Problem Theory (original) (raw)

Abstract

The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here is why, Why are we Turing complete? Being Turing complete also means that somehow our brain implements the function that a universal Turing machine implements. The point is that evolution achieved Turing completeness, and then the explanation should be evolutionary, but our explanation is mathematical. The trick is to introduce a mathematical theory of problems, under the basic assumption that solving more problems provides more survival opportunities. So we build a problem theory by fusing set and computing theories. Then we construct a series of resolvers, where each resolver is defined by its computing capacity, that exhibits the following property: all problems solved by a resolver are also solved by the next resolver in the series if certain condition is satisfied. The last of the conditions is to be Turing complete. This series defines a resolvers hierarchy that could be seen as a framework for the evolution of cognition. Then the answer to our question would be: to solve most problems. By the way, the problem theory defines adaptation, perception, and learning, and it shows that there are just three ways to resolve any problem: routine, trial, and analogy. And, most importantly, this theory demonstrates how problems can be used to found mathematics and computing on biology.

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References (12)

  1. Cantor (1895): Georg Cantor, "Contributions to the Founding of the Theory of Transfinite Numbers (First Article)"; in Mathematische Annalen, vol. xlvi, pp. 481-512, 1895. Translated by P.E.B. Jourdain in Contributions to the Founding of the Theory of Transfinite Numbers, Dover, New York, 1955. isbn: 978-0-486-60045-1.
  2. Casares (2017a): Ramón Casares, "Proof of Church's Thesis"; doi: 10.6084/m9.figshare.4955501, arXiv:1209.5036.
  3. Casares (2017b): Ramón Casares, "Syntax Evolution: Problems and Recursion"; doi: 10.6084/m9.figshare.4956359, arXiv:1508.03040.
  4. Curry & Feys (1958): Haskell B. Curry, and Robert Feys, with William Craig, Combi- natory Logic, Vol. I. North-Holland Publishing Company, Amsterdam, 1958. isbn: 978-0-7204-2207-8.
  5. Gandy (1980): Robin Gandy, "Church's Thesis and Principles for Mechanisms"; doi: 10.1016/s0049-237x(08)71257-6. In The Kleene Symposium (editors: J. Barwise, H.J. Keisler & K. Kunen), Volume 101 of Studies in Logic and the Foundations of Math- ematics; North-Holland, Amsterdam, 1980, pp. 123-148; isbn: 978-0-444-55719-3.
  6. Kleene (1936): Stephen C. Kleene, "λ-Definability and Recursiveness"; in Duke Mathe- matical Journal , vol. 2, pp. 340-353, 1936, doi: 10.1215/s0012-7094-36-00227-2.
  7. Lettvin et al. (1959): Jerome Y. Lettvin, Humberto R. Maturana, Warren S. McCulloch, and Walter H. Pitts, "What the Frog's Eye Tells the Frog's Brain"; in Proceedings of the IRE , vol. 47, no. 11, pp. 1940-1951, November 1959, doi: 10.1109/jrproc.1959.287207.
  8. Post (1936): Emil L. Post, "Finite Combinatory Processes -Formulation 1"; in The Journal of Symbolic Logic, vol. 1, no. 3, pp. 103-105, September 1936, doi: 10.2307/2269031. Received October 7, 1936.
  9. Post (1944): Emil L. Post, "Recursively Enumerable Sets of Positive Integers and their Decision Problems"; in Bulletin of the American Mathematical Society, vol. 50, no. 5, pp. 284-316, 1944, doi: 10.1090/s0002-9904-1944-08111-1.
  10. Turing (1936): Alan M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem"; in Proceedings of the London Mathematical Society, vol. s2- 42, issue 1, pp. 230-265, 1937, doi: 10.1112/plms/s2-42.1.230. Received 28 May, 1936. Read 12 November, 1936.
  11. Turing (1937): Alan M. Turing, "Computability and λ-Definability"; in The Journal of Symbolic Logic, vol. 2, no. 4, pp. 153-163, December 1937, doi: 10.2307/2268280. Received September 11, 1937.
  12. Turing (1938): Alan M. Turing, "Systems of Logic Based on Ordinals"; Princeton Uni- versity PhD dissertation. Submitted 17 May, 1938. Oral examination 31 May, 1938. Printed in Proceedings of the London Mathematical Society, vol. s2-45, issue 1, pp. 161-228, 1939, doi: 10.1112/plms/s2-45.1.161. Received 31 May, 1938. Read 16 June, 1938.