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Formalisation of Computability of Operators and Real-Valued Functionals via Domain Theory

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Computability and Complexity in Analysis (CCA 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2064))

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Abstract

Based on an effective theory of continuous domains, notions of computability for operators and real-valued functionals de fined on the class of continuous functions are introduced. Definability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a suitable tool for formalisation of complex systems, such as hybrid systems. In this framework the trajectories of continuous parts of hybrid systems can be represented by computable functionals.

This research was supported in part by the RFBR (grants N 99-01-00485, N 00-01- 00810) and by the Siberian Division of RAS (a grant for young researchers, 2000).

Acknowledgement

We are grateful to Yu. L. Ershov and K. Weihrauch for stimulating discussions. Many thanks to referees for helpful comments and suggestions for revision of the earlier version of this work.

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References

  1. S. Abramsky, A. Jung, Domain theory, Handbook of Logic in Computer Science, v. 3, Clarendon Press, 1994.

    Google Scholar 

  2. J. Blanck, Domain representability of metric space, Annals of Pure and Applied Logic, 83, 1997, pages 225–247.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Barwise, Admissible sets and structures, Berlin, Springer-Verlag, 1975.

    MATH  Google Scholar 

  4. A. Brown, C. Pearcy, Introduction to Analysis, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  5. L. Blum and M. Shub and S. Smale, On a theory of computation and complexity over the reals: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc., (N.S.), v. 21, no. 1, 1989, pages 1–46.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Edalat, Domain Theory and integration, Theoretical Computer Science, 151, 1995, pages 163–193.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Edalat, P. Sünderhauf, A domain-theoretic approach to computability on the real line, Theoretical Computer Science, 210, 1998, pages 73–98.

    Article  Google Scholar 

  8. Yu. L. Ershov, Computable functionals of nite types, Algebra and Logic, 11(4), 1996 pages 367–437.

    Google Scholar 

  9. Yu. L. Ershov, Definability and computability, Plenum, New York, 1996.

    Google Scholar 

  10. M. H. Escardó, PCF extended with real numbers: a domain-theoretic approach to hair-order exact real number computation, PhD thesis, Imperial College, University of London, London, 1997.

    Google Scholar 

  11. T. Erker, M. H. Escardó, K. Keimel, The way-below relation of function spaces over semantic domain, Topology and its Applications, 89(1-2), pages 61–74, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. H. Escardó, Function-space compactifications of function spaces, To appear in Topology and its Applications, 2000.

    Google Scholar 

  13. H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci., v. 20, 1982, pages 323–352.

    Article  MathSciNet  Google Scholar 

  14. G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, A Compendium Of Continuous Lattices, Springer Verlag, Berlin, 1980.

    MATH  Google Scholar 

  15. A. Grzegorczyk, On the denitions of computable real continuous functions, Fund. Math., N 44, 1957, pages 61–71.

    Google Scholar 

  16. T.A. Henzinger, Z. Manna, A. Pnueli, Towards refining Temporal Specifications into Hybrid Systems, LNCS N 736, 1993, pages 36–60.

    Google Scholar 

  17. T.A. Henzinger, V. Rusu, Reachability Verification for Hybrid Automata, LNCS N 1386, 1998, pages 190–205.

    Google Scholar 

  18. A. Jung, Cartesian Closed Categories of Domains, CWI Tract. Centrum voor Wiskunde en Informatica, Amsterdam v. 66, 1989.

    Google Scholar 

  19. M. Korovina, Generalized computability of real functions, Siberian Advance of Mathematics, v. 2, N 4, 1992, pages 1–18.

    MathSciNet  Google Scholar 

  20. M. Korovina, O. Kudinov, A New Approach to Computability over the Reals, SibAM, v. 8, N 3, 1998, pages 59–73.

    MATH  MathSciNet  Google Scholar 

  21. M. Korovina, O. Kudinov, Characteristic Properties of Majorant-Computability over the Reals, Proc. of CSL’98, LNCS, 1584, 1999, pages 188–204.

    Google Scholar 

  22. M. Korovina, O. Kudinov, Computability via Approximations, Bulletin of Symbolic Logic, v. 5, N 1, 1999

    Google Scholar 

  23. M. Korovina, O. Kudinov, A Logical approach to Specifications of Hybrid Systems, Proc. of PSI’99, to appear in LNCS, 2000, pages 10–16.

    Google Scholar 

  24. M. Korovina, O. Kudinov, Computability over the reals without equality, Proceedings of Mal’sev conference on Mathematical Logic, Novosibirsk, p 47, 1999.

    Google Scholar 

  25. Z. Manna, A. Pnueli, Verifying Hybrid Systems, LNCS N 736, 1993, pages 4–36.

    Google Scholar 

  26. R. Montague, Recursion theory as a branch of model theory, Proc. of the third international congr. on Logic, Methodology and the Philos. of Sc., 1967, Amsterdam, 1968, pages 63–86.

    Google Scholar 

  27. Y. N. Moschovakis, Abstract first order computability, Trans. Amer. Math. Soc., v. 138, 1969, pages 427–464.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Nerode, W. Kohn, Models for Hybrid Systems, Automata, Topologies, Controllability, Observability, LNCS N 736, 1993, pages 317–357.

    Google Scholar 

  29. Pietro Di Gianantonio, Real number computation and domain theory, Information and Computation, N 127, 1996, pages 11–25.

    Google Scholar 

  30. M. B. Pour-El, J. I. Richards, Computability in Analysis and Physics, Springer-Verlag, 1988.

    Google Scholar 

  31. D. Scott, Outline of a mathematical theory of computation, In 4th Annual Princeton Conference on Information Sciences and Systems, 1970, pages 169–176.

    Google Scholar 

  32. D. Scott, Continuous lattices, Lecture Notes in Mathematics, 274, Toposes, Algebraic geometry and Logic, Springer-Verlag, 1972, pages 97–136.

    Google Scholar 

  33. E. Schechter, Handbook of Analysis and Its Foundations, Academic Pressbook, 1996.

    Google Scholar 

  34. V. Stoltenberg-Hansen and J. V. Tucker, Complete local rings as domains, Journal of Symbolic Logic, 53, 1988, pages 603–624.

    Google Scholar 

  35. V. Stoltenberg-Hansen and J. V. Tucker, Effective algebras, Handbook of Logic in computer Science, v. 4, Clarendon Press, 1995, pages 375–526.

    MathSciNet  Google Scholar 

  36. H. Tong, Some characterizations of normal and perfectly normal space, Duke Math. J. N 19, 1952, pages 289–292.

    Google Scholar 

  37. B.A. Trakhtenbrot, Yu. Barzdin, Finite automata: Behaviour and Syntheses, North-Holland, 1973.

    Google Scholar 

  38. K. Weihrauch, Computability, volume 9 of EATCS Monographs on Theoretical Computer Science, Springer, Berlin, 1987.

    Google Scholar 

  39. K. Weihrauch, A simple introduction to computable analysis, Informatik Berichte 171, FernUniversitat, Hagen, 1995, 2-nd edition.

    Google Scholar 

  40. C. Kreitz, K. Weihrauch, Complexity Theory on Real Numbers and Functions, LNCS, 145, 1983, pages 165–175.

    Google Scholar 

  41. K. Weihrauch, X. Zheng Computability on Continuous, Lower Semi-Continuous and Upper Semi-Continuous real Functions, LNCS, 1276, 1997, pages 166–186.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Informatics Systems, Lavrent’ev pr., 6, Novosibirsk, Russia

    Margarita V. Korovina

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

    Oleg V. Kudinov

Authors
  1. Margarita V. Korovina
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  2. Oleg V. Kudinov
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Editor information

Editors and Affiliations

  1. University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK

    Jens Blanck

  2. Informatikzentrum, FernUniversität Hagen, 58084, Hagen, Germany

    Vasco Brattka  & Peter Hertling  & 

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© 2001 Springer-Verlag Berlin Heidelberg

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Korovina, M.V., Kudinov, O.V. (2001). Formalisation of Computability of Operators and Real-Valued Functionals via Domain Theory. In: Blanck, J., Brattka, V., Hertling, P. (eds) Computability and Complexity in Analysis. CCA 2000. Lecture Notes in Computer Science, vol 2064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45335-0_10

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  • DOI: https://doi.org/10.1007/3-540-45335-0_10

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  • Print ISBN: 978-3-540-42197-9

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