{"title":"Chapter Seven. Embeddings of the 1-3-1 lattice","authors":"","doi":"10.1515/9780691200217-008","DOIUrl":"https://doi.org/10.1515/9780691200217-008","url":null,"abstract":"","PeriodicalId":297672,"journal":{"name":"A Hierarchy of Turing Degrees","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131214740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Chapter Three. The hierarchy of totally ɑ-c.a. degrees","authors":"","doi":"10.1515/9780691200217-004","DOIUrl":"https://doi.org/10.1515/9780691200217-004","url":null,"abstract":"","PeriodicalId":297672,"journal":{"name":"A Hierarchy of Turing Degrees","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124395237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"m-topped degrees","authors":"R. Downey, Noam Greenberg","doi":"10.2307/j.ctvss3wsw.9","DOIUrl":"https://doi.org/10.2307/j.ctvss3wsw.9","url":null,"abstract":"This chapter assesses m-topped degrees. The notion of m-topped degrees comes from a general study of the interaction between Turing reducibility and stronger reducibilities among c.e. sets. For example, this study includes the contiguous degrees. A c.e. Turing degree d is m-topped if it contains a greatest degree among the many one degrees of c.e. sets in d. Such degrees were constructed in Downey and Jockusch. The dynamics of the cascading phenomenon occurring in the construction of m-topped degrees strongly resemble the dynamics of the embedding of the 1–3–1 lattice in the c.e. degrees. Similar dynamics occurred in the original construction of a noncomputable left–c.e. real with only computable presentations, which was discussed in the previous chapter.","PeriodicalId":297672,"journal":{"name":"A Hierarchy of Turing Degrees","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123344814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Prompt permissions","authors":"R. Downey, Noam Greenberg","doi":"10.2307/j.ctvss3wsw.11","DOIUrl":"https://doi.org/10.2307/j.ctvss3wsw.11","url":null,"abstract":"This chapter explores prompt versions of all levels in the hierarchy. This generalises the already familiar notion of prompt permitting, which is the prompt version of simple permitting. Prompt array noncomputable permission, for example, allows one to construct a pair of separating classes whose elements form minimal pairs (Theorem 8.22). Meanwhile, traditional (non-prompt) array noncomputable permission only gives Turing incomparability. There are at least two ways to get such an embedding: either by quickly getting the precise number of permissions required; or by getting many permissions (cofinitely many), in which case one can wait for the permissions and do not need them promptly.","PeriodicalId":297672,"journal":{"name":"A Hierarchy of Turing Degrees","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127128377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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