Skip to main content Accessibility help
×
Home

The importance of Π1 0 classes in effective randomness

  • George Barmpalias (a1), Andrew E.M. Lewis (a2) and Keng Meng Ng (a3)

Abstract

We prove a number of results in effective randomness, using methods in which Π1 0 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

Copyright

References

Hide All
[AK00] Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies' in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000.
[Bar06] Barmpalias, George, Random non-cupping revisited, Journal of Complexity, vol. 22 (2006), no. 6, pp. 850857.
[BLS08a] Barmpalias, George, Lewis, Andrew E. M., and Soskova, Mariya, Randomness, lowness and degrees, this Journal, vol. 73 (2008), no. 2, pp. 559577.
[BLS08b] Barmpalias, George, Lewis, Andrew E. M., and Stephan, Frank, Π1 0 classes, LR degrees and Turing degrees, Annals of Pure and Applied Logic, vol. 156, Issuel (2008), pp. 2138.
[JS72a] Jr.Jockusch, Carl G. and Soare, Robert I., Degrees of members of Π1 0 classes, Pacific Journal of Mathematics, vol. 40 (1972), pp. 605616.
[JS72b] Jr.Jockusch, Carl G. and Soare, Robert I., Π1 0 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[Cen99] Cenzer, Douglas, Π1 0 classes in computability theory, Handbook of computability theory, Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland, Amsterdam, 1999, pp. 3785.
[CS07] Cole, Joshua A. and Simpson, Stephen G., Mass problems and hyperarithmeticity, Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 125143.
[DHMN05] Downey, Rod, Hirschfeldt, Denis R., Miller, Joseph S., and Nies, André, Relativizing Chaitins halting probability, Journal of Mathematical Logic, vol. 5 (2005), no. 2, pp. 167192.
[DHNT06] Downey, Rod, Hirschfeldt, Denis R., Nies, André, and Terwijn, Sebastiaan A., Calibrating randomness, Bullin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 411491.
[DH09] Downey, Rod and Hirshfeldt, Denis R., Algorithmic randomness and complexity, Springer-Verlag, 2009, in preparation.
[FMN] Figueira, Santiago, Miller, Joseph, and Nies, André, Indifferent sets, to appear.
[KH07] Kjos-Hanssen, Bjørn, Low for random reals and positive-measure domination, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 11, pp. 37033709, electronic.
[Kuč85] Kučera, Antonín, Measure, Π1 0-classes and complete extensions of PA, Recursion theory week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245259.
[Kuč86] Kučera, Antonín, An alternative, priority-free, solution to Post's problem, Mathematical foundations of computer science 1986 (Bratislava, 1986), Lecture Notes in Computer Science, vol. 233, Springer, Berlin, 1986, pp. 493500.
[KS07] Kučera, Antonín and Slaman, Theodore, Low upper bounds of ideals, Preprint, 2007.
[Kur81] Kurtz, S., Randomness andgenericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois, Urbana, 1981.
[ML66] Martin-Löf, Per, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.
[Mil] Miller, Joseph S., The K-degrees, low for K degrees, and weakly low for K sets, preprint.
[Nie05a] Nies, André, Eliminating concepts, Proceedings of the IMS workshop on computational aspects of infinity, Singapore, 2005, in press.
[Nie05b] Nies, André, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274305.
[Nie07] Nies, André, Non-cupping and randomness, Proceedings of the American Mathematical Society, vol. 135 (2007), no. 3, pp. 837844, electronic.
[Nie09] Nies, André, Computability and randomness, Oxford University Press, 2009.
[Odi89] Odifreddi, P. G., Classical recursion theory, vol. I, North-Holland Publishing Co., Amsterdam, 1989.
[Sac63] Sacks, Gerald E., Degrees of unsolvability, Princeton University Press, Princeton, N.J., 1963.
[Sim07] Simpson, Stephen G., Almost everywhere domination and superhighness, Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 462482.
[Ste06] Stephan, Frank, Martin-Löf random and PA-complete sets, Logic colloquium '02, Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 342348.
[Sti72] Stillwell, John, Decidability of the “almost all” theory of degrees, this Journal, vol. 37 (1972), pp. 501506.

Keywords

Metrics

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