Π10 classes and strong degree spectra of relations
John Chisholm, Jennifer Chubb, Valentina S. Harizanov, Denis R. Hirschfeldt, Carl G., Jr. Jockusch, Timothy McNicholl, and Sarah Pingrey
Source: J. Symbolic Logic Volume 72, Issue 3
(2007), 1003-1018.
Abstract
We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable Π10 subsets of 2ω and Kolmogorov complexity play a major role in the proof.
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1191333852
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Mathematical Reviews number (MathSciNet): MR2354911
References
C. J. Ash, P. Cholak, and J. F. Knight, Permitting, forcing, and copies of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219--236.
Mathematical Reviews (MathSciNet): MR1460310
Digital Object Identifier: doi:10.1016/S0168-0072(96)00011-5
Zentralblatt MATH: 0883.03029
C. J. Ash and J. F. Knight, Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.
Mathematical Reviews (MathSciNet): MR1767842
Zentralblatt MATH: 0960.03001
C. J. Ash and A. Nerode, Intrinsically recursive relations, Aspects of effective algebra (J. N. Crossley, editor), U.D.A. Book Co., Yarra Glen, Victoria, Australia, 1981, (Proceedings of the Conference at Monash University, Clayton, Australia, Aug. 1--4, 1979), pp. 26--41.
Mathematical Reviews (MathSciNet): MR629248
Zentralblatt MATH: 0467.03041
E. Barker, Intrinsically $\Sigma _\alpha ^0$ relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105--130.
Mathematical Reviews (MathSciNet): MR955520
Digital Object Identifier: doi:10.1016/0168-0072(88)90014-0
Zentralblatt MATH: 0651.03034
G. Barmpalias, Hypersimplicity and semicomputability in the weak truth table degrees, Archive for Mathematical Logic, vol. 44 (2005), pp. 1045--1065.
Mathematical Reviews (MathSciNet): MR2193189
Digital Object Identifier: doi:10.1007/s00153-005-0288-9
Zentralblatt MATH: 1077.03026
M. Bickford and C. F. Mills, Lowness properties of r.e. sets, unpublished preprint.
D. Cenzer, P. Clote, R. L. Smith, R. I. Soare, and S. S. Wainer, Members of countable $\Pi _1^0$ classes, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145--163.
Mathematical Reviews (MathSciNet): MR854290
Digital Object Identifier: doi:10.1016/0168-0072(86)90067-9
Zentralblatt MATH: 0605.03020
D. Cenzer and R. L. Smith, On the ranked points of a $\Pi _1^0$ set, Journal of Symbolic Logic, vol. 54 (1989), pp. 975--991.
Mathematical Reviews (MathSciNet): MR1011184
Digital Object Identifier: doi:10.2307/2274757
Project Euclid: euclid.jsl/1183743032
Zentralblatt MATH: 0689.03022
R. Downey, On $\Pi _1^0$ classes and their ranked points, Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 499--512.
Mathematical Reviews (MathSciNet): MR1146607
Digital Object Identifier: doi:10.1305/ndjfl/1093635924
Project Euclid: euclid.ndjfl/1093635924
Zentralblatt MATH: 0753.03016
R. Downey, N. Greenberg, and R. Weber, Totally $\omega$-computably enumerable degrees I: bounding critical triples, submitted.
R. Downey, C. Jockusch, Jr., and M. Stob, Array nonrecursive sets and multiple permitting arguments, Recursion Theory Week (Oberwolfach, 1989) (K. Ambos-Spies, G. H. Müller, and G. E. Sacks, editors), Lecture Notes in Mathematics 1432, Springer--Verlag, Berlin, 1990, pp. 141--173.
Mathematical Reviews (MathSciNet): MR1071516
Zentralblatt MATH: 0713.03020
Digital Object Identifier: doi:10.1007/BFb0086116
R. G. Downey, $\Delta _2^0$ degrees and transfer theorems, Illinois Journal of Mathematics, vol. 31 (1987), pp. 419--427.
Mathematical Reviews (MathSciNet): MR892177
R. G. Downey, S. S. Goncharov, and D. R. Hirschfeldt, Degree spectra of relations on Boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105--111.
Mathematical Reviews (MathSciNet): MR2003628
R. G. Downey, C. Jockusch, Jr., and M. Stob, Array nonrecursive degrees and genericity, Computability, enumerability, unsolvability: Directions in recursion theory (S. B. Cooper, T. A. Slaman, and S. S. Wainer, editors), London Mathematical Society Lecture Notes Series 224, Cambridge University Press, Cambridge, 1996, pp. 93--104.
Mathematical Reviews (MathSciNet): MR1395872
S. S. Goncharov and B. Khoussainov, On the spectrum of degrees of decidable relations, Doklady Mathematics, vol. 55 (1997), pp. 55--57.
Mathematical Reviews (MathSciNet): MR1445858
V. S. Harizanov, Some effects of Ash--Nerode and other decidability conditions on degree spectra, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 51--65.
Mathematical Reviews (MathSciNet): MR1134916
Digital Object Identifier: doi:10.1016/0168-0072(91)90097-6
Zentralblatt MATH: 0756.03022
--------, Uncountable degree spectra, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 255--263.
Mathematical Reviews (MathSciNet): MR1133007
Digital Object Identifier: doi:10.1016/0168-0072(91)90049-R
Zentralblatt MATH: 0744.03035
--------, Turing degrees of certain isomorphic images of recursive relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103--113.
