Abstract
Recently, functions over the reals that extend elementarily computable functions over the integers have been proved to correspond to the smallest class of real functions containing some basic functions and closed by composition and linear integration.
We extend this result to all computable functions: functions over the reals that extend total recursive functions over the integers are proved to correspond to the smallest class of real functions containing some basic functions and closed by composition, linear integration and a very natural unique minimization schema.
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References
Arnold, V.I.: Ordinary Differential Equations. MIT Press, Cambridge (1978)
Asarin, E., Bouajjani, A.: Perturbed Turing machines and hybrid systems. In: Logic in Computer Science, pp. 269–278 (2001)
Asarin, E., Maler, O.: Achilles and the tortoise climbing up the arithmetical hierarchy. Journal of Computer and System Sciences 57(3), 389–398 (1998)
Bournez, O.: Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoretical Computer Science 210(1), 21–71 (1999)
Bournez, O.: Complexité Algorithmique des Systémes Dynamiques Continus et Hybrides. PhD thesis, Ecole Normale Supérieure de Lyon, Janvier (1999)
Bournez, O., Hainry, E.: An analog characterization of elementary computable functions over the real numbers. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 269–280. Springer, Heidelberg (2004)
Brattka, V.: Recursive characterizations of computable real-value functions and relations. Theoretical Computer Science 162(1), 45–77 (1996)
Brattka, V.: Computability over topological structures. In: Cooper, S.B., Goncharov, S.S. (eds.) Computability and Models, pp. 93–136. Kluwer Academic Publishers, New York (2003)
Campagnolo, M., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity 18(4), 977–1000 (2002)
Campagnolo, M.L.: Computational complexity of real valued recursive functions and analog circuits. PhD thesis, Universidade Técnica de Lisboa (2001)
Etesi, G., Németi, I.: Non-Turing computations via Malament-Hogarth spacetimes. International Journal Theoretical Physics 41, 341–370 (2002)
Graça, D., Costa, J.F.: Analog computers and recursive functions over the reals. Journal of Complexity 19, 644–664 (2003)
Grzegorczyk, A.: Computable functionals. Fundamenta Mathematicae 42, 168–202 (1955)
Henzinger, T., Raskin, J.-F.: Robust undecidability of timed and hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569. Springer, Heidelberg (1999)
Hogarth, M.L.: Does general relativity allow an observer to view an eternity in a finite time? Foundations of Physics Letters 5, 173–181 (1992)
Kalmár, L.: Egyszerü példa eldönthetetlen aritmetikai problémára. Mateés fizikai lapok 50, 1–23 (1943)
Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles III. Comptes rendus de l’Académie des Sciences Paris 241, 151–153 (1955)
Lipshitz, L., Rubel, L.A.: A differentially algebraic replacement theorem, and analog computability. Proceedings of the American Mathematical Society 99(2), 367–372 (1987)
Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44 (1996)
Mycka, J.: Infinite limits and R-recursive functions. Acta Cybernetica 16, 83–91 (2003)
Mycka, J.: μ-recursion and infinite limits. Theoretical Computer Science 302, 123–133 (2003)
Odifreddi, P.: Classical recursion theory II. North-Holland, Amsterdam (1999)
Ord, T.: Hypercomputation: computing more than the Turing machine. Technical report, University of Melbourne, (September 2002), See, http://www.arxiv.org/abs/math.lo/0209332
Orponen, P.: A survey of continuous-time computation theory. In: Du, D.-Z., Ko, K.-I. (eds.) Advances in Algorithms, Languages, and Complexity, pp. 209–224. Kluwer Academic Publishers, Dordrecht (1997)
Pour-El, M.B.: Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers). Transactions of the American Mathematical Society 199, 1–28 (1974)
Ramis, E., Deschamp, C., Odoux, J.: Cours de mathématiques spéciales, tome 3, topologie et éléments d’analyse. Masson (February 1995)
Rose, H.: Subrecursion: functions and hierarchies. Clarendon Press (1984)
Shannon, C.E.: Mathematical theory of the differential analyser. Journal of Mathematics and Physics MIT 20, 337–354 (1941)
Siegelmann, H.: Neural networks and analog computation - beyond the Turing limit, Birkauser (1998)
Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 2, 230–265 (1936)
Weihrauch, K.: Computable analysis. Springer, Heidelberg (2000)
Zhou, Q.: Subclasses of computable real valued functions. In: Jiang, T., Lee, D.T. (eds.) COCOON 1997. LNCS, vol. 1276, pp. 156–165. Springer, Heidelberg (1997)
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Bournez, O., Hainry, E. (2005). Real Recursive Functions and Real Extensions of Recursive Functions. In: Margenstern, M. (eds) Machines, Computations, and Universality. MCU 2004. Lecture Notes in Computer Science, vol 3354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31834-7_9
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DOI: https://doi.org/10.1007/978-3-540-31834-7_9
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