Skip to main content
SpringerLink
Log in

Recent Advances in Σ-Definability over Continuous Data Types

  • Conference paper
Perspectives of System Informatics (PSI 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2890))

Abstract

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.

This research was partially supported by the Danish Natural Science Research Council, Grant no. 21-02-0474, RFFI-DFG Grant no. 01-01-04003 and Grant Scientific School-2112.2003.1

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barwise, J.: Admissible sets and structure. Springer, Berlin (1975)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  3. Blass, A., Gurevich, Y., Shelah, S.: Choiceless polynomial time. APAL, 141–187 (1999)

    Google Scholar 

  4. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Berlin (1996)

    Google Scholar 

  5. Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1999)

    Google Scholar 

  6. Brattka, V., Hertling, P.: Topological properties of real number representations. TCS 284(2), 1–17 (2002)

    Article  MathSciNet  Google Scholar 

  7. Dahlhaus, E., Makowsky, J.A.: Query languages for hierarchic databases. Information and Computation, 1–32 (1992)

    Google Scholar 

  8. Davar, A., Gurevich, Y.: Fixed-point logics. BSL, 65–88 (2002)

    Google Scholar 

  9. Ebbinghaus, H., Flum, J.: Finite Model Theory. Springer, Berlin (1999)

    MATH  Google Scholar 

  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. 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. Engeler, E.: Formal Languages: Automata and Structures. Markham Publishing Co (1968)

    Google Scholar 

  13. Ershov, Y.L.: Definability and computability. Plenum, New-York (1996)

    Google Scholar 

  14. Freedman, H., Ko, K.: Computational complexity of real functions. TCS, 323–352 (1992)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  16. Gandy, R.: Inductive definitions. In: Fenstad, J.E., Hinman, P.D. (eds.) Generalized Recursion Theory, pp. 265–300. North-Holland, Amsterdam (1974)

    Chapter  Google Scholar 

  17. Grzegorczyk, A.: On the definitions of computable real continuous function. Fundamenta Mathematik, 61–71 (1957)

    Google Scholar 

  18. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. The MIT press, Cambridge (2002)

    Google Scholar 

  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. Immerman, N.: Descriptive Complexity. Springer, New-York (1999)

    MATH  Google Scholar 

  21. Kohlenbach, U.: Proof theory and computational analysis. Electronic Notes in Theoretical Computer Science (1998)

    Google Scholar 

  22. Korovina, M.: Computability and Σ-definability over the reals. Technical report, BRICS (2003), http://www.brics.dk/~korovina/compreals.ps

  23. Korovina, M.: Computational aspects of Σ-definability over the real numbers without the equality test (to appear), http://www.brics.dk/~korovina/definability.ps

  24. Korovina, M.: Fixed points on the abstract structures without the equality test. Technical report, BRICS (2002), http://www.brics.dk/RS/02/26/index.html

  25. Korovina, M.: Fixed points on the reals numbers without the equality test. Electronic Notes in TCS 66(1) (2002)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  29. Meer, K.: Counting problems over the reals. TCS 242(1-2), 41–58 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Moschovakis, Y.N.: Abstract first order computability. i, ii. Transactions of the American Mathematical Society, 427–446 (1969)

    Google Scholar 

  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. Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1988)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  34. Takeuti, G., Kino, A.: On predicates with constructive infinitary long expressions. J. Math. Soc. Japan, 176–190 (1963)

    Google Scholar 

  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. 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. Weihrauch, K.: Computable analysis. Springer, Berlin (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. BRICS , Department of Computer Science, University of Aarhus, Ny Munkegade, DK-8000, Aarhus C, Denmark

    Margarita Korovina

  2. A. P. Ershov Institute of Informatics Systems, Lavrent’ev ave., 6, 630090, Novosibirsk, Russia

    Margarita Korovina

Authors
  1. Margarita Korovina
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Technische Universität München, Germany

    Manfred Broy

  2. A.P. Ershov Institute of Informatics Systems, Siberian Branch of Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Alexandre V. Zamulin

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Korovina, M. (2004). Recent Advances in Σ-Definability over Continuous Data Types. In: Broy, M., Zamulin, A.V. (eds) Perspectives of System Informatics. PSI 2003. Lecture Notes in Computer Science, vol 2890. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39866-0_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-39866-0_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20813-6

  • Online ISBN: 978-3-540-39866-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation