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The Euclid Abstract Machine: Trisection of the Angle and the Halting Problem

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Unconventional Computation (UC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4135))

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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.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Maria Curie-Sklodowska, Lublin, Poland

    Jerzy Mycka

  2. Department of Mathematics, Universidade de Évora, Portugal

    Francisco Coelho

  3. Department of Mathematics, I.S.T., Universidade Técnica de Lisboa, and CMAF – Centro de Matemática e Aplicações Fundamentais, Lisboa, Portugal

    José Félix Costa

Authors
  1. Jerzy Mycka
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  2. Francisco Coelho
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  3. José Félix Costa
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

  2. Department of Computer Science, University of Auckland, 92019, Auckland, New Zealand

    Michael J. Dinneen

  3. Institute of Mathematics of the Romanian Academy, PO Box 1-764, 014700, Bucureşti, Romania

    Gheorghe Păun

  4. Leiden Center of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

  5. Dept. of Computer Science, University of York,  

    Susan Stepney

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© 2006 Springer-Verlag Berlin Heidelberg

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Mycka, J., Coelho, F., Costa, J.F. (2006). The Euclid Abstract Machine: Trisection of the Angle and the Halting Problem. In: Calude, C.S., Dinneen, M.J., Păun, G., Rozenberg, G., Stepney, S. (eds) Unconventional Computation. UC 2006. Lecture Notes in Computer Science, vol 4135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11839132_16

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  • DOI: https://doi.org/10.1007/11839132_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38593-6

  • Online ISBN: 978-3-540-38594-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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