{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T14:23:58Z","timestamp":1777645438388,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"2-3","license":[{"start":{"date-parts":[[2009,1,1]],"date-time":"2009-01-01T00:00:00Z","timestamp":1230768000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fundamenta Informaticae"],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p>We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer n \u2a7e 1, there is a D _{&amp;omega;^n} (\u03a3 ^1_1 )-complete tree language L _n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous B\u00fcchi tree automaton must be Borel. Then we consider the game tree languages W _{(&amp;inodot;,&amp;kappa;)} , for Mostowski-Rabin indices (\u0131\u03ba). We prove that the D _{&amp;omega;^n} (\u03a3 ^1_1 )-complete tree languages L _n are Wadge reducible to the game tree language W _{(&amp;inodot;,&amp;kappa;)} for \u03ba\u2212\u0131\u2a7e 2. In particular these languages W _{(&amp;inodot;,&amp;kappa;)} are not in any class D _{&amp;alpha;} (\u03a3 ^1_1 ) for \u03b1&lt;\u03c9 ^{&amp;omega;} .<\/jats:p>","DOI":"10.3233\/fi-2009-151","type":"journal-article","created":{"date-parts":[[2019,12,2]],"date-time":"2019-12-02T22:55:12Z","timestamp":1575327312000},"page":"287-303","source":"Crossref","is-referenced-by-count":5,"title":["On Recognizable Tree Languages Beyond the Borel Hierarchy"],"prefix":"10.1177","volume":"95","author":[{"given":"Olivier","family":"Finkel","sequence":"first","affiliation":[{"name":"Equipe de Logique Math\u00e9matique CNRS et\r\t\t\t Universit\u00e9 Paris 7, France. E-mail: finkel@logique.jussieu.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"Simonnet","sequence":"additional","affiliation":[{"name":"Syst\u00e8mes physiques pour l'environnement,\r\t\t\t Facult\u00e9 des Sciences, Universit\u00e9 de Corse Quartier\r\t\t\t Grossetti BP52 20250, Corte, France. E-mail: simonnet@univ-corse.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2009,1,1]]},"container-title":["Fundamenta Informaticae"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2009-151","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2009-151","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T06:32:13Z","timestamp":1777444333000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/FI-2009-151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,1]]},"references-count":0,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["10.3233\/FI-2009-151"],"URL":"https:\/\/doi.org\/10.3233\/fi-2009-151","relation":{},"ISSN":["0169-2968","1875-8681"],"issn-type":[{"value":"0169-2968","type":"print"},{"value":"1875-8681","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1,1]]}}}