{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T04:07:18Z","timestamp":1750392438398,"version":"3.41.0"},"publisher-location":"Cham","reference-count":28,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031959073","type":"print"},{"value":"9783031959080","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025]]},"DOI":"10.1007\/978-3-031-95908-0_11","type":"book-chapter","created":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T08:54:16Z","timestamp":1750323256000},"page":"143-158","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Learning Existentially Definable Subsets in\u00a0a\u00a0Computable Structure"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5834-2770","authenticated-orcid":false,"given":"Nikolay","family":"Bazhenov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8259-4441","authenticated-orcid":false,"given":"Manat","family":"Mustafa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,20]]},"reference":[{"issue":"1\u20132","key":"11_CR1","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/S00153-022-00842-6","volume":"62","author":"S Aleksandrova","year":"2023","unstructured":"Aleksandrova, S., Bazhenov, N., Zubkov, M.: Complexity of $$\\Sigma ^0_n$$-classifications for definable subsets. Arch. Math. Log. 62(1\u20132), 239\u2013256 (2023). https:\/\/doi.org\/10.1007\/S00153-022-00842-6","journal-title":"Arch. Math. Log."},{"key":"11_CR2","unstructured":"Ash, C.J., Knight, J.: Computable Structures and the Hyperarithmetical Hierarchy, Stud. Logic Found. Math., vol.\u00a0144. Elsevier, Amsterdam (2000)"},{"key":"11_CR3","doi-asserted-by":"publisher","unstructured":"Bazhenov, N., Fokina, E., Rossegger, D., Soskova, A., Vatev, S.: Learning families of algebraic structures from text. In: Levy Patey, L., Pimentel, E., Galeotti, L., Manea, F. (eds.) CiE 2024. LNCS, vol. 14773, pp. 166\u2013178. Springer, Cham (2024). https:\/\/doi.org\/10.1007\/978-3-031-64309-5_14","DOI":"10.1007\/978-3-031-64309-5_14"},{"key":"11_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/J.IC.2020.104590","volume":"275","author":"N Bazhenov","year":"2020","unstructured":"Bazhenov, N., Fokina, E., San Mauro, L.: Learning families of algebraic structures from informant. Inf. Comput. 275, 104590 (2020). https:\/\/doi.org\/10.1016\/J.IC.2020.104590","journal-title":"Inf. Comput."},{"issue":"1","key":"11_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/S00224-024-10201-Y","volume":"69","author":"N Bazhenov","year":"2025","unstructured":"Bazhenov, N., Mustafa, M.: On learning down-sets in quasi-orders, and ideals in Boolean algebras. Theory Comput. Syst. 69(1), 1 (2025). https:\/\/doi.org\/10.1007\/S00224-024-10201-Y","journal-title":"Theory Comput. Syst."},{"key":"11_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/J.IC.2022.104913","volume":"295","author":"D Belanger","year":"2023","unstructured":"Belanger, D., Gao, Z., Jain, S., Li, W., Stephan, F.: Learnability and positive equivalence relations. Inf. Comput. 295, 104913 (2023). https:\/\/doi.org\/10.1016\/J.IC.2022.104913","journal-title":"Inf. Comput."},{"issue":"2","key":"11_CR7","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/S0019-9958(75)90261-2","volume":"28","author":"L Blum","year":"1975","unstructured":"Blum, L., Blum, M.: Toward a mathematical theory of inductive inference. Inf. Control 28(2), 125\u2013155 (1975). https:\/\/doi.org\/10.1016\/S0019-9958(75)90261-2","journal-title":"Inf. Control"},{"issue":"5","key":"11_CR8","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s10469-019-09559-7","volume":"58","author":"S Boyadzhiyska","year":"2019","unstructured":"Boyadzhiyska, S., Lange, K., Raz, A., Scanlon, R., Wallbaum, J., Zhang, X.: Classifications of definable subsets. Algebra Log. 58(5), 383\u2013404 (2019). https:\/\/doi.org\/10.1007\/s10469-019-09559-7","journal-title":"Algebra Log."