Skip to main content Accessibility help
×
Home

Degree Spectra of Relations on Computable Structures

  • Denis R. Hirschfeldt (a1)

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.

Copyright

References

Hide All
[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. 219236.
[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. 2641.
[3] Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105130.
[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. 1337.
[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. 125151.
[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. 3248.
[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.
[8] Goncharov, S. S., The quantity of nonautoequivalent constructivizations, Algebra and Logic, vol. 16 (1977), pp. 169185.
[9] Goncharov, S. S., Computable single-valued numerations, Algebra and Logic, vol. 19 (1980), pp. 325356.
[10] Goncharov, S. S., Problem of the number of non-self-equivalent constructivizations, Algebra and Logic, vol. 19 (1980), pp. 401414.
[11] Goncharov, S. S., Groups with a finite number of constructivizations, Soviet Math.Dokl., vol. 23 (1981), pp. 5861.
[12] Goncharov, S. S., Limit equivalent constructivizations, Mathematical logic and the theory of algorithms, “Nauka” Sibirsk. Otdel., Novosibirsk, 1982, in Russian, pp. 412.
[13] Goncharov, S. S., Countable boolean algebras and decidability, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1997.
[14] Goncharov, S. S. and Dzgoev, V. D., Autostability of models, Algebra and Logic, vol. 19 (1980), pp. 2837.
[15] Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady Math., vol. 55 (1997), pp. 5557, research announcement.
[16] Goncharov, S. S., Molokov, A. V., and Romanovskii, N. S., Nilpotent groups of finite algorithmic dimension, Siberian Mathematics Journal, vol. 30 (1989), pp. 6368.
[17] Harizanov, V. S., Degree spectrum of a recursive relation on a recursive structure, Ph.D. Thesis , University of Wisconsin, Madison, WI, 1987.
[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. 5165.
[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. 130.
[20] Harizanov, V. S., Turing degrees of certain isomorphic images of computable relations, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 103113.
[21] Hirschfeldt, D. R., Degree spectra of intrinsically c. e. relations, to appear in Journal of Symbolic Logic.
[22] Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, to appear.
[23] Hodges, W., Model theory, Encyclopedia Math. Appl., vol. 42, Cambridge University Press, Cambridge, 1993.
[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.
[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. 153193.
[26] Kudinov, O., An integral domain with finite algorithmic dimension, personal communication.
[27] LaRoche, P., Recursively represented Boolean algebras, Notices of the American Mathematical Society, vol. 24 (1977), pp. A–552, research announcement.
[28] Metakides, G. and Nerode, A., Effective content of field theory, Annals of Mathematical Logic, vol. 17 (1979), pp. 289320.
[29] Millar, T., Recursive categoricity and persistence, The Journal of Symbolic Logic, vol. 51 (1986), pp. 430434.
[30] Moses, M., Relations intrinsically recursive in linear orders, Z. Math. Logik Grundlag. Math., vol. 32 (1986), pp. 467472.
[31] Nurtazin, A. T., Strong and weak constructivizations and enumerable families, Algebra and Logic, vol. 13 (1974), pp. 177184.
[32] Remmel, J. B., Recursively categorical linear orderings, Proceedings of the American Mathematical Society, vol. 83 (1981), pp. 387391.
[33] Soare, Robert I., Recursively enumerable sets and degrees, Perspect. Math. Logic, Springer–Verlag, Heidelberg, 1987.

Metrics

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