Non-traditional intervals and their use. Which ones really make sense?

TL;DR

This paper explores the rationale behind traditional interval choices in Interval Analysis, examines the potential of alternative interval types like improper intervals, and demonstrates their practical usefulness.

Contribution

It provides a theoretical and practical analysis of non-traditional intervals, especially improper intervals, and advocates for their inclusion in Interval Analysis.

Findings

Improper intervals and Kaucher arithmetic are highly useful.

Traditional intervals are closed and contain endpoints, but alternatives can offer advantages.

Expanding interval types enhances the flexibility and applicability of Interval Analysis.

Abstract

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are closed, i.\,e. contain their endpoints, and also what is wrong with an empty interval. The second question considered in the work is how expedient it is to expand the set of traditional intervals by some other objects. We show that improper ("reversed") intervals and the arithmetic of such intervals (Kaucher complete interval arithmetic) are very useful from many different points of view.