},{"issue":"6","key":"11_CR9","doi-asserted-by":"publisher","first-page":"1941","DOI":"10.1137\/S0097539793249694","volume":"28","author":"J Case","year":"1999","unstructured":"Case, J.: The power of vacillation in language learning. SIAM J. Comput. 28(6), 1941\u20131969 (1999). https:\/\/doi.org\/10.1137\/S0097539793249694","journal-title":"SIAM J. Comput."},{"issue":"4","key":"11_CR10","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/J.IC.2009.08.001","volume":"208","author":"M de Brecht","year":"2010","unstructured":"de Brecht, M., Yamamoto, A.: Topological properties of concept spaces (full version). Inf. Comput. 208(4), 327\u2013340 (2010). https:\/\/doi.org\/10.1016\/J.IC.2009.08.001","journal-title":"Inf. Comput."},{"key":"11_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-4305-3","volume-title":"Constructive Models","author":"YL Ershov","year":"2000","unstructured":"Ershov, Y.L., Goncharov, S.S.: Constructive Models. Kluwer Academic, New York (2000)"},{"key":"11_CR12","unstructured":"Fokina, E.B., K\u00f6tzing, T., San Mauro, L.: Limit learning equivalence structures. In: Garivier, A., Kale, S. (eds.) ALT 2019. Proceedings of Machine Learning Research, vol.\u00a098, pp. 383\u2013403 (2019)"},{"key":"11_CR13","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/978-3-642-27654-5_10","volume-title":"Computation, Physics and Beyond","author":"Z Gao","year":"2012","unstructured":"Gao, Z., Stephan, F., Wu, G., Yamamoto, A.: Learning families of closed sets in matroids. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds.) WTCS 2012. LNCS, vol. 7160, pp. 120\u2013139. Springer, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-27654-5_10"},{"issue":"5","key":"11_CR14","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1016\/S0019-9958(67)91165-5","volume":"10","author":"EM Gold","year":"1967","unstructured":"Gold, E.M.: Language identification in the limit. Inf. Control 10(5), 447\u2013474 (1967). https:\/\/doi.org\/10.1016\/S0019-9958(67)91165-5","journal-title":"Inf. Control"},{"key":"11_CR15","unstructured":"Goncharov, S.S., Kogabaev, N.T.: On $$\\Sigma ^0_1$$-classification of relations on computable structures. Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 8(4), 23\u201332 (2008). https:\/\/www.mathnet.ru\/eng\/vngu304. in Russian"},{"key":"11_CR16","volume-title":"Countable Boolean Algebras and Decidability","author":"SS Goncharov","year":"1997","unstructured":"Goncharov, S.S.: Countable Boolean Algebras and Decidability. Consultants Bureau, New York (1997)"},{"issue":"1","key":"11_CR17","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.jcss.2006.09.001","volume":"73","author":"VS Harizanov","year":"2007","unstructured":"Harizanov, V.S., Stephan, F.: On the learnability of vector spaces. J. Comput. Syst. Sci. 73(1), 109\u2013122 (2007). https:\/\/doi.org\/10.1016\/j.jcss.2006.09.001","journal-title":"J. Comput. Syst. Sci."},{"key":"11_CR18","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/6610.001.0001","volume-title":"Systems that Learn","author":"S Jain","year":"1999","unstructured":"Jain, S., Osherson, D., Royer, J.S., Sharma, A.: Systems that Learn, 2nd edn. MIT Press, Cambridge (1999)","edition":"2"},{"issue":"1\u20133","key":"11_CR19","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.tcs.2008.02.030","volume":"397","author":"S Lange","year":"2008","unstructured":"Lange, S., Zeugmann, T., Zilles, S.: Learning indexed families of recursive languages from positive data: a survey. Theor. Comput. Sci. 397(1\u20133), 194\u2013232 (2008). https:\/\/doi.org\/10.1016\/j.tcs.2008.02.030","journal-title":"Theor. Comput. Sci."