Advertisement

Outline of Partial Computability in Computable Topology

  • Margarita Korovina
  • Oleg Kudinov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

In the framework of computable topology we investigate properties of partial computable functions, in particular complexity of various problems in computable analysis in terms of index sets, the effective Borel and Lusin hierarchies.

References

  1. 1.
    Ershov, Y.L.: Theory of numberings. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 473–503. Elsevier Science B.V., Amsterdam (1999)CrossRefGoogle Scholar
  2. 2.
    Edalat, A.: Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbolic Log. 3(4), 401–452 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Gao, S.: Invariant Descriptive Set Theory. CRC Press, New York (2009)MATHGoogle Scholar
  4. 4.
    Gregoriades, V., Kispeter, T., Pauly, A.: A comparison of concepts from computable analysis and effective descriptive set theory. Math. Struct. Comput. Sci. 1–23(2016). https://doi.org/10.1017/S0960129516000128. (Published online: 23 June 2016)
  5. 5.
    Hemmerling, A.: Effective metric spaces and representations of the reals. Theor. Comput. Sci. 284(2), 347–372 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hemmerling, A.: On approximate and algebraic computability over the real numbers. Theor. Comput. Sci. 219(1–2), 185–223 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)CrossRefMATHGoogle Scholar
  8. 8.
    Korovina, M., Kudinov, O.: Complexity for partial computable functions over computable Polish spaces. Math. Struct. Comput. Sci. (2016). doi: 10.1017/S0960129516000438. (Published online: 19 December 2016)
  9. 9.
    Korovina, M., Kudinov, O.: Computable elements and functions in effectively enumerable topological spaces. Mathematical structure in Computer Science (2016). doi: 10.1017/S0960129516000141. (Published online: 23 June 2016)
  10. 10.
    Korovina, M., Kudinov, O.: Index sets as a measure of continuous constraint complexity. In: Voronkov, A., Virbitskaite, I. (eds.) PSI 2014. LNCS, vol. 8974, pp. 201–215. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46823-4_17 Google Scholar
  11. 11.
    Korovina, M., Kudinov, O.: Towards computability over effectively enumerable topological spaces. Electr. Notes Theor. Comput. Sci. 221, 115–125 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Korovina, M., Kudinov, O.: 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). doi: 10.1007/11494645_30 CrossRefGoogle Scholar
  13. 13.
    Korovina, M., Kudinov, O.: 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). doi: 10.1007/10703163_14 CrossRefGoogle Scholar
  14. 14.
    Moschovakis, Y.N.: Descriptive set theory. North-Holland, Amsterdam (2009)CrossRefMATHGoogle Scholar
  15. 15.
    Moschovakis, Y.N.: Recursive metric spaces. Fund. Math. 55, 215–238 (1964)MathSciNetMATHGoogle Scholar
  16. 16.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)MATHGoogle Scholar
  17. 17.
    Soare, R.I.: Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer Science and Business Media, Heidelberg (1987)CrossRefMATHGoogle Scholar
  18. 18.
    Spreen, D.: On effective topological spaces. J. Symb. Log. 63(1), 185–221 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Selivanov, V.: Towards the effective descriptive set theory. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 324–333. Springer, Cham (2015). doi: 10.1007/978-3-319-20028-6_33 CrossRefGoogle Scholar
  20. 20.
    Weihrauch, K.: Computable Analysis. Springer, New York (2000)CrossRefMATHGoogle Scholar
  21. 21.
    Weihrauch, K.: Computability on computable metric spaces. Theor. Comput. Sci. 113(1), 191–210 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.A.P. Ershov Institute of Informatics Systems, SbRASNovosibirskRussia
  2. 2.Sobolev Institute of Mathematics, SbRASNovosibirskRussia

Personalised recommendations