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Computable classes of constructivizations for models of finite constructivizability type. (English. Russian original) Zbl 0860.03032
Sib. Math. J. 34, No. 5, 812-824 (1993); translation from Sib. Mat. Zh. 34, No. 5, 23-37 (1993).
Let \(\nu\) be an enumeration of a countable structure \(M\). Expanding \(M\) by the names of all elements \(\nu(n)\) gives a structure \(M^*\). The theory of \(M^*\) is denoted by Th\((M,\nu)\). The set of all formulas from Th\((M,\nu)\) with at most \(n\) blocks of quantifiers is denoted by Th\(_n(M,\nu)\). The structure \((M,\nu)\) is called \(n\)-constructivizable if Th\(_n (M, \nu)\) is decidable. It is called \(n\)-complete if for every \(\varphi ({\mathbf a})\in\text{Th}_n(M,\nu)\) there exists an \(\exists\)-formula \(\psi ({\mathbf x})\) realized by \({\mathbf a}\) in \(M\) such that \(\psi ({\mathbf x})\) implies \(\varphi ({\mathbf x})\) in \(M\).
The paper is devoted to proving the following theorem. Let \((M,\nu)\) be \((n+1)\)-constructivizable and let \(M\) be \(n\)-complete but not \((n+1)\)-complete in any finite expansion by constants. Then for every effective class \(S\) of constructivizations of \(M\) one can effectively find a constructivization \(\mu\) which is not an \((n+1)\)-constructivization and is not autoequivalent to any element of \(S\).

MSC:
03C57 Computable structure theory, computable model theory
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References:
[1] S. S. Goncharov, ?The problem, of the number of nonautoequivalent constructivizations,? Algebra i Logika,19, No. 6, 621-639 (1980).
[2] S. S. Goncharov, ?On the number of nonautoequivalent constructivizations,? Algebra i Logika,16, No. 3, 257-289 (1977).
[3] S. S. Goncharov, ?Autostability, and computable families of constructivizations,? Algebra i Logika,14, No. 6, 647-680 (1975).
[4] S. S. Goncharov, ?Autostability of models and Abelian groups,? Algebra i Logika,19, No. 1, 23-44 (1980).
[5] S. S. Goncharov and V. D. Dzgoev, ?Autostability of models,?. Algebra i Logika,19, No. 1, 45-58 (1980). · Zbl 0468.03023
[6] V. P. Dobritsa, ?Computability of certain classes of constructive algebras,? Sibirsk. Mat. Zh.,18, No. 3, 570-579 (1977).
[7] V. P. Dobritsa, ?Complexity of the index set of a constructive model,? Algebra i Logika,19, No. 1, 45-58 (1980).
[8] Yu. L. Ershov, Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).
[9] A. T. Nurtazin, ?Strong and weak constructivizations and computable families,? Algebra i Logika,13, No. 3, 311-323 (1974).
[10] C. J. Ash and A. Nerode, ?Intrinsically recursive relations,? in: Aspects of Effective Algebra: Proc. Conf. at Monash. Univ., Australia, Upside Down a Book Company, 1981, pp. 24-41. · Zbl 0467.03041
[11] S. S. Goncharov, ?Certain properties of constructivizations of Boolean algebras,? Sibirsk. Mat. Zh.,16, No. 2, 264-278 (1975).
[12] S. S. Goncharov, ?Bounded theories of constructive Boolean algebras,? Sibirsk. Mat. Zh.,17, No. 4, 797-812 (1976). · Zbl 0361.02066
[13] Yu. L. Ershov, The Theory, of Enumerations, [in Russian], Nauka, Moscow (1977).
[14] H. Rogers, Theory of Recursive Functions and Effective Computability [Russian translation], Mir, Moscow (1972).
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