Semicomputable Geometry
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.
Computable set
Semicomputable set
Solovay reducibility
Left-ce real
Genericity
Theory of computation~Computability
129:1-129:13
Regular Paper
A full version of the article including all the proofs is available at https://hal.inria.fr/hal-01770562.
Mathieu
Hoyrup
Mathieu Hoyrup
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Diego
Nava Saucedo
Diego Nava Saucedo
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Supported by LORIA.
Don M.
Stull
Don M. Stull
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Supported by Inria Nancy Grand-Est.
10.4230/LIPIcs.ICALP.2018.129
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Mathieu Hoyrup, Diego Nava Saucedo, and Donald M. Stull
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