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RANK AND RANDOMNESS

  • RUPERT HÖLZL (a1) and CHRISTOPHER P. PORTER (a2)

Abstract

We show that for each computable ordinal $\alpha > 0$ it is possible to find in each Martin-Löf random ${\rm{\Delta }}_2^0 $ degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R’s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.

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[1]Ash, C. J. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North Holland, Amsterdam, 2000.
[2]Bienvenu, L. and Porter, C., Strong reductions in effective randomness. Theoretical Computer Science, vol. 459 (2012), pp. 5568.
[3]Binns, S., ${\rm{\Delta }}_2^0 $ classes with complex elements , this Journal, vol. 73 (2008), no. 4, pp. 13411353.
[4]Cenzer, D., ${\rm{\Pi }}_1^0 $classes in computability theory, Handbook of Computability Theory (Griffor, E. R., editor), Studies in Logic and the Foundations of Mathematics, vol. 140, North Holland, Amsterdam, 1999, pp. 3785.
[5]Cenzer, D. and Remmel, J., Index sets for ${\rm{\Pi }}_1^0 $ classes. Annals of Pure and Applied Logic , vol. 93 (1998), no. 1–3, pp. 361.
[6]Cenzer, D. and Smith, R. L., On the ranked points of a ${\rm{\Pi }}_1^0 $ set , this Journal, vol. 54 (1989), no. 3, pp. 975991.
[7]Cholak, P. and Downey, R., On the Cantor-Bendixon rank of recursively enumerable sets, this Journal, vol. 58 (1993), no. 2, pp. 629640.
[8]Downey, R. G., Wu, G., and Yang, Y., The members of thin and minimal ${\rm{\Pi }}_1^0 $ classes, their ranks and Turing degrees. Annals of Pure and Applied Logic , vol. 166 (2015), no. 7–8, pp. 755766.
[9]Hölzl, R. and Porter, C. P., Randomness for computable measures and initial segment complexity. Annals of Pure and Applied Logic, vol. 168 (2017), no. 4, pp. 860886.
[10]Kautz, S. M., Degrees of random sets, Ph.D thesis, Cornell University, 1991, p. 129.
[11]Kreisel, G., Analysis of the Cantor-Bendixson theorem by means of the analytic hierarchy. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1959), pp. 621626.
[12]Odifreddi, P., Classical Recursion Theory Vol. II, Studies in Logic and the Foundations of Mathematics, vol. 143, North Holland, Amsterdam, 1999.
[13]Owings, J., Rank, join, and Cantor singletons. Archive for Mathematical Logic, vol. 36 (1997), no. 4–5, pp. 313320.
[14]Porter, C. P., Trivial measures are not so trivial. Theory of Computing Systems, vol. 56 (2015), no. 3, pp. 487512.
[15]Zvonkin, A. K. and Levin, L. A., The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms. Uspekhi Matematicheskikh Nauk, vol. 25 (1970), no. 6(156), pp. 85127.

Keywords

RANK AND RANDOMNESS

  • RUPERT HÖLZL (a1) and CHRISTOPHER P. PORTER (a2)

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