Recent Advances in Σ-Definability over Continuous Data Types

  • Margarita Korovina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2890)


The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability on continuous data types. In the case of the real numbers we illustrate how computability can be expressed in the language of Σ-formulas.


Turing Machine Expressive Power Equality Test Atomic Formula Predicate Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Barwise, J.: Admissible sets and structure. Springer, Berlin (1975)Google Scholar
  2. 2.
    Blass, A., Gurevich, Y.: Background, reserve and gandy machines. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 1–17. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Blass, A., Gurevich, Y., Shelah, S.: Choiceless polynomial time. APAL, 141–187 (1999)Google Scholar
  4. 4.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Berlin (1996)Google Scholar
  5. 5.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1999)Google Scholar
  6. 6.
    Brattka, V., Hertling, P.: Topological properties of real number representations. TCS 284(2), 1–17 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dahlhaus, E., Makowsky, J.A.: Query languages for hierarchic databases. Information and Computation, 1–32 (1992)Google Scholar
  8. 8.
    Davar, A., Gurevich, Y.: Fixed-point logics. BSL, 65–88 (2002)Google Scholar
  9. 9.
    Ebbinghaus, H., Flum, J.: Finite Model Theory. Springer, Berlin (1999)zbMATHGoogle Scholar
  10. 10.
    Edalat, A., Escardo, M.: Integration in real pcf. In: Proc. IEEE Conference on Logic in Computer Science (LICS), pp. 382–393 (1996)Google Scholar
  11. 11.
    Edalat, A., Lieutie, A.: Domain theory and differential calculus (function of one variable. In: Proc. IEEE Conference on Logic in Computer Science (LICS), pp. 277–298 (2002)Google Scholar
  12. 12.
    Engeler, E.: Formal Languages: Automata and Structures. Markham Publishing Co (1968)Google Scholar
  13. 13.
    Ershov, Y.L.: Definability and computability. Plenum, New-York (1996)Google Scholar
  14. 14.
    Freedman, H., Ko, K.: Computational complexity of real functions. TCS, 323–352 (1992)Google Scholar
  15. 15.
    Friedman, H.: Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. In: Yates, C.M.E., Gandy, R.O. (eds.) Logic colloquium 1969, pp. 361–390. C.M.E, Hollang, Amsterdam (1971)CrossRefGoogle Scholar
  16. 16.
    Gandy, R.: Inductive definitions. In: Fenstad, J.E., Hinman, P.D. (eds.) Generalized Recursion Theory, pp. 265–300. North-Holland, Amsterdam (1974)CrossRefGoogle Scholar
  17. 17.
    Grzegorczyk, A.: On the definitions of computable real continuous function. Fundamenta Mathematik, 61–71 (1957)Google Scholar
  18. 18.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT press, Cambridge (2002)Google Scholar
  19. 19.
    Hinman, P.G.: Recursion on abstract structure. In: Griffor, E.R. (ed.) Handbook of Computability Theory, pp. 317–359. Elsevie, Amsterdam (1999)Google Scholar
  20. 20.
    Immerman, N.: Descriptive Complexity. Springer, New-York (1999)zbMATHGoogle Scholar
  21. 21.
    Kohlenbach, U.: Proof theory and computational analysis. Electronic Notes in Theoretical Computer Science (1998)Google Scholar
  22. 22.
    Korovina, M.: Computability and Σ-definability over the reals. Technical report, BRICS (2003),
  23. 23.
    Korovina, M.: Computational aspects of Σ-definability over the real numbers without the equality test (to appear),
  24. 24.
    Korovina, M.: Fixed points on the abstract structures without the equality test. Technical report, BRICS (2002),
  25. 25.
    Korovina, M.: Fixed points on the reals numbers without the equality test. Electronic Notes in TCS 66(1) (2002)Google Scholar
  26. 26.
    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. 188–204. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  27. 27.
    Korovina, M., Kudinov, O.: Some properties of majorant-computability. In: Arslanov, M., Lempp, S. (eds.) Recursion Theory and Complexity, Proceedings of the Kazan 1997 Workshop, Berlin - New York, July 14-19. de Gruyter Series in Logic and its Applications, pp. 97–115 (1999)Google Scholar
  28. 28.
    Locombe, D.: Recursion theoretical structure for relational systems. In: Gandy, R.O., Yates, C.M.E. (eds.) Proc. of Logic Colloquium 1969, pp. 3–17. North-Holland, Amsterdam (1971)CrossRefGoogle Scholar
  29. 29.
    Meer, K.: Counting problems over the reals. TCS 242(1-2), 41–58 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Moschovakis, Y.N.: Abstract first order computability. i, ii. Transactions of the American Mathematical Society, 427–446 (1969)Google Scholar
  31. 31.
    Moschovakis, Y.N.: Elementary Induction on Abstract Structur. Studies in Logic and the Foundations of Mathematics, vol. 77. North-Holland Publishing Co., American Elsevier Publishing Co, Amsterdam-London (1974)Google Scholar
  32. 32.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1988)Google Scholar
  33. 33.
    Sazonov, V.: Using agents for concurrent querying of web-like databases via hyperset- theoretic approach. In: Bjørner, D., Broy, M., Zamulin, A.V. (eds.) PSI 2001. LNCS, vol. 2244, pp. 378–394. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  34. 34.
    Takeuti, G., Kino, A.: On predicates with constructive infinitary long expressions. J. Math. Soc. Japan, 176–190 (1963)Google Scholar
  35. 35.
    Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many-sorted algebras. In: Maibaum, T.S.E., Abramsky, S., Gabbay, D.M. (eds.) Handbook of Logic in Computer Science, pp. 397–525. Oxford University Press, Oxford (2000)Google Scholar
  36. 36.
    Vardi, M.: The complexity of relational query languages. In: Proc. of the 14th ACM Symposium on the Theory of Computing, pp. 37–146 (1982)Google Scholar
  37. 37.
    Weihrauch, K.: Computable analysis. Springer, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Margarita Korovina
    • 1
    • 2
  1. 1.BRICS , Department of Computer ScienceUniversity of Aarhus, Ny MunkegadeAarhus CDenmark
  2. 2.A. P. Ershov Institute of Informatics SystemsNovosibirskRussia

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