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 Rd, 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 Jc is not computable; for all parameters c, the filled Julia set Kc 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.
