{"title":"可识别集的索引集在算术层次结构中的位置","authors":"Ulrike Brandt","doi":"10.1016/S0019-9958(86)80034-1","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in <em>Σ</em><sub>3</sub> ∩ <em>Π</em><sub>3</sub>. An identifiable set is presented with index set in <em>Σ</em><sub>2</sub> ∩ <em>Π</em><sub>3</sub> but neither in <em>Σ</em><sub>2</sub> nor in <em>Π</em><sub>2</sub>. Furthermore we show that there is no nonempty identifiable set with index set in <em>Σ</em><sub>1</sub>. In <em>Π</em><sub>1</sub> it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":"68 1","pages":"Pages 185-195"},"PeriodicalIF":0.0000,"publicationDate":"1986-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(86)80034-1","citationCount":"9","resultStr":"{\"title\":\"The position of index sets of identifiable sets in the arithmetical hierarchy\",\"authors\":\"Ulrike Brandt\",\"doi\":\"10.1016/S0019-9958(86)80034-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in <em>Σ</em><sub>3</sub> ∩ <em>Π</em><sub>3</sub>. An identifiable set is presented with index set in <em>Σ</em><sub>2</sub> ∩ <em>Π</em><sub>3</sub> but neither in <em>Σ</em><sub>2</sub> nor in <em>Π</em><sub>2</sub>. Furthermore we show that there is no nonempty identifiable set with index set in <em>Σ</em><sub>1</sub>. In <em>Π</em><sub>1</sub> it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.</p></div>\",\"PeriodicalId\":38164,\"journal\":{\"name\":\"信息与控制\",\"volume\":\"68 1\",\"pages\":\"Pages 185-195\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1986-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0019-9958(86)80034-1\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"信息与控制\",\"FirstCategoryId\":\"1093\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019995886800341\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995886800341","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
The position of index sets of identifiable sets in the arithmetical hierarchy
We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in Σ3 ∩ Π3. An identifiable set is presented with index set in Σ2 ∩ Π3 but neither in Σ2 nor in Π2. Furthermore we show that there is no nonempty identifiable set with index set in Σ1. In Π1 it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.
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