Mathematical Reviews (MathSciNet): MR1635602
Digital Object Identifier: doi:10.1016/S0168-0072(97)00056-0
Zentralblatt MATH: 0946.03051
--------, Relations on computable structures, Contemporary mathematics (N. Bokan, editor), University of Belgrade, 2000, pp. 65--81.
D. R. Hirschfeldt, Degree spectra of relations on computable structures, Bulletin of Symbolic Logic, vol. 6 (2000), pp. 197--212.
Mathematical Reviews (MathSciNet): MR1781623
Digital Object Identifier: doi:10.2307/421207
Project Euclid: euclid.bsl/1182353687
Zentralblatt MATH: 0968.03038
--------, Degree spectra of intrinsically c.e. relations, Journal of Symbolic Logic, vol. 66 (2001), pp. 441--469.
Mathematical Reviews (MathSciNet): MR1833490
Digital Object Identifier: doi:10.2307/2695024
Project Euclid: euclid.jsl/1183746453
Zentralblatt MATH: 0988.03065
--------, Degree spectra of relations on computable structures in the presence of $\Delta _2^0$ isomorphisms, Journal of Symbolic Logic, vol. 67 (2002), pp. 697--720.
Mathematical Reviews (MathSciNet): MR1905162
Digital Object Identifier: doi:10.2178/jsl/1190150105
Project Euclid: euclid.jsl/1190150105
Zentralblatt MATH: 1010.03033
D. R. Hirschfeldt and W. M. White, Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures, Notre Dame Journal of Formal Logic, vol. 43 (2002), pp. 51--64.
Mathematical Reviews (MathSciNet): MR2033315
Digital Object Identifier: doi:10.1305/ndjfl/1071505769
Project Euclid: euclid.ndjfl/1071505769
Zentralblatt MATH: 1048.03035
C. G. Jockusch, Jr., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420--436.
Mathematical Reviews (MathSciNet): MR220595
Digital Object Identifier: doi:10.2307/1994957
JSTOR: links.jstor.org
Zentralblatt MATH: 0198.32402
C. G. Jockusch, Jr. and T. G. McLaughlin, Countable retracing functions and $\Pi _2^0$ predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 67--93.
Mathematical Reviews (MathSciNet): MR269508
Project Euclid: euclid.pjm/1102978701
C. G. Jockusch, Jr. and R. A. Shore, Pseudojump operators II: transfinite iterations, hierarchies and minimal covers, Journal of Symbolic Logic, vol. 49 (1984), pp. 1205--1236.
Mathematical Reviews (MathSciNet): MR771789
Digital Object Identifier: doi:10.2307/2274273
Project Euclid: euclid.jsl/1183741700
C. G. Jockusch, Jr. and R. I. Soare, $\Pi _1^0$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
Mathematical Reviews (MathSciNet): MR316227
Digital Object Identifier: doi:10.2307/1996261
JSTOR: links.jstor.org
Zentralblatt MATH: 0262.02041
B. Khoussainov and R. A. Shore, Solutions of the Goncharov--Millar and degree spectra problems in the theory of computable models, Doklady Mathematics, vol. 61 (2000), pp. 178--179.
B. Kjos-Hanssen, W. Merkle, and F. Stephan, Kolmogorov complexity and the Recursion Theorem, Stacs 2006: Twenty-Third Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23--25, 2006, Proceedings, Lecture Notes in Computer Science 3884, Springer, pp. 149--161.
Mathematical Reviews (MathSciNet): MR2249365
Digital Object Identifier: doi:10.1007/11672142_11
Zentralblatt MATH: 1137.03026
G. Kreisel, Analysis of the Cantor--Bendixson Theorem by means of the analytic hierarchy, Bulletin de l'Académie Polonaise des Sciences, vol. 7 (1959), pp. 621--626.
Mathematical Reviews (MathSciNet): MR118673
M. Li and P. Vitányi, An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer--Verlag, New York, 1997.
Mathematical Reviews (MathSciNet): MR1438307
Zentralblatt MATH: 0866.68051
T. H. McNicholl, Intrinsic reducibilities, Mathematical Logic Quarterly, vol. 46 (2000), pp. 393--407.
Mathematical Reviews (MathSciNet): MR1774681
Digital Object Identifier: doi:10.1002/1521-3870(200008)46:3<393::AID-MALQ393>3.0.CO;2-H
Zentralblatt MATH: 0973.03060
J. Mohrherr, Index sets and truth-table degrees in recursion theory, PhD Dissertation, University of Illinois at Chicago, 1982.
--------, A refinement of low$_n$ and high$_n$ for the r.e. degrees, Zeitschrift für matematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 5--12.
Mathematical Reviews (MathSciNet): MR832857
P. Odifreddi, Classical recursion theory, North-Holland, Amsterdam, 1989.
Mathematical Reviews (MathSciNet): MR982269
Zentralblatt MATH: 0661.03029
B. Schaeffer, Dynamic notions of genericity and array noncomputability, Annals of Pure and Applied Logic, vol. 95 (1998), pp. 37--69.
Mathematical Reviews (MathSciNet): MR1650648
Digital Object Identifier: doi:10.1016/S0168-0072(98)00014-1
Zentralblatt MATH: 0932.03048
R. I. Soare, Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Springer--Verlag, Berlin, 1987.
Mathematical Reviews (MathSciNet): MR882921
