Computability of certain classes of constructive algebras

Добрица, В. П.

doi:10.1007/bf00967032

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The study of computable classes of constructivizations is a central trend in the present-day constructive model theory [1][2][3][4][5]. The concept is closely related to studies dealing With algorithmic dimensions of models and with the description of models of infinite algorithmic dimension and of noncomputable classes of constructivizations [5][6][7][8][9][10][11][12].In the present paper, we use the method to give a complete solution of the problem raised by Nurtazin in [12], which asks whether or not the class of weak constructivizations of strongly constructivizable models admitting weak constructivizations is computable. The range of problems mentioned is of interest from the standpoint of both mathematical logic and foundations of the theory of information processing, which deals with the construction of semantics for specification and programming languages with abstract data types.

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confidence: 99%

“…The concept is closely related to studies dealing With algorithmic dimensions of models and with the description of models of infinite algorithmic dimension and of noncomputable classes of constructivizations [5][6][7][8][9][10][11][12]. The concept is closely related to studies dealing With algorithmic dimensions of models and with the description of models of infinite algorithmic dimension and of noncomputable classes of constructivizations [5][6][7][8][9][10][11][12].…”

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Effectively infinite classes of weak constructivizations of models

Гончаров¹

1993

Algebr Logic

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The study of computable classes of constructivizations is a central trend in the present-day constructive model theory [1][2][3][4][5]. The key pursuit in this direction is to construct numerations with specified properties, of which the most interesting is a uniform construction of an effective representation of a model, given its specification. The range of problems mentioned is of interest from the standpoint of both mathematical logic and foundations of the theory of information processing, which deals with the construction of semantics for specification and programming languages with abstract data types. One of the main methods used to prove the infinity for classes of constructive models is based on proving that they are effectively infinite.The concept of an effectively infinite class of constructivizations was introduced in [6], where it was used to provide a characterization of nonautostable models, to study the computability of diverse classes of models, and to solve the problem of constructing a recursive model that satisfies a given specification or has a specified decidable problem. The concept is closely related to studies dealing With algorithmic dimensions of models and with the description of models of infinite algorithmic dimension and of noncomputable classes of constructivizations [5][6][7][8][9][10][11][12].In the present paper, we use the method to give a complete solution of the problem raised by Nurtazin in [12], which asks whether or not the class of weak constructivizations of strongly constructivizable models admitting weak constructivizations is computable. In [6], it was proved that for 1-constructivizable models, the class of constructivizations that are not 1-constructivizations is not computable if such a model has at least one constructivization that is not a 1-constructivization. The notion of n-constructivity, n E w, which is intermediate between constructivity and strong constructivity, was introduced in [13,14].We say that a model (~J~, ~) of signature E is constructive of finite type n if it is n-constructive, i.e., for the enrichment ff)~* of ~9~ by the constants {c,, I n E N} such that the value of c,, in ~" is equal to ~(n), there exists an effective procedure verifying whether the formulas in F,~ (formulas with n alternations of quantifiers in prenex normal form) are true.The class of formulas in the signature EU{c,~ I n E N}, true in ff~*, is denoted by Th (ff)~, v), and we write Tl~ (~, ~) for the class of formulas in F,,, true in ff~*. A sequence of numerated models (~, ~m'y(n, rn)) is called k-computable (strongly computable) if there exists an effective procedure enumerating all formulas from the corresponding set Wh~ (~, ~m~,(n, m)) [Th (~,,, ~rn'y(n, m))] uniformly in n.We note that the notions of a 0-constructivization and a 0-constructive model are equivalent to the standard notions of a constructivization and a constructive model, respectively. To study the algorithmic properties of the class of formulas in the signature ~, U {c,,] n E N}, we fix a GSdel numberi...

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