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Diagonally Non-Computable Functions and Bi-Immunity

  • Carl G. Jockusch (a1) and Andrew E. M. Lewis (a2)


We prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.



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[1] Ambos-Spies, K., Kjos-Hanssen, B., Lempp, S., and Slaman, T., Comparing DNR and WWKL, this Journal, vol. 69 (2004), pp. 10891104.
[2] Downey, R., Greenberg, N., Jockusch, C., and Milans, K., Binary subtrees with few labeled paths, Combinatorica, vol. 31 (2011), pp. 285303.
[3] Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, Springer, 2010.
[4] Jockusch, C., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.
[5] Jockusch, C., The degrees of bi-immune sets, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 135140.
[6] Jockusch, C., Upward closure of bi-immune degrees, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 18 (1972), pp. 285287.
[7] Jockusch, C., Degrees of functions with no fixed points, Logic, methodology, and philosophy of science VIII (Fenstad, J. E., Frolov, I. T., and Hilpinen, R., editors), North-Holland, Amsterdam, 1989, pp. 191201.
[8] Jockusch, C., Lerman, M., Solovay, R., and Soare, R., Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion, this Journal, vol. 54 (1989), pp. 12881323.
[9] Kumabe, M. and Lewis, A. E. M., A fixed point free minimal degree, Journal of the London Mathematical Society, vol. 80 (2009), no. 3, pp. 785797.
[10] Lachlan, A., Solution to a problem of Spector, Canadian Journal of Mathematics, vol. 23 (1971), pp. 247256.
[11] Nies, A., Computability and randomness, Oxford University Press, 2009.
[12] Simpson, S., Recursion theoretic aspects of the dual Ramsey theorem, Recursion theory week, Lecture Notes in Mathematics No. 1141, Springer-Verlag, Berlin, 1985, pp. 357371.
[13] Simpson, S., Mass problems and randomness, The Bulletin of Symbolic Logic, vol. 11 (2005), no. 1, pp. 127.
[14] Simpson, S., An extension of the recursively enumerable Turing degrees, Journal of the London Mathematical Society, vol. 75 (2007), pp. 287297.
[15] Soare, R., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.



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