A Qualitative Approach to Continuity
نویسنده
چکیده
The standard mathematical treatment of continuity for real-valued functions presupposes a model in which arbitrarily ne distinctions can be drawn between neighbouring states. In a qualitative model these distinctions are obliterated by treating whole ranges of states as qualitatively indistinguishable. In this paper we show how topological considerations can be used, in the form of a relation called`dominance', to handle continuity in discrete qualitative state-spaces derived from an underlying continuous space. A general theorem concerning the composition of such state-spaces is proved, and its potential usefulness illustrated by application to three diierent areas of spatio-temporal reasoning. Le traitement math ematique normal de la continuit e des fonctions aux valeurs r eels pr esuppose un mod ele dans lequel des distinctions arbitrairement nes peuvent ^ etre faites entre les etats avoisinants. Dans un mod ele qualitatif, ces distinctions sont eeac ees en consid erant une gamme des etats comme qualitativement indistinguibles. Dans cet article, nous montrons comment utiliser les consid erations topolo-giques, sous la forme d'une relation que nous appelons làdominance', aan de traiter la continuit e dans les espaces des etats qualitatifs, a condition que ces espaces soient d eriv es d'un espace quantitatif fondamental. Nous d emontrons un th eor eme g en eral relatif a la composition de tels espaces, et nous eclairons son utilit e potentielle par son application a trois domaines du raisonnement spatio-temporel. 1 Motivation In this paper we attempt to develop a theory of time, change, and continuity which is consistent with the standard mathematical model that has served generations of scientists well, and which nonetheless also takes seriously the disquiet that has led many philosophers and AI researchers to seek to replace this model by something more immediately adapted to the requirements of giving a qualitative rather than quantitative description of the phenomena. This is important for any application in 1
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