# Why Constraint Interval Arithmetic Works Well: A Theorem Explains Empirical Success

## Publication Date

4-1-2022

Technical Report: UTEP-CS-22-52

## Abstract

Often, we are interested in a quantity that is difficult or impossible to measure directly, e.g., tomorrow's temperature. To estimate this quantity, we measure auxiliary easier-to-measure quantities that are related to the desired ones by a known dependence, and use the known relation to estimate the desired quantity. Measurements are never absolutely accurate, there is always a measurement error, i.e., a non-zero difference between the measurement result and the actual (unknown) value of the corresponding quantity. In many practical situations, the only information that we have about each measurement error is the bound on its absolute value. In such situations, after each measurement, the only information that we gain about the actual (unknown) value of the corresponding quantity is that this value belongs to the corresponding interval. Thus, the only information that we have about the value of the desired quantity is that it belongs to the range of the values of the corresponding function when its inputs are in these intervals. Computing this range is one of the main problems of interval computations.

Lately, it was shown that in many cases, it is more efficient to compute the range if we first re-scale each input to the interval [0,1]; this is one of the main ideas behind Constraint Interval Arithmetic. In this paper, we explain the empirical success of this idea and even prove that, in some reasonable sense, this re-scaling is the best.

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