The Uniformity Principle for Σ-Definability with Applications to Computable Analysis

  • Margarita Korovina
  • Oleg Kudinov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

Abstract

In this paper we prove the Uniformity Principle for Σ–definability over the real numbers extended by open predicates. Using this principle we show that if we have a ΣK-formula, i.e. a formula with quantifier alternations where universal quantifiers are bounded by computable compact sets, then we can eliminate all universal quantifiers obtaining a Σ-formula equivalent to the initial one. We also illustrate how the Uniformity Principle can be employed for reasoning about computability over continuous data in an elegant way.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brattka, V., Weihrauch, K.: Computability on subsets of euclidean space I: Closed and compact sets. TCS 219, 65–93 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barwise, J.: Admissible sets and Structures. Springer, Berlin (1975)MATHGoogle Scholar
  3. 3.
    Ershov, Y.L.: Definability and computability. Plenum, New-York (1996)Google Scholar
  4. 4.
    Gherardi, G.: Some results in computable analysis and effective Borel measurability. PhD thesis, Siena (2006)Google Scholar
  5. 5.
    Korovina, M.V.: Computational aspects of sigma-definability over the real numbers without the equality test. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 330–344. Springer, Heidelberg (2003)Google Scholar
  6. 6.
    Korovina, M.V.: Gandy’s theorem for abstract structures without the equality test. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 290–301. Springer, Heidelberg (2003)Google Scholar
  7. 7.
    Korovina, M.V., Kudinov, O.V.: Characteristic properties of majorant-computability over the reals. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 188–203. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Korovina, M.V., Kudinov, O.V.: Semantic characterisations of second-order computability over the real numbers. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 160–172. Springer, Heidelberg (2001)Google Scholar
  9. 9.
    Korovina, M.V., Kudinov, O.V.: Towards computability of higher type continuous data. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 235–241. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Margarita Korovina
    • 1
  • Oleg Kudinov
    • 2
  1. 1.Institute of Informatics Systems, pr. Lavrenteva 6, 630090, NovosibirskRussia
  2. 2.Sobolev Institute of Mathematics, pr. Koptuga 4, 630090, NovosibirskRussia

Personalised recommendations