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Automorphisms of the truth-table degrees are fixed on a cone

Published online by Cambridge University Press:  12 March 2014

Bernard A. Anderson
Bernard A. Anderson*
Affiliation:
Department of Theoretical Computer, Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské Náměstí 25, 118 00 Prague 1, Czech Republic, E-mail: beraugander@yahoo.com
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Abstract

Let Dtt denote the set of truth-table degrees. A bijection π: Dtt → Dtt is an automorphism if for all truth-table degrees x and y we have x ≤tty ⇔ π(x) ≤ttπ(y). We say an automorphism π is fixed on a cone if there is a degree b such that for all x ≥ttb we have π(x) = x. We first prove that for every 2-generic real X we have X′ ≰ttX ⊕ 0′. We next prove that for every real X ≥tt 0′ there is a real Y such that Y ⊕ 0′ ≡ttY′ ≡ttX. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on a cone.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 679 - 688
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Cooper, B. S., 01 2008, private communication.Google Scholar
[2]Kjos-Hanssen, B., A rigid cone in the truth-table degrees with jump, arXiv: 0901:3949 [math. LO], preprint.Google Scholar
[3]Kjos-Hanssen, B., 04 2008, private communication.Google Scholar
[4]Kjos-Hanssen, B., Merkle, W., and Stephan, F., Kolomogorov complexity and the recursion theorem, STACS 2006: 23rd annual Symposium on Theoretical Aspects of Computer Science (Marseilles, France), Lecture Notes in Computer Science, vol. 3884, Springer, 2006, pp. 149–161.CrossRefGoogle Scholar
[5]Kučera, A., 11 2007, private communication.Google Scholar
[6]Kučera, A. and Slaman, T. A., Low upper bounds of ideals, preprint.Google Scholar
[7]Kumabe, M. and Lewis, A., A fixed point free minimal degree, preprint.Google Scholar
[8]Miller, J. S., Π10 classes in computable analysis and topology, Ph.D. thesis, Cornell University, 2002.Google Scholar
[9]Mohrherr, J., Density of a final segment of the truth-table degrees, Pacific Journal of Mathematics, vol. 115 (1984), pp. 409–419.CrossRefGoogle Scholar
[10]Mytilinaios, M. E. and Slaman, T. A., Differences between resource bounded degree structures, Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 12, pp. 1–12.CrossRefGoogle Scholar
[11]Nerode, A. and Shore, R. A., Reducibility orderings: theories, definability, and automorphisms, Annals of Mathematics, vol. 18 (1980), pp. 61–89.Google Scholar
[12]Odifreddi, P. G., Classical recursion theory, Elsevier, 1999.Google Scholar
[13]Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), no. 2, pp. 223–239.CrossRefGoogle Scholar
[14]Shore, R. A., Rigidity and biinterpretability in the hyperdegrees, Recursion theory workshop: Proceedings of the IMS program, Computational prospects of infinity (Chong, C. T., Qi, F., and Yang, Y., editors), World Scientific Publishing Company, 2007, to appear.Google Scholar
[15]Shore, R. A., 02 2008, private communication.Google Scholar
[16]Shore, R. A. and Slaman, T. A.. Defining the Turing jump, Mathematical Research Letters, vol. 6 (1999), no. 5–6, pp. 711–722.CrossRefGoogle Scholar
[17]Slaman, T. A. and Woodin, W. H., Decidability in degree structures, to appear.Google Scholar
[18]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, 1987.CrossRefGoogle Scholar