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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:
  S. S. Goncharov, ?The problem, of the number of nonautoequivalent constructivizations,? Algebra i Logika,19, No. 6, 621-639 (1980).  S. S. Goncharov, ?On the number of nonautoequivalent constructivizations,? Algebra i Logika,16, No. 3, 257-289 (1977).  S. S. Goncharov, ?Autostability, and computable families of constructivizations,? Algebra i Logika,14, No. 6, 647-680 (1975).  S. S. Goncharov, ?Autostability of models and Abelian groups,? Algebra i Logika,19, No. 1, 23-44 (1980).  S. S. Goncharov and V. D. Dzgoev, ?Autostability of models,?. Algebra i Logika,19, No. 1, 45-58 (1980). · Zbl 0468.03023  V. P. Dobritsa, ?Computability of certain classes of constructive algebras,? Sibirsk. Mat. Zh.,18, No. 3, 570-579 (1977).  V. P. Dobritsa, ?Complexity of the index set of a constructive model,? Algebra i Logika,19, No. 1, 45-58 (1980).  Yu. L. Ershov, Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).  A. T. Nurtazin, ?Strong and weak constructivizations and computable families,? Algebra i Logika,13, No. 3, 311-323 (1974).  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  S. S. Goncharov, ?Certain properties of constructivizations of Boolean algebras,? Sibirsk. Mat. Zh.,16, No. 2, 264-278 (1975).  S. S. Goncharov, ?Bounded theories of constructive Boolean algebras,? Sibirsk. Mat. Zh.,17, No. 4, 797-812 (1976). · Zbl 0361.02066  Yu. L. Ershov, The Theory, of Enumerations, [in Russian], Nauka, Moscow (1977).  H. Rogers, Theory of Recursive Functions and Effective Computability [Russian translation], Mir, Moscow (1972).
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