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We present several results that rely on arguments involving the combinatorics of “bushy trees”. These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as well as an extension of a result of Kumabe in which we establish that there are DNC functions relative to arbitrary oracles that are of minimal Turing degree. Along the way, we survey some of the existing instances of bushy tree arguments in the literature.

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[1] Ambos-Spies K., Kjos-Hanssen B., Lempp S., and Slaman T. A., Comparing DNR and WWKL . The Journal of Symbolic Logic, vol. 69 (2004), no. 4, pp. 10891104.
[2] Beros A. A., A DNC function that computes no effectively bi-immune set . Archive for Mathematical Logic, vol. 54 (2015), no. 5–6, pp. 521530.
[3] Bienvenu L. and Patey L., Diagonally non-computable functions and fireworks, arXiv e-prints, 2014.
[4] Cai M., Elements of classical recursion theory: Degree-theoretic properties and combinatorial properties, Ph.D. thesis, Cornell University, 2011.
[5] Dorais F. G., Hirst J. L., and Shafer P., Comparing the strength of diagonally nonrecursive functions in the absence of ${\rm{\Sigma }}_2^0$ induction . The Journal of Symbolic Logic, vol. 80 (2015), no. 4, pp. 12111235.
[6] Downey R. G., Greenberg N., Jockusch C. G. Jr., and Milans K. G., Binary subtrees with few labeled paths . Combinatorica, vol. 31 (2011), no. 3, pp. 285303.
[7] Downey R. G. and Hirschfeldt D. R., Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.
[8] Giusto M. and Simpson S. G., Located sets and reverse mathematics . The Journal of Symbolic Logic, vol. 65 (2000), no. 3, pp. 14511480.
[9] Greenberg N. and Miller J. S., Diagonally non-recursive functions and effective Hausdorff dimension . Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 636654.
[10] Jockusch C. G. Jr., Degrees of functions with no fixed points , Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987), Studies in Logic and the Foundations of Mathematics, vol. 126, North-Holland, Amsterdam, 1989, pp. 191201.
[11] Jockusch C. G. Jr. and Lewis A. E. M., Diagonally non-computable functions and bi-immunity . The Journal of Symbolic Logic, vol. 78, (2013), no. 3, pp. 977988.
[12] Kučera A., Measure, ${\rm{\Pi }}_1^0$ -classes and complete extensions of PA, Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245259.
[13] Kumabe M., A fixed point free minimal degree, unpublished, 1996.
[14] Kumabe M. and Lewis A. E. M., A fixed-point-free minimal degree . Journal of the London Mathematical Society (2), vol. 80 (2009), no. 3, pp. 785797.
[15] Kurtz S. A., Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign; ProQuest LLC, Ann Arbor, MI, 1981.
[16] Sacks G. E., Some open questions in recursion theory , Recursion Theory Week (Ebbinghaus H.-D., Müller G. H., and Sacks G. E., editors), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, Heidelberg, 1985, pp. 333342.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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