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Effective choice of constructivizations and recursive consistency of problems on constructive models. (English. Russian original) Zbl 0790.03038
Algebra Logic 31, No. 1, 1-12 (1992); translation from Algebra Logika 31, No. 1, 3-20 (1992).
The author considers a constructive (recursive) model $$\mathfrak M$$ of some language $$L$$ is a domain over which problems are defined. A problem on $$\mathfrak M$$ is a relation on $$\mathfrak M$$ that is invariant under automorphisms of $$\mathfrak M$$. Suppose that $$\pi=\{\nu_ 0,\nu_ 1,\dots\}$$ is a class of constructivizations of $${\mathfrak M}$$, $$\Sigma=\{R_ 0,R_ 1,\dots\}$$ is a computable family of problems, and for each $$R\in\Sigma$$ there exists $$\nu\in\pi$$ such that $$\nu^{-1}(R)$$ is recursive. We say that the problem of the effective choice has a solution if there is a recursive function $$g$$ for which $$\nu^{- 1}_{g(i)}(R_ i)$$ is recursive for all $$i\in N$$, $$R_ i\in\Sigma$$, $$\nu_{g(i)}\in\pi$$. A criterion for the problem of the effective choice to have a solution is given.
MSC:
 03C57 Computable structure theory, computable model theory 03D45 Theory of numerations, effectively presented structures
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References:
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