Journal of Symbolic Computation
Implementing mathematics with the Nuprl proof development system
Implementing mathematics with the Nuprl proof development system
Information and Computation - Semantics of Data Types
Constructive mathematics: a foundation for computable analysis
Theoretical Computer Science - Special issue on computability and complexity in analysis
COLOG '88 Proceedings of the International Conference on Computer Logic
Discrete linear objects in dimension n: the standard model
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
DGCI'08 Proceedings of the 14th IAPR international conference on Discrete geometry for computer imagery
Linear discrete line recognition and reconstruction based on a generalized preimage
IWCIA'06 Proceedings of the 11th international conference on Combinatorial Image Analysis
Ω-Arithmetization: A Discrete Multi-resolution Representation of Real Functions
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Arithmetization of a circular arc
DGCI'09 Proceedings of the 15th IAPR international conference on Discrete geometry for computer imagery
CompIMAGE'10 Proceedings of the Second international conference on Computational Modeling of Objects Represented in Images
Exploring the foundations of discrete analytical geometry in Isabelle/HOL
ADG'10 Proceedings of the 8th international conference on Automated Deduction in Geometry
Foundational aspects of multiscale digitization
Theoretical Computer Science
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This article presents a synthetic and self contained presentation of the discrete model of the continuum introduced by Harthong and Reeb [J. Harthong, Elements pour une theorie du continu, Asterisque 109/110 (1983) 235-244.[1]; J. Harthong, Une theorie du continu, in: H. Barreau, J. Harthong (Eds.), La mathematiques non standard, Editions du CNRS, 1989, pp. 307-329.[2]] and the related arithmetization process which led Reveilles [J.-P. Reveilles, Geometrie discrete, calcul en nombres entiers et algorithmique, Ph.D. Thesis, Universite Louis Pasteur, Strasbourg, France, 1991.[3]; J.-P. Reveilles, 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-Lof, 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.