Abstract
A model of computation with access to ℝ is suggested that is more restricted than, say fapC(ℝ) in [10]. In particular ℝ is treated as an accessory to the computer rather than an internal component. This has the desirable properties of a definability theorem and a parameterisation theorem for real algebraic numbers. The geometric properties of halting sets indicate the weakness of the theory. Some examples are taken from Fractal Geometry.
I am very grateful to Prof. J. Shepherdson for his helpful comments; and to my supervisor Dr. J. Mayberry for his support. I am also grateful to Dr. J. Rickard, H. Lamba, Dr. K. Falconer and O. Springer for many fruitful discussions. This work was funded by SERC.
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References
F. Abramson. Effective computation over the real numbers. In Proceedings of the 12th Annual (IEEE) Symposium and Automata Theory, pages 33–37, 1971.
N. P. Bamber. Initial Segments and Theories about Rings. PhD thesis, University of Bristol, 1993. To appear.
L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin (New Series) of the American Mathematical Society, 21(1):1–46, July 1989.
N. Cutland. Computability. Cambridge University Press, 1980.
K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications. John Wiley &: Sons, 1990.
M. C. Fitting. Fundamentals of Generalised Recursion Theory, volume 105 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1981.
N. Jacobson. Theory of Fields and Galois Theory, volume 3 of Lectures in Abstract Algebra, chapter 6, pages 269–318. D. van Nostrand, 1964.
G. Mills and J. B. Paris. Regularity in models of arithmetic. Journal of Symbolic Logic, 49(1):272–279, March 1984.
J. C. Shepherdson. Computation over abstract structures. In H. E. Rose and J. C. Shepherdson, editors, Logic Colloquium 73, pages 445–513. North-Holland, Amsterdam, 1975.
J. V. Tucker. Computing in algebraic systems. In F. R. Drake and S. S. Wainer, editors, Recursion Theory: its Generalisations and Applications, volume 45 of London Mathematical Society Lecture Note Series, pages 215–235, Proceedings of Logic Colloquium '79, Leeds, August 1979, 1979. Cambridge University Press.
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Bamber, N.P. (1993). Computation with access to the reals, but using only classical machines. In: Gottlob, G., Leitsch, A., Mundici, D. (eds) Computational Logic and Proof Theory. KGC 1993. Lecture Notes in Computer Science, vol 713. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022558
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DOI: https://doi.org/10.1007/BFb0022558
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