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Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

Published online by Cambridge University Press:  12 March 2014

Denis R. Hirschfeldt
Denis R. Hirschfeldt*
Affiliation:
School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand
*
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA, E-mail: [email protected]
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Abstract

We give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.

We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 2 , June 2002 , pp. 697 - 720
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Ash, C. J., Isomorphic recursive structures, in [9], pp. 167–182.CrossRefGoogle Scholar
[2]Ash, C. J., Cholak, P., and Knight, J. F., Permitting, forcing, and copying of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219–236.CrossRefGoogle Scholar
[3]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Clayton, 1979) (Crossley, J. N., editor). Upside Down A Book Co., Yarra Glen, Australia, 1981, pp. 26–41.Google Scholar
[4]Cooper, S. B., Harrington, L., Lachlan, A. H., Lempp, S., and Soare, R. I., The d. r. e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125–151.CrossRefGoogle Scholar
[5]Downey, R. G., Computahility theory and linear orderings, in [9], pp. 823–976.CrossRefGoogle Scholar
[6]Downey, R. G., Goncharov, S. S., and Hirschfeldt, D. R., Degree spectra of relations on Boolean algebras, to appear.Google Scholar
[7]Epstein, R. L., Haas, R., and Kramer, R. L., Hierarchies of sets and degrees below 0ℙ, Logic year 1979–80 (Proceedings of seminars and conferences in mathematical logic, University of Connecticut, Storrs, Connecticut, 1979/80) (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Heidelberg, 1981, pp. 32–48.Google Scholar
[8]Ershov, Y. L. and Goncharov, S. S., Elementary theories and their constructive models, in [9], pp. 115–166.Google Scholar
[9]Ershov, Y. L., Goncharov, S. S., Nerode, A., and Remmel, J. B. (editors). Handbook of recursive mathematics, Studies in Logic and the Foundations of Mathematics, vol. 138-139, Elsevier Science, Amsterdam, 1998.Google Scholar
[10]Goncharov, S. S., Autostable models and algorithmic dimensions, in [9], pp. 261–288.CrossRefGoogle Scholar
[11]Goncharov, S. S., Limit equivalent constructivizations, Mathematical logic and the theory of algorithms, Trudy Inst. Mat. (“Nauka” Sibirsk. Otdel., Novosibirsk), vol. 2, 1982, in Russian, pp. 4–12.Google Scholar
[12]Harizanov, V. S., Pure computable model theory, in [9], pp. 3–114.CrossRefGoogle Scholar
[13]Harizanov, V. S., Degree spectrum of a recursive relation on a recursive structure, Ph. d. thesis, University of Wisconsin, Madison, Wisconsin, 1987.Google Scholar
[14]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. 51–65.CrossRefGoogle Scholar
[15]Harizanov, V. S., Turing degrees of certain isomorphic images of computable relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103–113.CrossRefGoogle Scholar
[16]Hirschfeldt, D. R., Degree spectra of intrinsically c. e. relations, to appear this Journal.Google Scholar
[17]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, to appear in Annals of Pure and Applied Logic.Google Scholar
[18]Hirschfeldt, D. R. and White, W. M., Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures, to appear.Google Scholar
[19]Hodges, W., Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[20]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998). pp. 153–193.CrossRefGoogle Scholar
[21]Khoussainov, B. and Shore, R. A., Effective model theory: the number of models and their complexity, Models and computability (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Note Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–239.CrossRefGoogle Scholar
[22]Moschovakis, Y. N., Elementary induction on abstract structures, Studies in Logic and the Foundations of Mathematics, vol. 77, North-Holland Publishing Company, Amsterdam-London, 1974.Google Scholar
[23]Moses, M., Relations intrinsically recursive in linear orders, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986). pp. 467–472.CrossRefGoogle Scholar
[24]Remmel, J. B., Recursive isomorphism types of recursive Boolean algebras, this Journal, vol. 46 (1981). pp. 572–594.Google Scholar
[25]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[26]Soskov, I. N., Intrinsically hyperarithmetical sets, Mathematical Logic Quarterly, vol. 42 (1996). pp. 469–480.CrossRefGoogle Scholar