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Digital simulation of analog computation and Church's thesis

Published online by Cambridge University Press:  12 March 2014

Lee A. Rubel
Lee A. Rubel*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
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Extract

Church's thesis, that all reasonable definitions of “computability” are equivalent, is not usually thought of in terms of computability by a continuous computer, of which the general-purpose analog computer (GPAC) is a prototype. Here we prove, under a hypothesis of determinism, that the analytic outputs of a C∞ GPAC are computable by a digital computer.

In [POE, Theorems 5, 6, 7, and 8], Pour-El obtained some related results. (The proof there of Theorem 7 depends on her Theorem 2, for which the proof in [POE] is incorrect, but for which a correct proof is given in [LIR]. Also, the proof in [POE] of Theorem 8 depends on the unproved assertion that a solution of an algebraic differential equation must be analytic on an open subset of its domain. However, this assertion was later proved in [BRR].) As in [POE], we reduce the problem to a problem about solutions of certain systems of algebraic differential equations (ADE's). If such a system is nonsingular (i.e. if the “separant” does not vanish along the given solution), then the argument is very easy (see [VSD] for an even simpler situation), so that the essential difficulties arise from singular systems. Our main tools in handling these difficulties are drawn from the excellent (and difficult) paper [DEL] by Denef and Lipshitz. The author especially wants to thank Leonard Lipshitz for his kind help in the preparation of the present paper.

We emphasize here that our proof of the simulation result applies only to the GPAC as described below. The GPAC's form a natural subclass of the class of all analog computers, and are based on certain idealized components (“black boxes”), mostly associated with the technology of past decades. One can easily envisage other kinds of black boxes of an input-output character that would lead to different kinds of analog computers. (For example, one could incorporate delays, or spatial integrators in addition to the present temporal integrators, etc.) Whether digital simulation is possible for these “extended” analog computers poses a rich and challenging set of research questions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 1011 - 1017
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[BRR]Bruckner, A. M., Rosenfeld, M., and Rubel, L. A., The Darboux property and solutions of algebraic differential equations, Pacific Journal of Mathematics, vol. 118 (1985), pp. 295–302.CrossRefGoogle Scholar
[DEL]Denef, J. and Lipshitz, L., Power series solutions of algebraic differential equations, Mathematische Annalen, vol. 267 (1984), pp. 213–238.CrossRefGoogle Scholar
[HAR]Hardy, G. H., Divergent series, Clarendon Press, Oxford, 1949.Google Scholar
[HUR]Hurwitz, A., Sur le développement des fonctions satisfaisant à une équation différentielle algébrique, Annales Scientifiques de l'École Normale Supérieure, vol. 6 (1889), pp. 327–332.CrossRefGoogle Scholar
[LIR]Lipshitz, L. and Rubel, L. A., A differentially algebraic replacement theorem, and analog computability, Proceedings of the American Mathematical Society, vol. 99 (1987), pp. 367–372.CrossRefGoogle Scholar
[MAH]Mahler, K., Lectures on transcendental numbers, Lecture Notes in Mathematics, vol. 546, Springer-Verlag, Berlin, 1976.CrossRefGoogle Scholar
[MAL]Malgrange, B., Sur les points singuliers des équations différentielles, L'Enseignement Mathématique, vol. 20 (1974), pp. 147–176.Google Scholar
[POE]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, vol. 199 (1974), pp. 1–29.CrossRefGoogle Scholar
[RUB]Rubel, L. A., The brain as an analog computer, Journal of Theoretical Neurobiology, vol. 4 (1985), pp. 73–81.Google Scholar
[SHA]Shannon, C. E., Mathematical theory of the differential analyzer, Journal of Mathematics and Physics of the Massachusetts Institute of Technology, vol. 20 (1941), pp. 337–354.CrossRefGoogle Scholar
[VSD]Vergis, A., Steiglitz, K., and Dickinson, B., The complexity of analog computation, Mathematics and Computers in Simulation, vol. 28 (1986), pp. 91–113.CrossRefGoogle Scholar