Abstract
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 http://fgc.math.ist.utl.pt/papers/hierarchy.pdf.)
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References
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)
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)
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)
Campagnolo, M., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18(4), 977–1000 (2002)
Graça, D., Costa, J.F.: Analog computers and recursive functions over the reals. Journal of Complexity 19(5), 644–664 (2003)
Lebesgue, H.: Sur les fonctions représentables analytiquement. J. de Math. 1, 139–216 (1905)
Loff, B.: A functional characterisation of the analytical hierarchy, (submitted 2007)
Loff, B., Costa, J.F., Mycka, J.: Computability on reals, infinite limits and differential equations, (accepted for publication 2006)
Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44 (1996)
Moschovakis, Y.N.: Descriptive set theory. North–Holland, Amsterdam (1980)
Mycka, J.: μ-recursion and infinite limits. Theoretical Computer Science 302, 123–133 (2003)
Mycka, J., Costa, J.F.: Real recursive functions and their hierarchy. Journal of Complexity 20(6), 835–857 (2004)
Mycka, J., Costa, J.F.: The P ≠ NP conjecture in the context of real and complex analysis. Journal of Complexity 22(2), 287–303 (2006)
Mycka, J., Costa, J.F.: Undecidability over continuous-time. Logic Journal of the IGPL 14(5), 649–658 (2006)
Mycka, J., Costa, J.F.: A new conceptual framework for analog computation. Theoretical Computer Science, Accepted for publication (2007)
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Costa, J.F., Loff, B., Mycka, J. (2007). The New Promise of Analog Computation. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_20
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DOI: https://doi.org/10.1007/978-3-540-73001-9_20
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