{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:53:29Z","timestamp":1777517609725,"version":"3.51.4"},"reference-count":27,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T00:00:00Z","timestamp":1761955200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>\n                    In a recent article, we introduced and studied a precise class of dynamical systems called\n                    <jats:italic toggle=\"yes\">solvable systems<\/jats:italic>\n                    that present a dynamic ruled by discontinuous ordinary differential equations with\n                    <jats:italic toggle=\"yes\">solvable<\/jats:italic>\n                    right-hand terms. These systems are interesting because, when they exhibit a unique evolution, a transfinite method always exists to define such evolution as a limit of a sequence of continuous functions. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of real functions with bounded, solvable derivatives and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris\u2013Woodin, Denjoy, and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.\n                  <\/jats:p>","DOI":"10.1177\/22113568241310506","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T10:44:19Z","timestamp":1766486659000},"page":"178-197","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Set descriptive complexity of solvable functions"],"prefix":"10.1177","volume":"14","author":[{"given":"Olivier","family":"Bournez","sequence":"first","affiliation":[{"name":"LIX, Ecole Polytechnique, Institut Polytechnique de Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-0210-8508","authenticated-orcid":false,"given":"Riccardo","family":"Gozzi","sequence":"additional","affiliation":[{"name":"LACL, University Paris-Est Cr\u00e9teil Val de Marne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2025,12,23]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1941201337"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0885-064X(03)00034-7"},{"key":"e_1_3_2_4_2","doi-asserted-by":"crossref","unstructured":"Bournez O Campagnolo ML Gra\u00e7a DS et\u00a0al. The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation. In: Cai J Cooper SB and Li A (eds) International conference on theory and applications of models of computation. Berlin Heidelberg: Springer Berlin Heidelberg 2006.","DOI":"10.1007\/11750321_60"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/3127496"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.3233\/COM-210384"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2023.101755"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","unstructured":"Bournez O Gozzi R. Solving discontinuous initial value problems with unique solutions is equivalent to computing over the transfinite. In: Symposium on theoretical aspects of computer science (STACS) Clermont-Ferrand France. Saarbr\u00fccken\/Wadern Germany: Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik GmbH Dagstuhl Publishing pp.1\u201320. 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