Semicomputable Points in Euclidean Spaces

Authors Mathieu Hoyrup, Donald M. Stull

Thumbnail PDF


  • Filesize: 469 kB
  • 13 pages

Document Identifiers

Author Details

Mathieu Hoyrup
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Donald M. Stull
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Cite AsGet BibTex

Mathieu Hoyrup and Donald M. Stull. Semicomputable Points in Euclidean Spaces. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Semicomputable point
  • Left-c.e. real
  • Convex cone
  • Solovay reducibility
  • Genericity


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. George Barmpalias and Andrew Lewis-Pye. Differences of halting probabilities. J. Comput. Syst. Sci., 89:349-360, 2017. Google Scholar
  2. Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, 2004. Google Scholar
  3. Vasco Brattka and Gero Presser. Computability on subsets of metric spaces. Theoretical Computer Science, 305(1-3):43-76, 2003. Google Scholar
  4. Mathieu Hoyrup. Genericity of Weakly Computable Objects. Theory Comput. Syst., 60(3):396-420, 2017. Google Scholar
  5. Mathieu Hoyrup, Diego Nava Saucedo, and Don M. Stull. Semicomputable Geometry. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 129:1-129:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  6. Martin Kummer and Marcus Schäfer. Computability of convex sets. In Ernst W. Mayr and Claude Puech, editors, STACS 95, pages 550-561, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. Google Scholar
  7. Joseph S. Miller. On Work of Barmpalias and Lewis-Pye: A Derivation on the D.C.E. Reals. In Adam R. Day, Michael R. Fellows, Noam Greenberg, Bakhadyr Khoussainov, Alexander G. Melnikov, and Frances A. Rosamond, editors, Computability and Complexity - Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday, volume 10010 of Lecture Notes in Computer Science, pages 644-659. Springer, 2017. Google Scholar
  8. André Nies. Computability and randomness. Oxford logic guides. Oxford University Press, 2009. Google Scholar
  9. Martin Ziegler. Computability on Regular Subsets of Euclidean Space. Mathematical Logic Quarterly, 48(S1):157-181, 2002. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail