The New Promise of Analog Computation

  • José Félix Costa
  • Bruno Loff
  • Jerzy Mycka
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


We show that, using our more or less established framework of inductive definition of real-valued functions (work started by Cristopher Moore in [9]) together with ideas and concepts of standard computability we can prove theorems of Analysis. Then we will consider our ideas as a bridging tool between the standard Theory of Computability (and Complexity) on one side and Mathematical Analysis on the other, making real recursive functions a possible branch of Descriptive Set Theory. What follows is an Extended Abstract directed to a large audience of CiE 2007, Special Session on Logic and New Paradigms of Computability. (Proofs of statements can be found in a detailed long paper at the address


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  1. 1.
    Bournez, O., Campagnolo, M., Graça, D., Hainry, E.: The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 631–643. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Bournez, O., Hainry, E.: Real recursive functions and real extensions of recursive functions. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 116–127. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Bournez, O., Hainry, E.: Elementarily computable functions over the real numbers and ℝ-sub-recursive functions. Theoretical Computer Science 348(2–3), 130–147 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Campagnolo, M., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18(4), 977–1000 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Graça, D., Costa, J.F.: Analog computers and recursive functions over the reals. Journal of Complexity 19(5), 644–664 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lebesgue, H.: Sur les fonctions représentables analytiquement. J. de Math. 1, 139–216 (1905)Google Scholar
  7. 7.
    Loff, B.: A functional characterisation of the analytical hierarchy, (submitted 2007)Google Scholar
  8. 8.
    Loff, B., Costa, J.F., Mycka, J.: Computability on reals, infinite limits and differential equations, (accepted for publication 2006)Google Scholar
  9. 9.
    Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Moschovakis, Y.N.: Descriptive set theory. North–Holland, Amsterdam (1980)Google Scholar
  11. 11.
    Mycka, J.: μ-recursion and infinite limits. Theoretical Computer Science 302, 123–133 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mycka, J., Costa, J.F.: Real recursive functions and their hierarchy. Journal of Complexity 20(6), 835–857 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mycka, J., Costa, J.F.: The PNP conjecture in the context of real and complex analysis. Journal of Complexity 22(2), 287–303 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mycka, J., Costa, J.F.: Undecidability over continuous-time. Logic Journal of the IGPL 14(5), 649–658 (2006)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mycka, J., Costa, J.F.: A new conceptual framework for analog computation. Theoretical Computer Science, Accepted for publication (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • José Félix Costa
    • 1
    • 2
  • Bruno Loff
    • 2
  • Jerzy Mycka
    • 3
  1. 1.Department of Mathematics, Instituto Superior Técnico Universidade Técnica de Lisboa, LisboaPortugal
  2. 2.Centro de Matemática e Aplicações Fundamentais do Complexo Interdisciplinar, Universidade de Lisboa, LisbonPortugal
  3. 3.Institute of Mathematics, University of Maria Curie-Skłodowska, LublinPoland

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