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Recursive Unsolvability of a problem of Thue

  • Emil L. Post (a1)
Extract

Alonzo Church suggested to the writer that a certain problem of Thue [6] might be proved unsolvable by the methods of [5]. We proceed to prove the problem recursively unsolvable, that is, unsolvable in the sense of Church [1], but by a method meeting the special needs of the problem.

Thue's (general) problem is the following. Given a finite set of symbols α1, α2, …, αμ, we consider arbitrary strings (Zeichenreihen) on those symbols, that is, rows of symbols each of which is in the given set. Null strings are included.

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[1]Church, Alonzo, An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345363.
[2]Post, Emil L., Finite combinatory processes—formulation 1, this Journal, vol. 1 (1936), pp. 103105.
[3]Post, Emil L., Formal reductions of the general combinatorial decision problem, American journal of mathematics, vol. 65 (1943), pp. 197215.
[4]Post, Emil L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284316.
[5]Post, Emil L., A variant of a recursively unsolvable problem, Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 264268.
[6]Thue, Axel, Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln, Skrifter utgit av Videnskapsselskapet i Kristiania, I. Matematisk-naturvidenskabelig Klasse 1914, no. 10 (1914), 34 pp.
[7]Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, ser. 2 vol. 42 (1937), pp. 230265.
[8]Turing, A. M., Computability and λ-definability, this Journal, vol. 2 (1937), pp. 153163.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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