{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T15:35:18Z","timestamp":1747150518719,"version":"3.40.5"},"reference-count":14,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2021,11,20]],"date-time":"2021-11-20T00:00:00Z","timestamp":1637366400000},"content-version":"vor","delay-in-days":19,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11725103"],"award-info":[{"award-number":["11725103"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2021,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond \u03c9<jats:sup>2<\/jats:sup>. In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov\u00a0hierarchy.<\/jats:p>","DOI":"10.1002\/malq.202100004","type":"journal-article","created":{"date-parts":[[2021,11,20]],"date-time":"2021-11-20T20:15:28Z","timestamp":1637439328000},"page":"506-513","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Normalizing notations in the Ershov hierarchy"],"prefix":"10.1002","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8342-1964","authenticated-orcid":false,"given":"Cheng","family":"Peng","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences Nankai University Tianjin City 300071 China"}]}],"member":"311","published-online":{"date-parts":[[2021,11,20]]},"reference":[{"key":"e_1_2_6_2_1","doi-asserted-by":"crossref","unstructured":"M. M.Arslanov The Ershov hierarchy in:Computability in Context. Computation and Logic in the Real World edited byS. 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