### Schnorr trivial sets and truth-table reducibility

Johanna N. Y. Franklin and Frank Stephan
Source: J. Symbolic Logic Volume 75, Issue 2 (2010), 501-521.

#### Abstract

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.

First Page:
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1268917492
Digital Object Identifier: doi:10.2178/jsl/1268917492
Zentralblatt MATH identifier: 05725861
Mathematical Reviews number (MathSciNet): MR2648153

### References

Marat Arslanov, On some generalizations of a fixed-point theorem, Soviet Mathematics, vol. 25 (1981), no. 5, pp. 9--16, English translation, Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, vol. 25 (1981), no. 5, pp. 1--10, Russian.
Mathematical Reviews (MathSciNet): MR630478
Zentralblatt MATH: 0358.02046
Cristian S. Calude and André Nies, Chaitin $\Omega$ numbers and strong reducibilities, Journal of Universal Computer Science, vol. 3 (1997), pp. 1162--1166.
Mathematical Reviews (MathSciNet): MR1661774
Rod Downey, Evan Griffiths, and Geoffrey LaForte, On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, vol. 50 (2004), pp. 613--627.
Mathematical Reviews (MathSciNet): MR2096175
Digital Object Identifier: doi:10.1002/malq.200310121
Rod Downey, Denis R. Hirschfeldt, André Nies, and Sebastiaan A. Terwijn, Calibrating randomness, Bulletin of Symbolic Logic, vol. 12 (2006), pp. 411--491.
Mathematical Reviews (MathSciNet): MR2248591
Zentralblatt MATH: 1113.03037
Digital Object Identifier: doi:10.2178/bsl/1154698741
Project Euclid: euclid.bsl/1154698741
Johanna N. Y. Franklin, Aspects of Schnorr randomness, Ph.D. dissertation, University of California, Berkeley, 2007.
--------, Hyperimmune-free degrees and Schnorr triviality, Journal of Symbolic Logic, vol. 73 (2008), pp. 999--1008.
Mathematical Reviews (MathSciNet): MR2444282
Zentralblatt MATH: 1181.03045
Digital Object Identifier: doi:10.2178/jsl/1230396761
Project Euclid: euclid.jsl/1230396761
--------, Schnorr trivial reals: A construction, The Archive for Mathematical Logic, vol. 46 (2008), pp. 665--678.
Mathematical Reviews (MathSciNet): MR2395564
Zentralblatt MATH: 1142.03020
Digital Object Identifier: doi:10.1007/s00153-007-0061-3
--------, Schnorr triviality and genericity, Journal of Symbolic Logic, vol. 75 (2010), pp. 191--207.
Zentralblatt MATH: 05681298
Project Euclid: euclid.jsl/1264433915
Denis Hirschfeldt, André Nies, and Frank Stephan, Using random sets as oracles, Journal of the London Mathematical Society, vol. 75 (2007), pp. 610--622.
Mathematical Reviews (MathSciNet): MR2352724
Zentralblatt MATH: 1128.03036
Digital Object Identifier: doi:10.1112/jlms/jdm041
Carl G. Jockusch and Frank Stephan, A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39 (1993), pp. 515--530, A corrective note (keeping the main results intact) appeared in the same journal, vol. 43 (1997), p. 569.
Mathematical Reviews (MathSciNet): MR1270396
Zentralblatt MATH: 0799.03048
Digital Object Identifier: doi:10.1002/malq.19930390153
Ming Li and Paul Vitányi, An introduction to Kolmogorov complexity and its applications, Springer, Heidelberg, 1993.
Mathematical Reviews (MathSciNet): MR1238938
Per Martin-Löf, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602--619.
Mathematical Reviews (MathSciNet): MR223179
Nenad Mihailović, Algorithmic randomness, Inaugural-Dissertation, University of Heidelberg, 2007.
Webb Miller and Donald Martin, The degrees of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159--166.
Mathematical Reviews (MathSciNet): MR228341
André Nies, Lowness properties of reals and randomness, Advances in Mathematics, vol. 197 (2005), pp. 274--305.
Mathematical Reviews (MathSciNet): MR2166184
Zentralblatt MATH: 1141.03017
Digital Object Identifier: doi:10.1016/j.aim.2004.10.006
André Nies, Frank Stephan, and Sebastiaan A. Terwijn, Randomness, relativization and Turing degrees, Journal of Symbolic Logic, vol. 70 (2005), pp. 515--535.
Mathematical Reviews (MathSciNet): MR2140044
Zentralblatt MATH: 1090.03013
Digital Object Identifier: doi:10.2178/jsl/1120224726
Project Euclid: euclid.jsl/1120224726
Piergiorgio Odifreddi, Classical recursion theory, North-Holland, Amsterdam, 1989.
Mathematical Reviews (MathSciNet): MR982269
--------, Classical recursion theory, Volume II, North-Holland, Amsterdam, 1999.
Mathematical Reviews (MathSciNet): MR1718169
Zentralblatt MATH: 0931.03057
Hartley Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
Mathematical Reviews (MathSciNet): MR224462
Gerald E. Sacks, A maximal set which is not complete, Michigan Mathematical Journal, vol. 11 (1964), pp. 193--205.
Mathematical Reviews (MathSciNet): MR166090
Digital Object Identifier: doi:10.1307/mmj/1028999130
Project Euclid: euclid.mmj/1028999130
Claus Peter Schnorr, Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer, 1971.
Mathematical Reviews (MathSciNet): MR414225
--------, Process complexity and effective random tests, Journal of Computer and System Sciences, vol. 7 (1973), pp. 376--388.
Mathematical Reviews (MathSciNet): MR325366
Zentralblatt MATH: 0273.68036
Digital Object Identifier: doi:10.1016/S0022-0000(73)80030-3
Robert Soare, Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.
Mathematical Reviews (MathSciNet): MR882921
Frank Stephan, The complexity of the set of nonrandom numbers, Randomness and complexity, from Leibnitz to Chaitin (Cristian S. Calude, editor), vol. 217--230, World Scientific, 2007.
Mathematical Reviews (MathSciNet): MR2427551
Zentralblatt MATH: 1138.03038
Digital Object Identifier: doi:10.1142/9789812770837_0012