Interpretability and Optimality

Abstract

The Semantic Theorems show that ${f}^{{\ast}}(\boldsymbol{X})$ and ${f}^{{\ast}{\ast}}(\boldsymbol{X})$ are optimal from a semantic point of view, and clarify which ⊆ -sense of rounding is the right one when *-semantic or **-semantic are to be applied. They provide, therefore, a general norm that computational functions F from ${I}^{{\ast}}({\mathbb{R}}^{k})$ to ${I}^{{\ast}}(\mathbb{R})$ must satisfy to conform to the f ^∗ or the f ^∗∗-semantic, but this is still not a general procedure by which these functions may be effectively computed. These procedures will be provided by the modal syntactic extension of continuous real functions, as far as they satisfy certain suitability conditions.