Skip to main content Accessibility help

Schnorr trivial sets and truth-table reducibility

  • Johanna N. Y. Franklin (a1) and Frank Stephan (a2)


We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey. Griffiths and LaForte.



Hide All
[1]Arslanov, Marat, On some generalizations of a fixed-point theorem, Soviet Mathematics, vol. 25 (1981), no. 5, pp. 916, English translation, Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika. vol. 25 (1981), no. 5, pp. 1–10, Russian.
[2]Calude, Cristian S. and Nies, André. Chaitin Ω numbers and strong rcducibililies, Journal of Universal Computer Science, vol. 3 (1997), pp. 11621166.
[3]Downey, Rod, Griffiths, Evan, and LaForte, Geoffrey, On Schnorr and computable randomness, martingales, andmachines, Mathematical Logic Quarterly, vol. 50 (2004). pp. 613627.
[4]Downey, Rod, Hirschfeldt, Denis R., Nies, André, and Terwijn, Sebastiaan A.. Calibrating randomness, The Bulletin of Symbolic Logic, vol. 12 (2006). pp. 411491.
[5]Franklin, Johanna N. Y., Aspects of Schnorr randomness, Ph.D. dissertation, University of California, Berkeley, 2007.
[6]Franklin, Johanna N. Y., Hyperimmune-free degrees and Schnorr triviality, this Journal, vol. 73 (2008). pp. 9991008.
[7]Franklin, Johanna N. Y., Schnorr trivial reals: A construction. The Archive for Mathematical Logic, vol. 46 (2008), pp. 665678.
[8]Franklin, Johanna N. Y., Schnorr triviality andgenericity, this Journal, vol. 75 (2010), pp. 191207.
[9]Hirschfeldt, Denis, Nies, André, and Stephan, Frank, Using random sets as oracles, Journal of the London Mathematical Society, vol. 75 (2007). pp. 610622.
[10]Jockusch, Carl G. and Stephan, Frank, A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39 (1993), pp. 515530. A corrective note (keeping the main results intact) appeared in the same journal, vol. 43 (1997). p. 569.
[11]Li, Ming and Vitányi, Paul, An introduction to Kolmogorov complexity and its applications, Springer, Heidelberg, 1993.
[12]Martin-Löf, Per, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.
[13]Mihailović, Nenad, Algorithmic randomness, Inaugural-Dissertation, University of Heidelberg, 2007.
[14]Miller, Webb and Martin, Donald, The degrees of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159166.
[15]Nies, André, Lowness properties of reals and randomness, Advances in Mathematics, vol. 197, (2005), pp. 274305.
[16]Nies, André, Stephan, Frank, and Terwijn, Sebastiaan A.. Randomness, relativization and Turing degrees, this Journal, vol. 70 (2005). pp. 515535.
[17]Odifreddi, Piergiorgio, Classical recursion theory, North-Holland, Amsterdam, 1989.
[18]Odifreddi, Piergiorgio, Classical recursion theory, Volume II, North-Holland, Amsterdam, 1999.
[19]Rogers, Hartley, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[20]Sacks, Gerald E., A maximal set which is not complete, Michigan Mathematical Journal, vol. 11 (1964), pp. 193205.
[21]Schnorr, Claus Peter, Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer, 1971.
[22]Schnorr, Claus Peter, Process complexity and effective random tests, Journal of Computer and System Sciences, vol. 7 (1973), pp. 376388.
[23]Soare, Robert, Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.
[24]Stephan, Frank, The complexity of the set of nonrandom numbers, Randomness and complexity, from Leibnitz to Chaitin (Calude, Cristian S., editor), vol. 217-230, World Scientific, 2007.

Related content

Powered by UNSILO

Schnorr trivial sets and truth-table reducibility

  • Johanna N. Y. Franklin (a1) and Frank Stephan (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.