},{"key":"11_CR20","volume-title":"Hilbert\u2019s Tenth Problem","author":"YV Matiyasevich","year":"1993","unstructured":"Matiyasevich, Y.V.: Hilbert\u2019s Tenth Problem. MIT Press, Cambridge (1993)"},{"key":"11_CR21","doi-asserted-by":"publisher","DOI":"10.1017\/9781108525749","volume-title":"Computable Structure Theory: Within the Arithmetic","author":"A Montalb\u00e1n","year":"2021","unstructured":"Montalb\u00e1n, A.: Computable Structure Theory: Within the Arithmetic. Cambridge University Press, Cambridge (2021)"},{"key":"11_CR22","unstructured":"Odifreddi, P.: Classical Recursion Theory, Stud. Logic Found. Math., vol.\u00a0125. Elsevier, Amsterdam (1992)"},{"key":"11_CR23","volume-title":"Systems that Learn","author":"DN Osherson","year":"1986","unstructured":"Osherson, D.N., Stob, M., Weinstein, S.: Systems that Learn. MIT Press, Cambridge (1986)"},{"issue":"2","key":"11_CR24","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/S0019-9958(82)80025-9","volume":"52","author":"DN Osherson","year":"1982","unstructured":"Osherson, D.N., Weinstein, S.: Criteria of language learning. Inf. Control 52(2), 123\u2013138 (1982). https:\/\/doi.org\/10.1016\/S0019-9958(82)80025-9","journal-title":"Inf. Control"},{"issue":"1","key":"11_CR25","doi-asserted-by":"publisher","first-page":"49","DOI":"10.2307\/2270581","volume":"30","author":"H Putnam","year":"1965","unstructured":"Putnam, H.: Trial and error predicates and the solution to a problem of Mostowski. J. Symb. Log. 30(1), 49\u201357 (1965). https:\/\/doi.org\/10.2307\/2270581","journal-title":"J. Symb. Log."},{"issue":"3","key":"11_CR26","doi-asserted-by":"publisher","first-page":"572","DOI":"10.2307\/2273757","volume":"46","author":"JB Remmel","year":"1981","unstructured":"Remmel, J.B.: Recursive isomorphism types of recursive Boolean algebras. J. Symb. Log. 46(3), 572\u2013594 (1981). https:\/\/doi.org\/10.2307\/2273757","journal-title":"J. Symb. Log."},{"issue":"2","key":"11_CR27","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/S0304-3975(00)00272-3","volume":"268","author":"F Stephan","year":"2001","unstructured":"Stephan, F., Ventsov, Y.: Learning algebraic structures from text. Theor. Comput. Sci. 268(2), 221\u2013273 (2001). https:\/\/doi.org\/10.1016\/S0304-3975(00)00272-3","journal-title":"Theor. Comput. Sci."},{"issue":"1\u20133","key":"11_CR28","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1016\/j.tcs.2008.02.021","volume":"397","author":"T Zeugmann","year":"2008","unstructured":"Zeugmann, T., Zilles, S.: Learning recursive functions: a survey. Theor. Comput. Sci. 397(1\u20133), 4\u201356 (2008). https:\/\/doi.org\/10.1016\/j.tcs.2008.02.021","journal-title":"Theor. Comput. Sci."}],"container-title":["Lecture Notes in Computer Science","Crossroads of Computability and Logic: Insights, Inspirations, and Innovations"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-95908-0_11","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T08:54:18Z","timestamp":1750323258000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-95908-0_11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9783031959073","9783031959080"],"references-count":28,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-95908-0_11","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"20 June 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Disclosure of Interests"}},{"value":"CiE","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Conference on Computability in Europe","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Lisbon","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Portugal","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"14 July 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"18 July 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"21","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"cie2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/sites.google.com\/view\/cie2025-computabilityineurope\/home","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}