Skip to main content Accessibility help

Π10 classes and strong degree spectra of relations

  • John Chisholm (a1), Jennifer Chubb (a2), Valentina S. Harizanov (a3), Denis R. Hirschfeldt (a4), Carl G. Jockusch (a5), Timothy McNicholl (a6) and Sarah Pingrey (a7)...


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.



Hide All
[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.
[2]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.
[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.
[4]Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105130.
[5]Barmpalias, G., Hypersimplicity and semicomputability in the weak truth table degrees, Archive for Mathematical Logic, vol. 44 (2005), pp. 10451065.
[6]Bickford, M. and Mills, C. F., Lowness properties of r.e. sets, unpublished preprint.
[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.
[8]Cenzer, D. and Smith, R. L., On the ranked points of a set, this Journal, vol. 54 (1989), pp. 975991.
[9]Downey, R., On classes and their ranked points, Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 499512.
[10]Downey, R., Greenberg, N., and Weber, R., Totally ω-computably enumerable degrees I: bounding critical triples, submitted.
[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.
[12]Downey, R. G., degrees and transfer theorems, Illinois Journal of Mathematics, vol. 31 (1987), pp. 419427.
[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.
[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.
[15]Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady Mathematics, vol. 55 (1997), pp. 5557.
[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.
[17]Harizanov, V. S., Uncountable degree spectra, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 255263.
[18]Harizanov, V. S., Turing degrees of certain isomorphic images of recursive relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103113.
[19]Harizanov, V. S., Relations on computable structures, Contemporary mathematics (Bokan, N., editor), University of Belgrade, 2000, pp. 6581.
[20]Hirschfeldt, D. R., Degree spectra of relations on computable structures, The Bulletin of Symbolic Logic, vol. 6 (2000), pp. 197212.
[21]Hirschfeldt, D. R., Degree spectra of intrinsically ce. relations, this Journal, vol. 66 (2001), pp. 441469.
[22]Hirschfeldt, D. R., Degree spectra of relations on computable structures in the presence of isomorphisms, this Journal, vol. 67 (2002), pp. 697720.
[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.
[24]Jockusch, C. G. Jr., Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 420436.
[25]Jockusch, C. G. Jr., and McLaughlin, T. G., Countable retracing functions and predicates, Pacific Journal of Mathematics, vol. 30 (1969), pp. 6793.
[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.
[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.
[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.
[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.
[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.
[31]Li, M. and Vitányi, P., An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, New York, 1997.
[32]McNicholl, T. H., Intrinsic reducibilities, Mathematical Logic Quarterly, vol. 46 (2000), pp. 393407.
[33]Mohrherr, J., Index sets and truth-table degrees in recursion theory, PhD Dissertation, University of Illinois at Chicago, 1982.
[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.
[35]Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam, 1989.
[36]Schaeffer, B., Dynamic notions of genericity and array noncomputability, Annals of Pure and Applied Logic, vol. 95 (1998), pp. 3769.
[37]Soare, R. I., Recursively enumerable sets and degrees. A study of computable functions and computably generated sets, Springer-Verlag, Berlin, 1987.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


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