On the Lattices of Effectively Open Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

We show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is hereditarily undecidable. Moreover, for several important spaces (e.g., finite-dimensional Euclidean spaces and the domain \(P\omega \)) this theory is m-equivalent to the first-order arithmetic.

Keywords

Lattice Effectively open set First-order theory Interpretation 

Notes

Acknowledgement

We thank André Nies for a discussion of the lower bound problem for \(Th(\varSigma ^0_1(\mathcal {N}))\), and the referees for valuable comments.

References

  1. 1.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Oxford University Press, Oxford (1994)Google Scholar
  2. 2.
    Ershov, Y.: Computable functionals of finite types. Algebra Log. 11(4), 367–433 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ershov, Y.: Theory of domains and nearby. In: Bjørner, D., Broy, M., Pottosin, I.V. (eds.) Formal Methods in Programming and Their Applications. LNCS, vol. 735, pp. 1–7. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  4. 4.
    Grzegorczyk, A.: Undecidability of some topological theories. Fundamenta Mathematicae 38, 137–152 (1951)MathSciNetGoogle Scholar
  5. 5.
    Herrmann, E.: Definable boolean pairs in the lattice of recursively enumerable sets. In: Proceedings of 1st Easter Conference in Model Theory, Diedrichshagen, pp. 42–67 (1983)Google Scholar
  6. 6.
    Herrmann, E.: The undecidability of the elementary theory of the lattice of recursively enumerable sets. In: Proceedings of 2nd Frege Conference at Schwerin, GDR, vol. 20, pp. 66–72 (1984)Google Scholar
  7. 7.
    Harrington, L., Nies, A.: Coding in the lattice of enumerable sets. Adv. Math. 133, 133–162 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Korovina, M.V., Kudinov, O.V.: The uniformity principle for sigma-definability. J. Log. Comput. 19(1), 159–174 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Moschovakis, Y.N.: Descriptive Set Theory. North Holland, Amsterdam (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Nies, A.: Effectively dense Boolean algebras and their applications. Trans. Am. Math. Soc. 352(11), 4989–5012 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Selivanov, V.L.: Towards a descriptive set theory for domain-like structures. Theoret. Comput. Sci. 365, 258–282 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.S.L. Sobolev Institute of Mathematics SB RASNovosibirskRussia
  2. 2.A.P. Ershov Institute of Informatics Systems SB RASNovosibirsk State UniversityNovosibirskRussia

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