Abstract
To study physical the realizability of “computational” procedures, the notion of “inaccessibility” is introduced. As specific examples, the halting set of a universal Turing machine, the Mandelbrot set, and a riddled basin, all of which are defined by decision procedures, are studied. Decision procedures of a halting set of a universal Turing machine and the Mandelbrot set are shown to be inaccessible, that is, the precision of the decision in these procedures cannot be increased asymptotically as the error is decreased to 0. On the other hand, the decision procedure of a riddled basin is shown to have different characteristics regarding (in) accessibility, from the other two instances. The physical realizability of computation models is discussed in terms of the inaccessibility.
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References
A. M. Turing, “On computable numbers with an application to the Entscheidungsproblem,” Proc. London Math. Soc. 42 (1936), 230.
J. E. Hopcroft and J. D. Ullman, Introduction to automata theory, languages and computation (Addison-Wesley, Reading, Mass., 1979).
M. Davis, Computability & unsolvability (Dover, New York, 1982).
H. T. Siegelmann and S. Fishman, “Analog computation with dynamical systems,” Physica D 120 (1998),214.
B. B. Mandelbrot, The fractal geometry of nature (W.H. Freeman, New York, 1983).
M. F. Barnsley, Fractals everywhere (Academic Press, Boston, 1988).
K. J. Falconer, The geometry of fractal sets (Cambridge University Press, Cambridge, 1985).
E. Ott, Chaos in dynamical systems (Cambridge University Press, Cambridge, 1993).
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, New York, 1983).
R. L. Devaney, A first course in chaotic dynamical systems: theory and experiment (Addison-Wesley, Reading, Mass., 1992); An introduction to chaotic dynamical systems (Addison-Wesley, Redwood City, Calif., 1989).
C. Moore, “Generalized shifts: unpredictability and undecidability in dynamical systems,” Nonlinearity 4 (1991), 199; “Unpredictability and undecidability in dynamical systems,” Phys. Rev. Lett. 64 (1990),2354.
I. Shimada, Talk at international symposium on information physics 1992, Kyushu institute of technology.
S. Wolfram, “Undecidability and intractability in theoretical physics,” Phys. Rev. Lett. 54 (1985),735.
A. Saito and K. Kaneko, “Geometry of undecidable systems,” Prog. Theor. Phys. 99 (1998),885; “Inaccessibility and Undecidability in Computation, Geometry and Dynamical Systems,” submitted to Physica D.
T. Kamae and S. Takahashi, Ergodic theory and fractals (Springer, Tokyo, 1993).
G. de Rham, “Sur quelques courbes definies par des équations fonctionnelles,” Rend. Sem. Mat. Torino 16 (1957),101.
P. Grassberger, “Generalized dimensions of strange attractors,” Phys. Lett. 97A (1983),227.
N. Chomsky, “Three models for the description of language,” IRE Trans. on Information Theory 2 (1956), 113; “On certain formal properties of grammars,” Information and Control 2 (1959), 137.
Y. Rogozhin, “Small universal Turing machines,” Theor. Comput. Sci. 168 (1996), 215.
L. Staiger, “w-Languages,” in Handbook of formal languages 3, edited by G. Rozenberg and A. Salomaa (Springer, Berlin, 1997).
W. Thomas, “Automata on infinite objects,” in Handbook of theoretical computer science B, edited by J. van Leeuwen (Elsevier, Amsterdam, 1990).
J. D. Farmer, “Sensitive dependence on parameters in nonlinear dynamics,” Phys. Rev. Lett. 55 (1985),351.
C. Grebogi et al., “Exterior dimension of fat fractals,” Phys. Lett. A 110 (1985), 1.
R. Penrose, The emperor’s new mind (Oxford University Press, Oxford, 1989).
L. Blum, et al., Complexity and real computation (Springer, New York, 1998).
M. Shishikura, “The boundary of the Mandelbrot set has Hausdorff dimension two,” Astérisque 222 (1994), 389.
E. Ott, et al., “The transition to chaotic attractors with riddled basins,” Physica D 76 (1994), 384.
J. C. Sommerer and E. Ott, “A physical system with qualitatively uncertain dynamics,” Nature 365 (1993), 138.
E.Ott et al., “Scaling behavior of chaotic systems with riddled basins,” Phys. Rev. Lett. 71 (1993),4134.
J. Milnor, “On the concept of attractor,” Commun. Math. Phys. 99 (1985), 177.
M. B. Pour-EI and J. I. Richards, Computability in analysis and physics (Springer Verlag, Berlin, 1989).
H. T. Siegelmann and E. D. Sontag, “Analog computation via neural networks,” Theor. Comput. Sci. 131 (1994),331.
C. Moore, “Recursion theory on the reals and continuous-time computation,” Theor. Comput. Sci. 162 (1996),23.
R. Penrose, Shadows of the mind (Oxford University Press, Oxford, 1994).
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Saito, A., Kaneko, K. (2001). Inaccessibility in Decision Procedures. In: Antoniou, I., Calude, C.S., Dinneen, M.J. (eds) Unconventional Models of Computation, UMC’2K. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0313-4_17
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DOI: https://doi.org/10.1007/978-1-4471-0313-4_17
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