Semicomputable Geometry

Authors Mathieu Hoyrup, Diego Nava Saucedo, Don M. Stull

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Author Details

Mathieu Hoyrup
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Diego Nava Saucedo
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Don M. Stull
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

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Mathieu Hoyrup, Diego Nava Saucedo, and Don M. Stull. Semicomputable Geometry. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 129:1-129:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Computability and semicomputability of compact subsets of the Euclidean spaces are important notions, that have been investigated for many classes of sets including fractals (Julia sets, Mandelbrot set) and objects with geometrical or topological constraints (embedding of a sphere). In this paper we investigate one of the simplest classes, namely the filled triangles in the plane. We study the properties of the parameters of semicomputable triangles, such as the coordinates of their vertices. This problem is surprisingly rich. We introduce and develop a notion of semicomputability of points of the plane which is a generalization in dimension 2 of the left-c.e. and right-c.e. numbers. We relate this notion to Solovay reducibility. We show that semicomputable triangles admit no finite parametrization, for some notion of parametrization.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Computable set
  • Semicomputable set
  • Solovay reducibility
  • Left-ce real
  • Genericity


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