set Y is an interval, but more complicated sets are also
possible: e.g., if we know that y
4], then the set of possible
values of y is the union of two intervals [−2