Symmetric enumeration reducibility

Abstract. Symmetric Enumeration reducibility (≤se) is a subrelation of Enu-meration reducibility (≤e) in which both the positive and negative informa-tion 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 Enumera-tion 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. 1