The euclid abstract machine: trisection of the angle and the halting problem

  • Authors:

  • Jerzy Mycka;Francisco Coelho;José Félix Costa

  • Affiliations:

  • Institute of Mathematics, University of Maria Curie-Sklodowska, Lublin, Poland;Department of Mathematics, Universidade de Évora, Portugal;Department of Mathematics, I.S.T., Universidade Técnica de Lisboa, and CMAF – Centro de Matemática e Aplicações Fundamentais, Lisboa, Portugal

  • Venue:
  • UC'06 Proceedings of the 5th international conference on Unconventional Computation
  • Year:
  • 2006

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Abstract

What is the meaning of hypercomputation, the meaning of computing more than the Turing machine? Concrete non-computable functions always hide the halting problem as far as we know. Even the construction of a function that grows faster than any recursive function — the Busy Beaver — a more natural function, hides the halting function, that can easily be put in relation with the Busy Beaver. Is this super-Turing computation concept related only with the halting problem and its derivatives? We built an abstract machine based on the historic concept of compass and ruler construction which reveals the existence of non-computable functions not related with the halting problem. These natural, and the same time, non-computable functions can help to understand the nature of the uncomputable and the purpose, the goal, and the meaning of computing beyond Turing.