{"title":"多项式跳转算子","authors":"Mike Townsend","doi":"10.1016/S0019-9958(86)80032-8","DOIUrl":null,"url":null,"abstract":"<div><p>For recursive sets <em>A</em>, define a complexity theoretic version of the ordinary recursion theoretic jump by setting <em>A′</em> equal to the canonical <em>NP<sup>A</sup></em>-complete set. Thus <em>A</em> < <em><sup>P</sup><sub>T</sub> A′</em> iff <em>P<sup>A</sup></em> ≠ <em>NP<sup>A</sup></em>. The <em>n</em>th jump, <em>A</em><sup>(<em>n</em>)</sup>, is defined by iteration. A jumps <em>n</em> times if <em>A</em> < <em><sup>P</sup><sub>T</sub> A′</em> < <em><sup>P</sup><sub>T</sub></em>… < <em><sup>P</sup><sub>T</sub> A</em><sup>(<em>n</em>)</sup>. It is straightforward that the jump operation is monotone. Post's theorem holds for the (relativized) polynomial hierarchy. We establish the following analogues of results in ordinary recursion theory: all relationships between pairs of polynomial Turing degrees and their jumps consistent with monotonicity can be realized by degrees which jump at least twice. For example, there are polynomially incomparable <em>A</em> and <em>B</em> with <em>A′</em> ≡ <em><sup>P</sup><sub>T</sub> B′</em>. Moreover, if for each recursive <em>D</em> the set of <em>E</em> such that <em>D</em> join <em>E</em> jumps at least <em>n</em> times is effectively comeager, then these relationships can be realized by degrees jumping at least <em>n</em> times. We also relativize some well-known results by showing that if <em>A′</em> is polynomially many-one reducible to the join of <em>A</em> and a (co-)sparse set, then <em>P<sup>A</sup></em> = <em>NP<sup>A</sup></em>; and if <em>A′</em> is polynomially Turing reducible to the join of <em>A</em> and an <em>NP<sup>A</sup></em>-(co-)sparse set, then the relativized polynomial hierarchy collapses to <em>Δ<sup>P,A</sup></em><sub>2</sub>.</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":"68 1","pages":"Pages 146-169"},"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)80032-8","citationCount":"1","resultStr":"{\"title\":\"A polynomial jump operator\",\"authors\":\"Mike Townsend\",\"doi\":\"10.1016/S0019-9958(86)80032-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For recursive sets <em>A</em>, define a complexity theoretic version of the ordinary recursion theoretic jump by setting <em>A′</em> equal to the canonical <em>NP<sup>A</sup></em>-complete set. Thus <em>A</em> < <em><sup>P</sup><sub>T</sub> A′</em> iff <em>P<sup>A</sup></em> ≠ <em>NP<sup>A</sup></em>. The <em>n</em>th jump, <em>A</em><sup>(<em>n</em>)</sup>, is defined by iteration. A jumps <em>n</em> times if <em>A</em> < <em><sup>P</sup><sub>T</sub> A′</em> < <em><sup>P</sup><sub>T</sub></em>… < <em><sup>P</sup><sub>T</sub> A</em><sup>(<em>n</em>)</sup>. It is straightforward that the jump operation is monotone. Post's theorem holds for the (relativized) polynomial hierarchy. We establish the following analogues of results in ordinary recursion theory: all relationships between pairs of polynomial Turing degrees and their jumps consistent with monotonicity can be realized by degrees which jump at least twice. For example, there are polynomially incomparable <em>A</em> and <em>B</em> with <em>A′</em> ≡ <em><sup>P</sup><sub>T</sub> B′</em>. Moreover, if for each recursive <em>D</em> the set of <em>E</em> such that <em>D</em> join <em>E</em> jumps at least <em>n</em> times is effectively comeager, then these relationships can be realized by degrees jumping at least <em>n</em> times. We also relativize some well-known results by showing that if <em>A′</em> is polynomially many-one reducible to the join of <em>A</em> and a (co-)sparse set, then <em>P<sup>A</sup></em> = <em>NP<sup>A</sup></em>; and if <em>A′</em> is polynomially Turing reducible to the join of <em>A</em> and an <em>NP<sup>A</sup></em>-(co-)sparse set, then the relativized polynomial hierarchy collapses to <em>Δ<sup>P,A</sup></em><sub>2</sub>.</p></div>\",\"PeriodicalId\":38164,\"journal\":{\"name\":\"信息与控制\",\"volume\":\"68 1\",\"pages\":\"Pages 146-169\"},\"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)80032-8\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"信息与控制\",\"FirstCategoryId\":\"1093\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019995886800328\",\"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/S0019995886800328","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
For recursive sets A, define a complexity theoretic version of the ordinary recursion theoretic jump by setting A′ equal to the canonical NPA-complete set. Thus A < PT A′ iff PA ≠ NPA. The nth jump, A(n), is defined by iteration. A jumps n times if A < PT A′ < PT… < PT A(n). It is straightforward that the jump operation is monotone. Post's theorem holds for the (relativized) polynomial hierarchy. We establish the following analogues of results in ordinary recursion theory: all relationships between pairs of polynomial Turing degrees and their jumps consistent with monotonicity can be realized by degrees which jump at least twice. For example, there are polynomially incomparable A and B with A′ ≡ PT B′. Moreover, if for each recursive D the set of E such that D join E jumps at least n times is effectively comeager, then these relationships can be realized by degrees jumping at least n times. We also relativize some well-known results by showing that if A′ is polynomially many-one reducible to the join of A and a (co-)sparse set, then PA = NPA; and if A′ is polynomially Turing reducible to the join of A and an NPA-(co-)sparse set, then the relativized polynomial hierarchy collapses to ΔP,A2.