Symmetric enumeration reducibility

  • Authors:

  • Charles M. Harris

  • Affiliations:

  • Department of Pure Mathematics, University of Leeds, Leeds, England

  • Venue:
  • CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
  • Year:
  • 2005

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Abstract

Symmetric Enumeration reducibility (≤se) is a subrelation of Enumeration reducibility (≤e) in which both the positive and negative information content of sets is compared. In contrast with Turing reducibility (≤T) however, the positive and negative parts of this relation are separate. A basic classification of ≤se in terms of standard reducibilities is carried out and it is shown that the natural embedding of the Turing degrees into the Enumeration degrees easily translates to this context. A generalisation of the relativised Arithmetical Hierarchy is achieved by replacing the relation c.e. in by ≤e and ≤T by ≤se in the underlying framework of the latter.