Sub-computabilities

  • Fabien Givors
  • Gregory Lafitte
Conference paper

DOI: 10.1007/978-3-642-22953-4_28

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6914)
Cite this paper as:
Givors F., Lafitte G. (2011) Sub-computabilities. In: Owe O., Steffen M., Telle J.A. (eds) Fundamentals of Computation Theory. FCT 2011. Lecture Notes in Computer Science, vol 6914. Springer, Berlin, Heidelberg

Abstract

Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be one-one, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a sub-computability as an enumeration of the r.e. sets that are themselves one-one enumerable by total functions of the given collection. Notions similar to the classical computability ones are introduced and variants of the classical theorems are shown. We also introduce sub-reducibilities and study the related completeness notions. One of the striking results is the existence of natural (recursive) sets which play the role of low (non-recursive) solutions to Post’s problem for these sub-reducibilities. The similarity between sub-computabilities and (complete) computability is surprising, since there are so many missing r.e. sets in sub-computabilities. They can be seen as toy models of computability.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fabien Givors
    • 1
  • Gregory Lafitte
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille (LIF)CNRS – Aix-Marseille UniversitéMarseille Cedex 13France

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