Reliable Computing 4: 109–112, 1998.
1998 Kluwer Academic Publishers. Printed in the Netherlands.
Intervals in Space-Time:
A. D. Alexandrov is 85
Department of Mathematics, Boston University, Boston, MA 02215, USA,
Department of Electrical and Computer Engineering, University of Texas at El Paso, El Paso, TX
79968, USA, e-mail: firstname.lastname@example.org
On August 12, 1997, Alexander Danilovich Alexandrov, the well-known geometer,
a member of the Russian Academy of Sciences and of several foreign academies,
will turn 85.
A. D. Alexandrov’s seminal research papers cover areas ranging from quan-
tum mechanics to geometry to philosophy of science. In particular, his innovative
approach to foundations of space-time geometry is directly related to intervals and
interval computations. In this short text, we will brieﬂy describe his interval-related
Before Alexandrov: 1-dimensional intervals that correspond to measuring
time. Let us ﬁrst describe a non-relativistic situation. In non-relativistic (New-
tonian) physics, every event is characterized by a single number t (its moment of
time). An event with a larger value of t occurs after the event with a smaller value
of time. (In this sense, in Newtonian physics, temporal order is 1-dimensional.)
Measurement inaccuracies make this simple picture a little bit more complicated.
Indeed, measurement is never 100% accurate, there is usually some uncertainty
associated with it. As a result, if, e.g., we measure time, we do not get the exact
actual value t of the time. Instead, we get an approximate measurement result
based on which, we can determine the interval t =[t
t] of possible values of t,where
t+∆,and∆is an upper bound on the measurement error (this upper
bound is usually given by a manufacturer of the time measuring instrument).
From the physical viewpoint, the interval means that the event, whose time we
are measuring, occured after the moment t
and before the moment t. In other words,
we have two events that bound the event in whose timing we are interested.
As a result, to describe the actual information about the events, we must use
intervals t instead of numbers t.
A. Levichev is on leave from the Institute of Mathematics, Novosibirsk, 630090, Russia.