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Π10 classes and strong degree spectra of relations

Published online by Cambridge University Press:  12 March 2014

John Chisholm
Affiliation:
Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA, E-mail: JA-Chisholm@wiu.edu
Jennifer Chubb
Affiliation:
Department of Mathematics, The George Washington University, Washington, D.C. 20052, USA, E-mail: jchubb@gwu.edu
Valentina S. Harizanov
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC. 20052, USA, E-mail: harizanv@gwu.edu
Denis R. Hirschfeldt
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA, E-mail: drh@math.uchicago.edu
Carl G. Jockusch Jr.
Affiliation:
Department of Mathematics, University of Illinois, 1409 W Green St.. Urbana, IL 61801, USA, E-mail: jockusch@math.uiuc.edu
Timothy McNicholl
Affiliation:
Department of Mathematics, Lamar University, Beaumont Texas 77710, USA, E-mail: timothy.h.mcnicholl@gmail.com
Sarah Pingrey
Affiliation:
Department of Mathematics, The George Washington University, Washington, D.C. 20052, USA, E-mail: spingrey@gwu.edu

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 subsets of 2ω and Kolmogorov complexity play a major role in the proof.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Ash, C. J., Cholak, P., and Knight, J. F., Permitting, forcing, and copies of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219236.CrossRefGoogle Scholar
[2]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.Google Scholar
[3]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Crossley, J. N., 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. 2641.Google Scholar
[4]Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105130.CrossRefGoogle Scholar
[5]Barmpalias, G., Hypersimplicity and semicomputability in the weak truth table degrees, Archive for Mathematical Logic, vol. 44 (2005), pp. 10451065.CrossRefGoogle Scholar
[6]Bickford, M. and Mills, C. F., Lowness properties of r.e. sets, unpublished preprint.Google Scholar
[7]Cenzer, D., Clote, P., Smith, R. L., Soare, R. I., and Wainer, S. S., Members of countable classes, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145163.CrossRefGoogle Scholar
[8]Cenzer, D. and Smith, R. L., On the ranked points of a set, this Journal, vol. 54 (1989), pp. 975991.Google Scholar
[9]Downey, R., On classes and their ranked points, Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 499512.CrossRefGoogle Scholar
[10]Downey, R., Greenberg, N., and Weber, R., Totally ω-computably enumerable degrees I: bounding critical triples, submitted.Google Scholar
[11]Downey, R., Jockusch, C. Jr., and Stob, M., Array nonrecursive sets and multiple permitting arguments, Recursion Theory Week (Oberwolfach, 1989) (Ambos-Spies, K., Müller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics 1432, Springer-Verlag, Berlin, 1990, pp. 141173.CrossRefGoogle Scholar
[12]Downey, R. G., degrees and transfer theorems, Illinois Journal of Mathematics, vol. 31 (1987), pp. 419427.Google Scholar
[13]Downey, R. G., Goncharov, S. S., and Hirschfeldt, D. R., Degree spectra of relations on Boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105111.CrossRefGoogle Scholar
[14]Downey, R. G., Jockusch, C. Jr., and Stob, M., Array nonrecursive degrees andgenericity, Computability, enumerability, unsolvability: Directions in recursion theory (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), London Mathematical Society Lecture Notes Series 224, Cambridge University Press, Cambridge, 1996, pp. 93104.CrossRefGoogle Scholar
[15]Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady Mathematics, vol. 55 (1997), pp. 5557.Google Scholar
[16]Harizanov, V. S., Some effects of Ash–Nerode and other decidability conditions on degree spectra, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 5165.CrossRefGoogle Scholar
[17]Harizanov, V. S., Uncountable degree spectra, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 255263.CrossRefGoogle Scholar
[18]Harizanov, V. S., Turing degrees of certain isomorphic images of recursive relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103113.CrossRefGoogle Scholar
[19]Harizanov, V. S., Relations on computable structures, Contemporary mathematics (Bokan, N., editor), University of Belgrade, 2000, pp. 6581.Google Scholar
[20]Hirschfeldt, D. R., Degree spectra of relations on computable structures, The Bulletin of Symbolic Logic, vol. 6 (2000), pp. 197212.CrossRefGoogle Scholar
[21]Hirschfeldt, D. R., Degree spectra of intrinsically ce. relations, this Journal, vol. 66 (2001), pp. 441469.Google Scholar
[22]Hirschfeldt, D. R., Degree spectra of relations on computable structures in the presence of isomorphisms, this Journal, vol. 67 (2002), pp. 697720.Google Scholar
[23]Hirschfeldt, D. R. and White, W. M., Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures, Notre Dame Journal of Formal Logic, vol. 43 (2002), pp. 5164.Google Scholar
[24]Jockusch, C. G. Jr., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.CrossRefGoogle Scholar
[25]Jockusch, C. G. Jr., and McLaughlin, T. G., Countable retracing functions and predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 6793.CrossRefGoogle Scholar
[26]Jockusch, C. G. Jr., and Shore, R. A., Pseudojump operators IT. transfinite iterations, hierarchies and minimal covers, this Journal, vol. 49 (1984), pp. 12051236.Google Scholar
[27]Jockusch, C. G. Jr. and Soare, R. I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.Google Scholar
[28]Khoussainov, B. and Shore, R. A., Solutions of the Goncharov–Millar and degree spectra problems in the theory of computable models, Doklady Mathematics, vol. 61 (2000), pp. 178179.Google Scholar
[29]Kjos-Hanssen, B., Merkle, W., and Stephan, F., 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. 149161.Google Scholar
[30]Kreisel, G., Analysis of the Cantor-Bendixson Theorem by means of the analytic hierarchy, Bulletin de l'Académie Polonaise des Sciences, vol. 7 (1959), pp. 621626.Google Scholar
[31]Li, M. and Vitányi, P., An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
[32]McNicholl, T. H., Intrinsic reducibilities, Mathematical Logic Quarterly, vol. 46 (2000), pp. 393407.3.0.CO;2-H>CrossRefGoogle Scholar
[33]Mohrherr, J., Index sets and truth-table degrees in recursion theory, PhD Dissertation, University of Illinois at Chicago, 1982.Google Scholar
[34]Mohrherr, J., A refinement of lown and highn for the r.e. degrees, Zeitschrift für matematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 512.Google Scholar
[35]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam, 1989.Google Scholar
[36]Schaeffer, B., Dynamic notions of genericity and array noncomputability, Annals of Pure and Applied Logic, vol. 95 (1998), pp. 3769.CrossRefGoogle Scholar
[37]Soare, R. I., Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Springer-Verlag, Berlin, 1987.Google Scholar
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