Insight in discrete geometry and computational content of a discrete model of the continuum

Abstract

This article presents a synthetic and self contained presentation of the discrete model of the continuum introduced by Harthong and Reeb [J. Harthong, Éléments pour une théorie du continu, Astérisque 109/110 (1983) 235–244. [1]; J. Harthong, Une théorie du continu, in: H. Barreau, J. Harthong (Eds.), La mathématiques non standard, Éditions du CNRS, 1989, pp. 307–329. [2]] and the related arithmetization process which led Reveillès [J.-P. Reveillès, Géométrie discrète, calcul en nombres entiers et algorithmique, Ph.D. Thesis, Université Louis Pasteur, Strasbourg, France, 1991. [3]; J.-P. Reveillès, D. Richard, Back and forth between continuous and discrete for the working computer scientist, Annals of Mathematics and Artificial Intelligence, Mathematics and Informatic 16(1–4) (1996) 89–152. [4]] to the definition of a discrete analytic line. We present then some basis on constructive mathematics [E. Bishop, D. Bridges, Constructive Analysis, Springer, Berlin, 1985. [5]], its link with programming [P. Martin-Löf, Constructive mathematics and computer programming, in: Logic, Methodology and Philosophy of Science, vol. VI, 1980, pp. 153–175. [6]; W.A. Howard, The formulae-as-types notion of construction, To H.B. Curry: Essays on Combinatory Logic, Lambda-calculus and Formalism, 1980, pp. 479–490. [7]] and we propose an analysis of the computational content of the so-called Harthong–Reeb line. More precisely, we show that a suitable version of this new model of the continuum partly fits with the constructive axiomatic of R proposed by Bridges [Constructive mathematics: a foundation for computable analysis, Theoretical Computer Science 219(1–2) (1999) 95–109. [8]]. This is the first step of a more general program on a constructive approach of the scaling transformation from discrete to continuous space.