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ISSN 1088-6826(e) ISSN 0002-9939(p)

A basis theorem for $\Pi_1^0$ classes of positive measure and jump inversion for random reals

Authors: Rod Downey and Joseph S. Miller
Journal: Proc. Amer. Math. Soc. 134 (2006), 283-288
MSC (2000): Primary 03D28, 68Q30
Posted: August 11, 2005
MathSciNet review: 2170569
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the Shoenfield jump inversion theorem to the members of any $\Pi^0_1$ class $\+ P\subseteq 2^\omega$ with nonzero measure; i.e., for every $\Sigma^0_2$ set $S\geq_T\emptyset'$ , there is a $\Delta^0_2$ real $A\in\+ P$ such that $A'\equiv_T S$ . In particular, we get jump inversion for $\Delta^0_2$ -random reals.

References

1. M. M. Arslanov, R. F. Nadyrov, and V. D. Solovev, A criterion for the completeness of recursively enumerable sets, and some generalizations of a fixed point theorem, Izv. Vyss. Ucebn. Zaved. Matematika 4 (179) (1977), 3-7. MR 0446930 (56:5247)
2. Douglas Cenzer, $\Pi\sp 0\sb 1$ classes in computability theory, Handbook of computability theory, Stud. Logic Found. Math., vol. 140, North-Holland, Amsterdam, 1999, pp. 37-85. MR 1720779 (2001d:03110)
3. R. Downey and D. Hirschfeldt, Algorithmic randomness and complexity, Springer-Verlag, Berlin, to appear.
4. Rodney G. Downey, Denis R. Hirschfeldt, André Nies, and Sebastiaan A. Terwijn, Calibrating randomness, Bulletin of Symbolic Logic, to appear.
5. Richard Friedberg, A criterion for completeness of degrees of unsolvability, J. Symb. Logic 22 (1957), 159-160. MR 0098025 (20:4488)
6. Péter Gács, Every sequence is reducible to a random one, Inform. and Control 70 (1986), no. 2-3, 186-192. MR 0859105 (87k:03043)
7. Carl G. Jockusch, Jr. and Robert I. Soare, $\Pi \sp{0}\sb{1}$ classes and degrees of theories, Trans. Amer. Math. Soc. 173 (1972), 33-56. MR 0316227 (47:4775)
8. S. Kautz, Degrees of random sets, Ph.D. Dissertation, Cornell University, 1991.
9. Antonín Kucera, Measure, $\Pi\sp 0\sb 1$ -classes and complete extensions of ${\rm PA}$ , Recursion theory week (Oberwolfach, 1984), Lecture Notes in Math., vol. 1141, Springer, Berlin, 1985, pp. 245-259. MR 0820784 (87e:03102)
10. -, On the use of diagonally nonrecursive functions, Logic Colloquium '87 (Granada, 1987), Stud. Logic Found. Math., vol. 129, North-Holland, Amsterdam, 1989, pp. 219-239. MR 1015310 (91c:03037)
11. S. Kurtz, Randomness and genericity in the degrees of unsolvability, Ph.D. Dissertation, University of Illinois, Urbana, 1981.
12. M. Li and P. Vitányi, An introduction to Kolmogorov complexity and its applications, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1993. MR 1238938 (94j:68121)
13. Per Martin-Löf, The definition of random sequences, Information and Control 9 (1966), 602-619. MR 0223179 (36:6228)
14. Joseph S. Miller and Liang Yu, On initial segment complexity and degrees of randomness, to appear.
15. Gerald E. Sacks, A minimal degree less than , Bull. Amer. Math. Soc. 67 (1961), 416-419. MR 0126380 (23:A3676)
16. Dana Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proc. Sympos. Pure Math., Vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 117-121. MR 0141595 (25:4993)
17. J. R. Shoenfield, On degrees of unsolvability, Ann. of Math. (2) 69 (1959), 644-653. MR 0105355 (21:4097)
18. Robert I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987. MR 0882921 (88m:03003)
19. Frank Stephan, Martin-Löf random sets and PA-complete sets, to appear.

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Additional Information

Rod Downey
Affiliation: School of Mathematical and Computing Sciences, Victoria University, P.O. Box 600, Wellington, New Zealand
Email: Rod.Downey@mcs.vuw.ac.nz

Joseph S. Miller
Affiliation: Department of Mathematics, University of Connecticut, U-3009, 196 Auditorium Road, Storrs, Connecticut 06269
Email: joseph.s.miller@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07901-3
PII: S 0002-9939(05)07901-3
Received by editor(s): April 13, 2004
Received by editor(s) in revised form: June 30, 2004
Posted: August 11, 2005
Additional Notes: Both authors were supported by the Marsden Fund of New Zealand. The first author was also partially supported by NSFC Grand International Joint Project Grant No. 60310213 ``New Directions in Theory and Applications of Models of Computation'' (China).
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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