Modal Intervals Revisited Part 1: A Generalized Interval Natural Extension
Résumé
The modal intervals theory is an extension of the classical intervals theory which provides richer interpretations (including in particular inner and outer approximations of the ranges of real functions). In spite of its promising potential, the modal intervals theory is not widely used today because of its original and complicated construction. The present paper proposes a new formulation of the modal intervals theory. New extensions of continuous real functions to generalized intervals (intervals whose bounds are not constrained to be ordered) are defined. They are called AE-extensions. These AE-extensions provide the same interpretations as the ones provided by the modal intervals theory, thus enhancing the interpretation of the classical interval extensions. The construction of AE-extensions strictly follows the model of the classical intervals theory: Starting from a generalization of the definition of the extensions to classical intervals, the minimal AE-extensions of the elementary operations are first built leading to a generalized interval arithmetic. This arithmetic is proved to coincide with the well known Kaucher arithmetic. Then the natural AE-extensions are constructed similarly to the classical natural extensions. The natural AE-extensions represent an important simplification of the formulation of the four "theorems of $\ast$ and $\ast\ast$ interpretation of a modal rational extension" and "theorems of coercion to $\ast$ and $\ast\ast$ interpretability" of the modal intervals 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 the classical intervals theory, the new formulation of the modal intervals 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 extension to generalized intervals will be introduced in the second part of this paper.