Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-22T14:25:58.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Degree Spectra of Relations on Computable Structures

Published online by Cambridge University Press:  15 January 2014

Denis R. Hirschfeldt
Denis R. Hirschfeldt*
Affiliation:
School of Mathematical and Computing Sciences, Victoria University of Wellington, New zealand E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)

Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned with computable structures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.

In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 2 , June 2000 , pp. 197 - 212
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ash, Chris J., Cholak, Peter, and Knight, Julia F., Permitting, forcing, and copying of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219–236.Google Scholar
[2] Ash, Chris J. and Nerode, Anil, Intrinsically recursive relations, Aspects of effective algebra (Clayton, 1979) (Yarra Glen, Australia) (Crossley, J. N., editor), Upside Down A Book Co., 1981, pp. 26–41.Google Scholar
[3] Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105–130.CrossRefGoogle Scholar
[4] Cholak, P., Goncharov, S. S., Khoussainov, B., and Shore, R. A., Computably categorical structures and expansions by constants, The Journal of Symbolic Logic, vol. 64 (1999), pp. 13–37.Google Scholar
[5] 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.Google Scholar
[6] Epstein, Richard L., Haas, Richard, and Kramer, Richard L., Hierarchies of sets and degrees below 0', Logic year 1979–80 (Proceedings of Seminars and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80) (Heidelberg) (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer–Verlag, 1981, pp. 32–48.Google Scholar
[7] Ershov, Yu. 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
[8] Goncharov, S. S., The quantity of nonautoequivalent constructivizations, Algebra and Logic, vol. 16 (1977), pp. 169–185.Google Scholar
[9] Goncharov, S. S., Computable single-valued numerations, Algebra and Logic, vol. 19 (1980), pp. 325–356.Google Scholar
[10] Goncharov, S. S., Problem of the number of non-self-equivalent constructivizations, Algebra and Logic, vol. 19 (1980), pp. 401–414.Google Scholar
[11] Goncharov, S. S., Groups with a finite number of constructivizations, Soviet Math.Dokl., vol. 23 (1981), pp. 58–61.Google Scholar
[12] Goncharov, S. S., Limit equivalent constructivizations, Mathematical logic and the theory of algorithms, “Nauka” Sibirsk. Otdel., Novosibirsk, 1982, in Russian, pp. 4–12.Google Scholar
[13] Goncharov, S. S., Countable boolean algebras and decidability, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1997.Google Scholar
[14] Goncharov, S. S. and Dzgoev, V. D., Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 28–37.Google Scholar
[15] Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady Math., vol. 55 (1997), pp. 55–57, research announcement.Google Scholar
[16] Goncharov, S. S., Molokov, A. V., and Romanovskii, N. S., Nilpotent groups of finite algorithmic dimension, Siberian Mathematics Journal, vol. 30 (1989), pp. 63–68.CrossRefGoogle Scholar
[17] Harizanov, V. S., Degree spectrum of a recursive relation on a recursive structure, Ph.D. Thesis , University of Wisconsin, Madison, WI, 1987.Google Scholar
[18] 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.Google Scholar
[19] Harizanov, V. S., The possible Turing degree of the nonzero member in a two element degree spectrum, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 1–30.Google Scholar
[20] Harizanov, V. S., Turing degrees of certain isomorphic images of computable relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103–113.Google Scholar
[21] Hirschfeldt, D. R., Degree spectra of intrinsically c. e. relations, to appear in Journal of Symbolic Logic.Google Scholar
[22] Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, to appear.Google Scholar
[23] Hodges, W., Model theory, Encyclopedia Math. Appl., vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
[24] Khoussainov, B. and Shore, R. A., Solutions of the Goncharov-Millar and degree spectra problems in the theory of computable models, to appear in Dokl. Akad. Nauk SSSR.Google Scholar
[25] 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.Google Scholar
[26] Kudinov, O., An integral domain with finite algorithmic dimension, personal communication.Google Scholar
[27] LaRoche, P., Recursively represented Boolean algebras, Notices of the American Mathematical Society, vol. 24 (1977), pp. A–552, research announcement.Google Scholar
[28] Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289–320.Google Scholar
[29] Millar, T., Recursive categoricity and persistence, The Journal of Symbolic Logic, vol. 51 (1986), pp. 430–434.Google Scholar
[30] Moses, M., Relations intrinsically recursive in linear orders, Z. Math. Logik Grundlag. Math., vol. 32 (1986), pp. 467–472.Google Scholar
[31] Nurtazin, A. T., Strong and weak constructivizations and enumerable families, Algebra and Logic, vol. 13 (1974), pp. 177–184.Google Scholar
[32] Remmel, J. B., Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387–391.Google Scholar
[33] Soare, Robert I., Recursively enumerable sets and degrees, Perspect. Math. Logic, Springer–Verlag, Heidelberg, 1987.Google Scholar