{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T11:19:31Z","timestamp":1772450371446,"version":"3.50.1"},"reference-count":7,"publisher":"Oxford University Press (OUP)","issue":"7","license":[{"start":{"date-parts":[[2021,6,14]],"date-time":"2021-06-14T00:00:00Z","timestamp":1623628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,10,22]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we introduce and characterize two \u2018analog reducibility\u2019 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\u2013427. 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 \u2018analog enumerate\u2019 any non-c.e. discrete set and that a sufficiently generic positive $[0,1]$-valued oracle neither \u2018analog enumerates\u2019 a non-c.e. discrete set nor \u2018analog computes\u2019 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 \u2018hereditarily compact superstructure\u2019, which was the original motivation for the concepts in this paper.<\/jats:p>","DOI":"10.1093\/logcom\/exab036","type":"journal-article","created":{"date-parts":[[2021,5,6]],"date-time":"2021-05-06T04:48:41Z","timestamp":1620276521000},"page":"1561-1597","source":"Crossref","is-referenced-by-count":1,"title":["Analog reducibility"],"prefix":"10.1093","volume":"31","author":[{"given":"James","family":"Hanson","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin\u2013Madison, 480 Lincoln Dr., Madison, WI 53706, USA"}]}],"member":"286","published-online":{"date-parts":[[2021,6,14]]},"reference":[{"key":"2021102214140497500_ref1","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/s11856-019-1943-x","article-title":"Characterizing the continuous degrees","volume":"234","author":"Andrews","year":"2019","journal-title":"Israel Journal of Mathematics"},{"key":"2021102214140497500_ref2","volume-title":"Computable Structures and the Hyperarithmetical Hierarchy. 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