Model Intervals

This paper summarizes the most important results and features of Modal Interval Analysis. The ground idea of Interval mathematics is that ordinary set-theoretical intervals are the consistent context for numerical computing. However, this interval context presents basic structural and semantic rigidity arising from its set-theoretical foundation. To correct this situation, Modal Interval Analysis defines intervals starting from the identification of real numbers with the sets of predicates they accept or reject. A modal interval is defined as a pair formed by a classical interval (i.e. a set of numbers) and a quantifier, following a similar method to that in which real numbers are associated in pairs having the same absolute value but opposite signs. Two different extensions for a continuous function (called semantic extensions, since both will have a meaning thanks to the important semantic theorems) are defined and their properties are established. The definition of the rational extension, and its relationships with the semantic extensions, it make possible to compute the semantic extensions and to give a logical meaning to the interval results of a rational computations.For some functions the semantic extensions are equal, for instance, for the arithmetic operators, which can be obtained through computations with the intervals' bounds, obtaining the definitions of the well-known Kaucher's completed interval arithmetic. It is important to remark that the process of the construction of the Modal Interval Analysis is absolutely different from the process followed by Kaucher.Similarly to the Kaucher's completed interval arithmetic, with modal intervals it is possible to solve the equations A + X = B or A * X = B but with a very important difference. To find the algebraic solution for the equation A + X = B or A * X = B when A, X and B are classical intervals is a single problem: a) to find and interval X which substituted in the corresponding equation, satisfies the equality. Modal Interval Analysis not only solves that problem but also it gives a logical meaning to the solution.As a conclusion, the most important difference between Modal Interval Analysis versus Classical Intervals + Kaucher's Completed Arithmetic is the logical-semantic ground of the modal intervals and the meaning for the interval results in the functional computations or in the solution of a linear equation, provided by the semantic theorems.