Skip to main content

Uniformity in Computable Structure Theory


We investigate the effects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity (of a structure) and intrinsic computability (of a relation on a computable structure). We consider and compare two different notions of uniformity, previously studied by Kudinov and by Ventsov. We discuss some of their results and establish new ones, while also exploring the connections with the relative computable structure theory of Ash, Knight, Manasse, and Slaman and Chisholm and with previous work of Ash, Knight, and Slaman on uniformity in a general computable structure-theoretical setting.

This is a preview of subscription content, access via your institution.


  1. 1.

    R. G. Downey, “Computability theory and linear orderings,” in Handbook of Recursive Mathematics, Stud. Log. Found. Math., Vol. 139, Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Elsevier, Amsterdam (1998), pp. 823–976.

    Google Scholar 

  2. 2.

    S. S. Goncharov, “The problem of the number of nonautoequivalent constructivizations,” Algebra Logika, 19, No. 6, 621–639 (1980).

    Google Scholar 

  3. 3.

    P. Cholak, S. Goncharov, B. Khoussainov, and R. A. Shore, “Computably categorical structures and expansions by constants,” J. Symb. Log., 64, No. 1, 13–37 (1999).

    Google Scholar 

  4. 4.

    S. S. Goncharov, “The quantity of nonautoequivalent constructivizations,” Algebra Logika, 16, No. 3, 257–282 (1977).

    Google Scholar 

  5. 5.

    T. Millar, “Recursive categoricity and persistence,” J. Symb. Log., 51, No. 2, 430–434 (1986).

    Google Scholar 

  6. 6.

    C. J. Ash, J. F. Knight, M. S. Manasse, and T. A. Slaman, “Generic copies of countable structures,” Ann. Pure Appl. Logic, 42, No. 3, 195–205 (1989).

    Google Scholar 

  7. 7.

    J. Chisholm, “Effective model theory vs. recursive model theory,” J. Symb. Log., 55, No. 3, 1168–1191 (1990).

    Google Scholar 

  8. 8.

    C. F. McCoy, “Finite computable dimension does not relativize,” Arch. Math. Log., 41, No. 4, 309–320 (2002).

    Google Scholar 

  9. 9.

    O. V. Kudinov, “An autostable 1-decidable model without a computable Scott family of ∃-formulas,” Algebra Logika, 35, No. 4, 458–467 (1996).

    Google Scholar 

  10. 10.

    O. V. Kudinov, “Some properties of autostable models,” Algebra Logika, 35, No. 6, 685–698 (1996).

    Google Scholar 

  11. 11.

    O. V. Kudinov, “A description of autostable models,” Algebra Logika, 36, No. 1, 26–36 (1997).

    Google Scholar 

  12. 12.

    Yu. G. Ventsov, “Effective choice for relations and reducibilities in classes of constructive and positive models,” Algebra Logika, 31, No. 2, 101–118 (1992).

    Google Scholar 

  13. 13.

    Yu. G. Ventsov, “Effective choice operations on constructive and positive models,” Algebra Logika, 32, No. 1, 45–53 (1993).

    Google Scholar 

  14. 14.

    C. J. Ash, J. F. Knight, and T. A. Slaman, “Relatively recursive expansions. II,” Fund. Math., 142, No. 2, 147–161 (1993).

    Google Scholar 

  15. 15.

    R. I. Soare, Recursively Enumerable Sets and Degrees, Persp. Math. Logic, Springer, Heidelberg (1987).

    Google Scholar 

  16. 16.

    W. Hodges, Model Theory, Enc. Math. Appl., Vol. 42, Cambridge University, Cambridge (1993).

    Google Scholar 

  17. 17.

    C. J. Ash and J. F. Knight, “Relatively recursive expansions,” Fund. Math., 140, No. 2, 137–155 (1992).

    Google Scholar 

  18. 18.

    J. C. Shepherdson and J. Myhill, “Effective operations on partial recursive functions,” Z. Math. Log. Grund. Math., 1, No. 4, 310–317 (1955).

    Google Scholar 

  19. 19.

    G. Kreisel, D. Lacombe, and J. R. Shoenfield, “Partial recursive functionals and effective operations,” in Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957, Stud. Log. Found. Math., A. Heyting (ed.), North-Holland, Amsterdam (1959), pp. 195–207.

  20. 20.

    H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, 2nd edn., MIT, Cambridge, Mass. (1987).

    Google Scholar 

  21. 21.

    B. Khoussainov and R. A. Shore, “Computable isomorphisms, degree spectra of relations, and Scott families,” Ann. Pure Appl. Log., 93, Nos. 1–3, 153–193 (1998).

    Google Scholar 

  22. 22.

    V. V. V'yugin, “Discrete classes of recursively enumerable sets,” Algebra Logika, 11, No. 3, 243–256 (1972).

    Google Scholar 

  23. 23.

    C. J. Ash and A. Nerode, “Intrinsically recursive relations. Aspects of effective algebra,” in Proc. Conf. Monash Univ., Clayton, Australia (1981), pp. 26–41.

  24. 24.

    M. S. Manasse, “Techniques and counterexamples in almost categorical recursive model theory,” Ph.D. Thesis, University of Wisconsin, Madison (1982).

    Google Scholar 

  25. 25.

    J. Chisholm, “The complexity of intrinsically r.e. subsets of existentially decidable models,” J. Symb. Log., 55, No. 3, 1213–1232 (1990).

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Downey, R.G., Hirschfeldt, D.R. & Khoussainov, B. Uniformity in Computable Structure Theory. Algebra and Logic 42, 318–332 (2003).

Download citation

  • computably categorical structure
  • intrinsically computable relation on a computable structure
  • relative computable structure
  • general computable structure