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eT-Reducibility of Sets

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Abstract—

This paper is devoted to the study of \(eT\)-reducibility, i.e., the most intuitively general algorithmic reducibility, which is both enumeration and decision reducibilities. The corresponding degree structure, that is, the upper semilattice \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) of \(eT\)-degrees is considered. It is shown that it is possible to correctly define a jump operation on \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) by using the \(T\)-jump or \(e\)-jump of sets. The local properties of \(eT\)-degrees, such as totality and computable enumerability, are considered. It is proven that all degrees between the smallest element and the first jump in \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) are computably enumerable; moreover, these degrees contain computably enumerable sets and only them. The existence of nontotal \(eT\)-degrees is established. Based on this, we obtain some results on the relations between degrees; in particular, the fact that every \(eT\)-degree is either completely contained in some \(T\)-degree or \(e\)-degree, or completely coincides with it, implies that nontotal \(e\)-degrees are contained in the \(T\)-degrees located above the second \(T\)-jump and coincide with the corresponding nontotal \(eT\)-degrees.

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Correspondence to R. R. Iarullin.

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The authors declare that they have no conflicts of interest.

ADDITIONAL INFORMATION

Roman R. Iarullin, orcid.org/0000-0003-1604-5599, graduate student.

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Translated by I. Tselishcheva

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Iarullin, R.R. eT-Reducibility of Sets. Aut. Control Comp. Sci. 54, 615–618 (2020). https://doi.org/10.3103/S0146411620070196

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  • DOI: https://doi.org/10.3103/S0146411620070196

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