Skip to main content
SpringerLink
Log in

Highlights of the Rice-Shapiro Theorem in Computable Topology

  • Conference paper
  • First Online:
Perspectives of System Informatics (PSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10742))

Abstract

Computable topological spaces naturally arise in computer science for continuous data type representations that have tools for effective reasoning about quite complex objects such as real numbers and functions, solutions of differential equations, functionals and operators. Algebraic and continuous domains, computable metric spaces, computable Polish spaces have been successfully used in the theoretical foundation of computer science. In this paper we consider generalisations of the famous Rice-Shapiro theorem in the framework of effectively enumerable topological spaces that contain the weakly-effective \(\omega \)–continuous domains and computable metric spaces as proper subclasses. We start with the classical case when the spaces admit principal computable numberings of computable elements and one can investigate arithmetical complexity of index sets. We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. It turns out that if we relax these requirements then the Rice-Shapiro theorem does not hold. Then we discuss the perspective of extensions of the Rice-Shapiro theorem to spaces that do not have computable numberings of computable elements, in particular to computable Polish spaces.

The research leading to these results has received funding from the DFG grant WERA MU 1801/5-1 and the DFG/RFBR grant CAVER BE 1267/14-1 and 14-01-91334 and the grant council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Edalat, A.: Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbolic Logic 3(4), 401–452 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berger, U.: Total sets and objects in domain theory. Ann. Pure Appl. Logic. 60(2), 91–117 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brattka, V.: Computable versions of Baire’s category theorem. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 224–235. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44683-4_20

    Chapter  Google Scholar 

  4. Brodhead, P., Cenzer, D.A.: Effectively closed sets and enumerations. Arch. Math. Log. 46(7–8), 565–582 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calvert, W., Fokina, E., Goncharov, S.S., Knight, J.F., Kudinov, O.V., Morozov, A.S., Puzarenko, V.: Index sets for classes of high rank structures. J. Symb. Log. 72(4), 1418–1432 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calvert, W., Harizanov, V.S., Knight, J.F., Miller, S.: Index sets of computable structures. J. Algebra Logic 45(5), 306–325 (2006)

    Article  MATH  Google Scholar 

  7. Ceitin, G.S.: Mean value theorems in constructive analysis. Transl. Am. Math. Soc. Transl. Ser. 2(98), 11–40 (1971)

    Google Scholar 

  8. Cenzer, D.A., Remmel, J.B.: Index sets for \(\Pi ^0_1\) classes. Ann. Pure Appl. Logic 93(1–3), 3–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cenzer, D.A., Remmel, J.B.: Index sets in computable analysis. Theor. Comput. Sci. 219(1–2), 111–150 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ershov, Y.L.: Model \(\mathbb{C}\) of partial continuous functionals. In: Logic Colloquium 76, pp. 455–467. North-Holland, Amsterdam (1977)

    Google Scholar 

  11. Ershov, Y.L.: Theory of numberings. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 473–503. Elsevier Science B.V., Amsterdam (1999)

    Google Scholar 

  12. Friedberg, R.M.: 4-quantifier completeness: a Banach-Mazur functional not uniformly partial recursive. Bulletin de l’Academie Polonaise des sciences, Serie des sci. math., astr. et phys. 6(1), 1–5 (1958)

    Google Scholar 

  13. Hertling, P.: A Banach-Mazur computable but not Markov computable function on the computable real numbers. Ann. Pure Appl. Logic 132(2–3), 227–246 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gierz, G., Heinrich Hofmann, K., Keime, K., Lawson, J.D., Mislove, M.W.: Continuous lattices and domain. In: Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  15. Grubba, T., Weihrauch, K.: On computable metrization. Electr. Notes Theor. Comput. Sci. 167, 345–364 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grubba, T., Weihrauch, K.: Elementary computable topology. J. UCS 15(6), 1381–1422 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Gregoriades, V., Kispeter, T., Pauly, A.: A comparison of concepts from computable analysis and effective descriptive set theory. Math. Struct. Comput. Sci. (2014). http://arxiv.org/abs/1403.7997

  18. Korovina, M., Kudinov, O.: Rice-Shapiro Theorem in Computable Topology (2017). http://arxiv.org/abs/1708.09820

  19. Korovina, M., Kudinov, O.: Computable elements and functions in effectively enumerable topological spaces. J. Math. Struct. Comput. Sci. (2016). https://doi.org/10.1017/S0960129516000141

  20. 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). https://doi.org/10.1007/978-3-662-46823-4_17

    Google Scholar 

  21. Korovina, M., Kudinov, O.: Rice’s theorem in effectively enumerable topological spaces. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 226–235. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20028-6_23

    Chapter  Google Scholar 

  22. Korovina, M., Kudinov, O.: Positive predicate structures for continuous data. J. Math. Struct. Comput. Sci. 25(8), 1669–1684 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Korovina, M., Kudinov, O.: Towards computability over effectively enumerable topological spaces. Electr. Notes Theor. Comput. Sci. 221, 115–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martin-Löf, P.: Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm (1970)

    MATH  Google Scholar 

  25. Mazur, S.: Computable Analysis, vol. 33. Razprawy Matematyczne, Warsaw (1963)

    Google Scholar 

  26. Moschovakis, Y.N.: Recursive metric spaces. Fund. Math. 55, 215–238 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  28. Soare, R.I.: Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer, Heidelberg (1987)

    Book  MATH  Google Scholar 

  29. Shoenfield, J.R.: Degrees of Unsolvability. North-Holland Publishing Company, Amsterdam (1971)

    MATH  Google Scholar 

  30. Spreen, D.: On effective topological spaces. J. Symb. Log. 63(1), 185–221 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Spreen, D.: Effective inseparability in a topological setting. Ann. Pure Appl. Logic 80(3), 257–275 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Spreen, D.: On some decision problems in programming. Inf. Comput. 122(1), 120–139 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Spreen, D.: On r.e. inseparability of CPO index sets. In: Börger, E., Hasenjaeger, G., Rödding, D. (eds.) LaM 1983. LNCS, vol. 171, pp. 103–117. Springer, Heidelberg (1984). https://doi.org/10.1007/3-540-13331-3_36

    Chapter  Google Scholar 

  34. Vjugin, V.V.: On some examples of upper semilattices of computable numberings. Algebra Logic 13(5), 512–529 (1973)

    Google Scholar 

  35. Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)

    Book  MATH  Google Scholar 

  36. Weihrauch, K., Deil, T.: Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik 63. RWTH Aachen (1980)

    Google Scholar 

  37. Weihrauch, K.: Computability on computable metric spaces. Theor. Comput. Sci. 113(1), 191–210 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems, SbRAS, Novosibirsk State University, Novosibirsk, Russia

    Margarita Korovina

  2. Sobolev Institute of Mathematics, SbRAS, Novosibirsk State University, Novosibirsk, Russia

    Oleg Kudinov

Authors
  1. Margarita Korovina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg Kudinov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Margarita Korovina .

Editor information

Editors and Affiliations

  1. Ivannikov Institute for System Programming of RAS, Moscow, Russia

    Alexander K. Petrenko

  2. The University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Korovina, M., Kudinov, O. (2018). Highlights of the Rice-Shapiro Theorem in Computable Topology. In: Petrenko, A., Voronkov, A. (eds) Perspectives of System Informatics. PSI 2017. Lecture Notes in Computer Science(), vol 10742. Springer, Cham. https://doi.org/10.1007/978-3-319-74313-4_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-74313-4_18

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74312-7

  • Online ISBN: 978-3-319-74313-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation