Mark Braverman
Abstract
How to test the usefulness of computation for understanding and predicting continuous phenomena.
- Blum, L., Cucker, F., Shub, M., and Smale, S. Complexity and Real Computation. Springer-Verlag, New York, 1998. Google Scholar
Digital Library
- Braverman, M. and Yampolsky, M. Computability of Julia Sets. Springer Verlag, Berlin Heidelberg, 2009. Google ScholarDigital Library
- Moore, C. unpredictability and undecidability in dynamical systems. Physical Review Letters 64, 20 (May 1990), 2354--2357.Google Scholar
- Turing, A.M. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42, 2 (Nov. 12, 1936), 230--265.Google Scholar
- Vardi, M. Solving the unsolvable. Commun. ACM 54, 7 (July 2011), 5. Google ScholarDigital Library
- Weihrauch, K. Computable Analysis. Springer-Verlag, Berlin, 2000. Google ScholarDigital Library
- Wolfram, S. A New Kind of Science. Wolfram Media, Champaign, IL, 2002. Google ScholarDigital Library
Index Terms
- Computing with real numbers, from Archimedes to Turing and beyond
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Reviews
Grigore Albeanu
Under a concise and a very attractive title, the author invites the reader to a beautiful journey over the computability field. Are natural processes computationally predictable__?__ Based on the computable analysis model, this paper describes important aspects to provide the answer. The author discusses the notions of computable real numbers, computable real functions, and computable sets of R d , and notes that graphs of computable functions are computable sets and "nice" sets are also computable. The author introduces dynamical systems, starting with a linear dynamic system corresponding to a harmonic oscillator and a computational simulation of a "computer dynamical system." The rest of the paper is dedicated to the computability of Julia sets. Based on previously published results, the paper highlights the existence of the computable parameters c , such that the Julia set J c is not computable; for all parameters c , the filled Julia set K c is computable. The paper concludes with lessons learned and bibliographic notes. The main highlights of the paper include the identification of natural systems that would be computationally impossible to predict (if implemented with no noise), but are statistically predictable in the presence of noise, and the adequacy of computable analysis for the study of the computational hardness of natural problems over real numbers. Readers interested in computability with real numbers and researchers working on natural systems as computational devices will find valuable ideas for future developments. Online Computing Reviews Service
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