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Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 1–15 | Cite as

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

  • Juan A. Nido Valencia
  • Julio E. Solís Daun
  • Luis M. Villegas SilvaEmail author
Article

Abstract

We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.

Keywords

Horn theory Primitive recursive set functions Recursively enumerable set \(\alpha \)-Recursion theory Primitive recursively closed ordinals Admissible recursion \(\beta \)-Recursion 

Mathematics Subject Classification

Primary 03C55 03C70 03D20 03D60 03D65 03E45 Secondary 03C62 03D75 

Notes

Acknowledgments

This research was partially supported by CONACYT grant 400200-5-32267-E. This work started during the third author’s sabbatical leave at Institut für Mathematik, Humboldt Universität, 10099 Berlin, Germany. We would like to thank the Referee for his/her interest in our work and for his/her helpful comments that will greatly improve the manuscript.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  • Juan A. Nido Valencia
    • 1
  • Julio E. Solís Daun
    • 2
  • Luis M. Villegas Silva
    • 2
    Email author
  1. 1.Posgrado en Ciencias de la ComplejidadUniversidad Autónoma de la Ciudad de MéxicoBenito JuarezMexico
  2. 2.Departamento de MatemáticasUniversidad Autónoma Metropolitana-IztapalapaIztapalapaMexico

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