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Abstract <p>The paper studies <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m1--> </InlineEquation>-computable families (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(u\geqslant 2\)</EquationSource> <!--LobJMat2560612Nodirov-m2--> </InlineEquation>) and their minimal numberings. It is proved that the class of all single-valued <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m3--> </InlineEquation>-computable numberings of any <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m4--> </InlineEquation>-computable infinite family of total functions is effectively infinite. It is established that for every <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u-1}\)</EquationSource> <!--LobJMat2560612Nodirov-m5--> </InlineEquation>-computable numbering <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nu\)</EquationSource> <!--LobJMat2560612Nodirov-m6--> </InlineEquation> of an infinite family of total functions there exists a uniformly <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u-1}\)</EquationSource> <!--LobJMat2560612Nodirov-m7--> </InlineEquation>-computable sequence of its single-valued numberings such that <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\nu\)</EquationSource> <!--LobJMat2560612Nodirov-m8--> </InlineEquation> is reducible to their direct sum. It is also shown that if <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq9.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(u&gt;2\)</EquationSource> <!--LobJMat2560612Nodirov-m9--> </InlineEquation>, then every <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m10--> </InlineEquation>-computable numbering of any infinite family is reducible to the direct sum of some uniformly <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m11--> </InlineEquation>-computable and uniformly <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8285_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Sigma^{0}_{u}\)</EquationSource> <!--LobJMat2560612Nodirov-m12--> </InlineEquation>-minimal sequence of numberings of the family.</p>

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Notes on Classes of Minimal Numberings of Arithmetical Set Families

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Sh. D. Nodirov,
  • M. Kh. Faizrahmanov

摘要

Abstract

The paper studies \(\Sigma^{0}_{u}\) -computable families ( \(u\geqslant 2\) ) and their minimal numberings. It is proved that the class of all single-valued \(\Sigma^{0}_{u}\) -computable numberings of any \(\Sigma^{0}_{u}\) -computable infinite family of total functions is effectively infinite. It is established that for every \(\Sigma^{0}_{u-1}\) -computable numbering \(\nu\) of an infinite family of total functions there exists a uniformly \(\Sigma^{0}_{u-1}\) -computable sequence of its single-valued numberings such that \(\nu\) is reducible to their direct sum. It is also shown that if \(u>2\) , then every \(\Sigma^{0}_{u}\) -computable numbering of any infinite family is reducible to the direct sum of some uniformly \(\Sigma^{0}_{u}\) -computable and uniformly \(\Sigma^{0}_{u}\) -minimal sequence of numberings of the family.