Part 1: A Generalized Interval Natural Extension
Alexandre Goldsztejn · 2012
Modal interval theory is an of classical interval theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, modal interval theory is not widely used today because of its original and complicated construction. The present paper proposes a new formulation of modal interval theory. New extensions of continuous real functions to generalized intervals (intervals whose bounds are not constrained to be ordered) are dened. They are called AE-extensions. These AE-extensions provide the same interpretations as the ones provided by modal interval theory, thus enhancing the interpretation of the classical interval extensions. The construction of AE-extensions strictly follows the model of classical interval theory: starting from a generalization of the denition of the extensions to classical intervals, the minimal AE-extensions of the elementary operations are rst built leading to a generalized interval arithmetic. This arithmetic is proved to coincide with the well known Kaucher arithmetic. Then natural AE-extensions are constructed similarly to the classical natural extensions. The natural AE-extensions represent an important simplication of the formulation of the four \theorems of and interpretation of a modal rational extension and \theorems of coercion to and interpretability of modal interval theory. New proofs are provided for the interpretation of these natural AE-extensions that correct the one proposed in the framework of modal intervals. With a construction similar to classical interval theory, the new formulation of modal interval theory proposed in this paper should facilitate the understanding of the underlying mechanisms, the addition of new items to the theory (e.g. new extensions) and its usage. In particular, a new mean-value to generalized intervals will be introduced in the second part of this paper.