Gradual interval arithmetic and fuzzy interval arithmetic

This paper proposes an analysis of and a reflection on interval arithmetic (IA) and its extension to gradual interval arithmetic (GIA). Through this reflection, an overview of a part of IA that is directly related to fuzzy interval arithmetic (FIA) is analyzed, compared, and categorized according to two main families of IA: standard interval arithmetic (SIA) and instantiated interval arithmetic (IIA). Furthermore, SIA and IIA visions represent two viewpoints of computation that are different and they will cause modifications in interval interpretation and manipulation. This vision is essential in understanding the philosophy of IA and GIA computational mechanisms. The contribution of this paper is twofold. First, according to SIA and IIA visions, an analysis and a classification of a part of IAs are given. Equivalences and links between these IAs are analyzed and established. Second, an extension of IA to the gradual context is proposed. The GIA extension provides a new interpretation of FIA according to the gradual representation.