## Abstract

In a lecture in Kazan (1977), Goncharov dubbed a number of problems regarding the classification of computable members of various classes of structures. Some of the problems seemed likely to have nice answers, while others did not. At the end of the lecture, Shore asked what would be a convincing negative result. The goal of the present article is to consider some possible answers to Shore's question. We consider structures Д of some computable language, whose universes are computable sets of constants. In measuring complexity, we identify Д with its atomic diagram D(Д), which, via the Gödel numbering, may be treated as a subset of ω. In particular, Д is computable if D(Д) is computable. If K is some class, then K^{c} denotes the set of computable members of K. A computable characterization for K should separate the computable members of K from other structures, that is, those that either are not in K or are not computable. A computable classification (structure theorem) should describe each member of K^{c} up to isomorphism, or other equivalence, in terms of relatively simple invariants. A computable non-structure theorem would assert that there is no computable structure theorem. We use three approaches. They all give the “correct” answer for vector spaces over Q, and for linear orderings. Under all of the approaches, both classes have a computable characterization, and there is a computable classification for vector spaces, but not for linear orderings. Finally, we formulate some open problems.

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## REFERENCES

- 1.W. Hodges, “What is a structure theory?,”
*Bull. London Math. Soc.,***19**, No. 3(78) 209-237 (1987).Google Scholar - 2.S. Shelah, “Classification of first order theories which have a structure theorem,”
*Bull. Am. Math. Soc.*,**12**, No. 2, 227-232 (1985).Google Scholar - 3.H. Friedman and L. Stanley, “On Borel reducibility theory for classes of computable structures,”
*J. Symb. Log.,***54**, No. 3, 894-914 (1989).Google Scholar - 4.C. J. Ash and J. F. Knight,
*Computable Structures and the Hyperarithmetical Hierarchy*, Elsevier, Amsterdam (2000).Google Scholar - 5.D. Scott, “Logic with denumerably long formulas and finite strings of quantifiers,” in
*The Theory of Models,*J. Addison, L. Henkin, and A. Tarski (eds.), North-Holland, Amsterdam (1970), pp. 329-341.Google Scholar - 6.H. J. Keisler,
*Model Theory for Infinitary Logic,*North-Holland, Amsterdam (1971).Google Scholar - 7.H. Rogers,
*Theory of Recursive Functions and Effective Computability,*McGraw-Hill, New York (1967).Google Scholar - 8.J. Harrison, “Recursive pseudo well-orderings,”
*Trans. Am. Math. Soc.,***131**, No. 2, 526-543 (1968).Google Scholar - 9.G. E. Sacks,
*Higher Type Recursion Theory,*Springer, Berlin (1990).Google Scholar - 10.S. S. Goncharov, “Autostability and computable families of constructivizations,”
*Algebra Logika,***14**, No. 6, 647-680 (1975).Google Scholar - 11.S. S. Goncharov, “The quantity of non-autoequivalent constructivizations,”
*Algebra Logika*,**16**, No. 6, 257-282 (1977).Google Scholar - 12.C. J. Ash, “Categoricity in hyperarithmetical degrees,”
*Ann. Pure Appl. Log.,***34**, No. 1, 1-14 (1987).Google Scholar - 13.E. Lopez-Escobar, “An addition to 'On definable well-orderings',”
*Fund. Math*.,**59**, No. 3, 299-300 (1966).Google Scholar - 14.M. Morley, “Omitting classes of elements,” in
*The Theory of Models,*M. Addison, L. Henkin, and A. Tarski (eds.), North-Holland, Amsterdam (1970), pp. 265-273.Google Scholar - 15.D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinco, “Degree spectra and computable dimensions in algebraic structures,” Preprint.Google Scholar
- 16.C. J. Ash and J. F. Knight, “Pairs of recursive structures,”
*Ann. Pure Appl. Log*.,**46**, No. 3, 211-234 (1990).Google Scholar - 17.C. J. Ash, C. G. Jockusch, and J. F. Knight, “Jumps of orderings,”
*Trans. Am. Math. Soc*.,**319**, No. 2, 573-599 (1990).Google Scholar - 18.C. J. Ash, “A construction for recursive linear orderings,”
*J. Symb. Log.,***56**, No. 2, 673-683 (1991).Google Scholar - 19.C. J. Ash, “Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees,”
*Trans. Am. Math. Soc*.,**298**, No. 2, 497-514 (1986); Corrections:*Ibid.*,**310**, No. 2, 851 (1988).Google Scholar - 20.A. T. Nurtazin, “Computable classes and algebraic criteria for autostability,” Ph.D. Thesis, Institute of Mathematics and Mechanics, Alma-Ata (1974).Google Scholar
- 21.C. G. Jockusch and R. I. Soare, “Degrees of orderings not isomorphic to recursive linear orderings,”
*Ann. Pure Appl. Log.,***52**, Nos. 1/2, 39-64 (1991).Google Scholar