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Characteristic Properties of Majorant-Computability Over the Reals

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Computer Science Logic (CSL 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1584))

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Abstract

Characteristic properties of majorant-computable real-valued functions are studied. A formal theory of computability over the reals which satisfies the requirements of numerical analysis used in Computer Science is constructed on the base of the definition of majorant-computability proposed in [13]. A model-theoretical characterization of majorant-computability real-valued functions and their domains is investigated. A theorem which connects the graph of a majorant-computable function with validity of a finite formula on the set of hereditarily finite sets on \({\bar{\mathbb{R} }}\), \({\rm\bf HF}({\bar{\mathbb{R} }})\) (where \({\bar{\mathbb{R} }}\) is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the definition of majorant-computability and the notions of computability earlier proposed by Blum et al., Edalat, Sünderhauf, Pour-El and Richards, Stoltenberg-Hansen and Tucker is given. Examples of majorant-computable real-valued functions are presented.

This research was supported in part by the RFBR (grant N 96-15-96878) and by the Siberian Division of RAS (a grant for young researchers, 1997)

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References

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

    MATH  Google Scholar 

  2. Bochnak, J., Coster, M., Roy, M.-F.: Géometrie algébrique réelle. Springer, Heidelberg (1987)

    MATH  Google Scholar 

  3. Bitsadze, A.V.: The bases of analytic function theory of complex variables, Nauka, Moskow (1972)

    Google Scholar 

  4. Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the reals: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.) 21(1), 1–46 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. van den Dries, L.: Remarks on Tarski’s problem concerning (IR,+, ∗, exp). In: Proc. Logic Colloquium 1982, pp. 240–266 (1984)

    Google Scholar 

  6. Edalat, A., Sünderhauf, P.: A domain-theoretic approach to computability on the real line. Theoretical Computer Science (to appear)

    Google Scholar 

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

    Google Scholar 

  8. Freedman, H., Ko, K.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)

    Article  MathSciNet  Google Scholar 

  9. Grzegorczyk, A.: On the definitions of computable real continuous functions. Fund. Math. 44, 61–71 (1957)

    MATH  MathSciNet  Google Scholar 

  10. Jacobson, N.: Basic Algebra, vol. II. Freedman and Company, New York (1995)

    Google Scholar 

  11. Korovina, M.: Generalized computability of real functions. Siberian Advance of Mathematics 2(4), 1–18 (1992)

    MathSciNet  Google Scholar 

  12. Korovina, M.: About an universal recursive function and abstract machines on the list superstructure over the reals. Vychislitel’nye Sistemy, Novosibirsk 156, 3–20 (1996)

    MATH  MathSciNet  Google Scholar 

  13. Korovina, M., Kudinov, O.: A New Approach to Computability over the Reals. SibAM 8(3), 59–73 (1998)

    MATH  MathSciNet  Google Scholar 

  14. Montague, R.: Recursion theory as a branch of model theory. In: Proc. of the third international congr. on Logic, Methodology and the Philos. of Sc., Amsterdam, pp. 63–86 (1968)

    Google Scholar 

  15. Moschovakis, Y.N.: Abstract first order computability. Trans. Amer. Math. Soc. 138, 427–464 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  17. Robinson, A.: Nonstandard Analysis. Studies in Logic and foundation of Mathematics. North-Holland, Amsterdam (1966)

    Google Scholar 

  18. Scott, D.: Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176 (1970)

    Google Scholar 

  19. Stoltenberg-Hansen, V., Tucker, J.V.: Effective algebras. Handbook of Logic in computer Science, vol. 4, pp. 375–526. Clarendon Press, Oxford (1995)

    Google Scholar 

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Authors and Affiliations

  1. Insitute of Informatics Systems, Lavrentév pr., 6, Novosibirsk, Russia

    M. V. Korovina

  2. Institute of Mathematics, University pr., 4, Novosibirsk, Russia

    O. V. Kudinov

Authors
  1. M. V. Korovina
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  2. O. V. Kudinov
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Editors and Affiliations

  1. Computing Laboratory, Oxford University, Wolfson Building Parks Road Oxford,, OX1 3QD, United Kingdom

    Georg Gottlob

  2. GREYC, Université de Caen, CNRS, Campus 2, F-14032, Caen cedex, France

    Etienne Grandjean

  3.  ,  

    Katrin Seyr

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Korovina, M.V., Kudinov, O.V. (1999). Characteristic Properties of Majorant-Computability Over the Reals. In: Gottlob, G., Grandjean, E., Seyr, K. (eds) Computer Science Logic. CSL 1998. Lecture Notes in Computer Science, vol 1584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10703163_14

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  • DOI: https://doi.org/10.1007/10703163_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65922-8

  • Online ISBN: 978-3-540-48855-2

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