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Journals ACM SIGACT News Volume 32 Issue 2

Review o f Complexity and Real Computation 4 Authors : Blum, Cucker, Shub, and Smale Published by Springer-Verlag $34.95 on amazon .com, but see bestbuys .com Hardcover, 432 pages ISBN 038798281 7 Reviewed by Timothy H . McNicholl 5 In [3], Penrose posed the question "Is the Mandelbrot set decidable?" That is, is there an algorithm that given a point as input will determine if that point is in the Mandelbrot set . Par t of the problem with this question is figuring out its proper interpretation . The traditional theory of computability deals only with computations over the natural numbers . Computations over the integers and rationals can be modeled as computations over the natural numbers . But since the reals are uncountable, Penrose's question is not amenable to this approach . One could answe r the question only for points with rational coordinates . However, this approach does not work fo r related questions. For example, the curve x 3 + y 3 = 1 has no rational points . Yet, the question of its decidability can not be dismissed as meaningless or trivial. A traditional approach would be t o use the field of computable

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Review of Complexity and real computation by Blum, Cucker, Shub, and Smale. Springer-Verlag.

Reviewer-McNicholl, Timothy H.
ACM SIGACT News , Volume 32 (2)
Association for Computing Machinery – Jun 1, 2001

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