Analog reducibility

Abstract

Abstract In this paper we introduce and characterize two ‘analog reducibility’ notions for $[0,1]$-valued oracles on $\omega $ obtained by applying the syntactic characterizations of Turing and enumeration reducibility in terms of (positive) relatively $\varSigma _1$ and $\varPi _1$ formulas to formulas in continuous logic (Ben Yaacov, Berenstein, Henson and Usvyatsov, 2008, Model Theory for Metric Structures, vol. 2 of London Mathematical Society Lecture Note Series, pp. 315–427. Cambridge University Press.). The resulting analog and analog enumeration degree structures, $\mathscr {D}_a$ and $\mathscr {D}_{ae}$, naturally extend $\mathscr {D}_T$ and $\mathscr {D}_e$ in a compatible way. To show that these extensions are proper we prove that a sufficiently generic total $[0,1]$-valued oracle does not ‘analog enumerate’ any non-c.e. discrete set and that a sufficiently generic positive $[0,1]$-valued oracle neither ‘analog enumerates’ a non-c.e. discrete set nor ‘analog computes’ a non-trivial total $[0,1]$-valued oracle. We also provide a characterization of the continuous degrees among $\mathscr {D}_{ae}$ as precisely $\mathscr {D}_e \cap \mathscr {D}_a$. Finally we characterize a generalization of r.i.c.e. relations to metric structures via $\varSigma _1$ formulas in the ‘hereditarily compact superstructure’, which was the original motivation for the concepts in this paper.