Computable Model Theory over the Reals

  • Andrey S. MorozovEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)


This paper is a survey of results together with a list of open questions on \(\Sigma \)–definability of structures over \(\mathbb {HF}(\mathbb {R})\), the hereditarily finite superstructure over the ordered field of the real numbers.


  1. 1.
    Morozov, A.S., Korovina, M.V.: On \(\Sigma \)-definability of countable structures over real numbers, complex numbers, and quaternions. Algebra Logic 47(3), 193–209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barwise, J.: Admissible Sets and Structures. Springer, Heidelberg (1975)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ershov, Y.L.: Definability and Computability. Plenum Publ. Co., New York (1996)zbMATHGoogle Scholar
  4. 4.
    Ershov, Y.L.: \(\Sigma \)-definability of algebraic structures. In: Studies in Logic and Foundations of Mathematics, vol. 1, pp. 235–260. Elsevier, Amsterdam (1998)Google Scholar
  5. 5.
    Korovina, M.V.: Generalized computability of functions on the reals. Vychislitel’nye Systemi (Computing Systems) 133, 38–68 (1990). In RussianMathSciNetGoogle Scholar
  6. 6.
    Morozov, A.S.: On some presentations of the real number field. Algebra Logic 51(1), 66–88 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Morozov, A.S.: On \(\Sigma \)-rigid presentations of the real order. Siberian Math. J. 55(3), 562–572 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Morozov, A.S.: One-dimensional \(\Sigma \)-presentations of structures over \(\mathbb{HF}{\mathbb{R}}\). In: Geschke, P.S.S., Löwe, B. (eds.) Infinity, Computability, and Metamathematics, Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch. Tributes Series, vol. 23, pp. 285–298. College Publications, London (2014)Google Scholar
  9. 9.
    Morozov, A.S.: \(\Sigma \)-presentations of the ordering on the reals. Algebra Logic 53(3), 217–237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Morozov, A.S.: A sufficient condition for nonpresentability of structures in hereditarily finite superstructures. Algebra Logic 55, 242–251 (2016)CrossRefGoogle Scholar
  11. 11.
    Morozov, A.S.: Nonpresentability of some structures of analysis in hereditarily finite superstructures. Algebra Logic (2017, to appear)Google Scholar
  12. 12.
    Korovina, M.V., Kudinov, O.V.: Positive predicate structures for continuous data. Math. Struct. Comput. Sci. 25(Special Issue 08), 1669–1684 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ershov, Y.L., Goncharov, S.S.: Constructive Models. Siberian School of Algebra and Logic. Kluwer Academic/Plenum, New York (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. The Rand Corporation, Santa Monica (1957)zbMATHGoogle Scholar
  15. 15.
    Weihrauch, K.: Computable Analysis. An Introduction, p. 285. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ershov, Y.L., Puzarenko, V.G., Stukachev, A.I.: \(\mathbb{HF}\)-Computability. In: Cooper, S.B., Andrea, S. (eds.) Computability in Context. Computation and Logic in the Real World, pp. 169–242. Imperial College Press, London (2011)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics SB RASNovosibirskRussia
  2. 2.The Novosibirsk State UniversityNovosibirskRussia

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