Affiliation:
1. Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706, USA
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.
Publisher
Oxford University Press (OUP)
Subject
Logic,Hardware and Architecture,Arts and Humanities (miscellaneous),Software,Theoretical Computer Science
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