{"title":"多项式时间图灵 Mitoticity 和算术层次结构","authors":"A. H. Mokatsian","doi":"10.1134/s1054661824010127","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Let <span>\\(\\omega \\)</span> be the set of all nonnegative integers. Let <b>P</b> be a class of problems recognized by deterministic Turing machines, which run in polynomial time. It is known that effective enumeration of the sets of the class <b>P</b> (namely, <span>\\({{P}_{0}},{{P}_{1}}\\)</span>, …, <span>\\({{P}_{i}}\\)</span>, …) exists and thus <span>\\({\\mathbf{P}} = \\{ {{P}_{i}}\\,|\\,i \\in \\omega \\} .\\)</span> Note that for each <span>\\(i\\)</span>, <span>\\({{P}_{i}}\\)</span> is a set of strings that are sequences of 0s and 1s. Based on available numbering of computably enumerable (c.e.) sets <span>\\({{\\{ {{W}_{i}}\\} }_{{i \\in \\omega }}}\\)</span>, a sequence of sets of non-negative numbers <span>\\({{\\hat {P}}_{i}}\\)</span> is constructed such that there is an effective enumeration of them. Let us define <span>\\({\\mathbf{\\hat {P}}}\\)</span> as follows: <span>\\(~{\\mathbf{\\hat {P}}} = \\{ {{\\hat {P}}_{i}}\\,|\\,i \\in \\omega \\} \\)</span>. It’s obvious that it is possible to define such relations between the elements of the set of mentioned strings and between the elements of the set of nonnegative integers that these two sets will be isomorphic (with respect to the relations in question). The article shows that it is possible to define such relations between the elements of <span>\\({\\mathbf{P}}\\)</span> and between the elements of <span>\\(\\hat {{\\mathbf{P}}}\\)</span> that there will be homomorphic mappings from <span>\\({\\mathbf{P}}\\)</span> to <span>\\(\\hat {{\\mathbf{P}}}\\)</span> and vice versa, from <span>\\(\\hat {{\\mathbf{P}}}\\)</span> to <span>\\({\\mathbf{P}}\\)</span> (with respect to the relations in question). Based on the notions of <i>T</i>-mitoticity and <i>T</i>-autoreducibility, Ambos-Spies introduced the notions of <i>P‑T</i>-mitoticity, weakly <i>P</i>-<i>T</i>-mitoticity and <i>P</i>-<i>T</i>-autoreducibility. By analogy with the mentioned notions we introduce the notions of <span>\\(\\hat {P}\\)</span>-<i>T</i>-mitoticity, weakly <span>\\(\\hat {P}\\)</span>-<i>T</i>-mitoticity and <span>\\(\\hat {P}\\)</span>-<i>T</i>-autoreducibility. It is proved in the article that the index sets {<span>\\({\\text{z}}\\,|\\,{{{\\text{W}}}_{{\\text{z}}}}\\)</span> is <span>\\({{\\hat {P}}}\\)</span><i>-T-</i>mitotic}, <span>\\({\\text{\\{ z}}\\,|\\,{{{\\text{W}}}_{{\\text{z}}}}\\)</span> is weakly <span>\\({{\\hat {P}}}\\)</span><i>-T-</i>mitotic}, <span>\\({\\text{\\{ }}~{\\text{z}}\\,|\\,{{{\\text{W}}}_{{\\text{z}}}}\\)</span> is <span>\\({{\\hat {P}}}\\)</span>-<i>T-</i>autoreducible} and <span>\\({\\text{\\{ z}}\\,|\\,{{{\\text{W}}}_{{\\text{z}}}} \\in {\\mathbf{\\hat {P}}}\\} \\)</span> are <span>\\({{{\\mathbf{\\Sigma }}}_{3}}\\)</span>-complete.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":"27 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Polynomial Time Turing Mitoticity and Arithmetical Hierarchy\",\"authors\":\"A. H. Mokatsian\",\"doi\":\"10.1134/s1054661824010127\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>Let <span>\\\\(\\\\omega \\\\)</span> be the set of all nonnegative integers. Let <b>P</b> be a class of problems recognized by deterministic Turing machines, which run in polynomial time. It is known that effective enumeration of the sets of the class <b>P</b> (namely, <span>\\\\({{P}_{0}},{{P}_{1}}\\\\)</span>, …, <span>\\\\({{P}_{i}}\\\\)</span>, …) exists and thus <span>\\\\({\\\\mathbf{P}} = \\\\{ {{P}_{i}}\\\\,|\\\\,i \\\\in \\\\omega \\\\} .\\\\)</span> Note that for each <span>\\\\(i\\\\)</span>, <span>\\\\({{P}_{i}}\\\\)</span> is a set of strings that are sequences of 0s and 1s. Based on available numbering of computably enumerable (c.e.) sets <span>\\\\({{\\\\{ {{W}_{i}}\\\\} }_{{i \\\\in \\\\omega }}}\\\\)</span>, a sequence of sets of non-negative numbers <span>\\\\({{\\\\hat {P}}_{i}}\\\\)</span> is constructed such that there is an effective enumeration of them. Let us define <span>\\\\({\\\\mathbf{\\\\hat {P}}}\\\\)</span> as follows: <span>\\\\(~{\\\\mathbf{\\\\hat {P}}} = \\\\{ {{\\\\hat {P}}_{i}}\\\\,|\\\\,i \\\\in \\\\omega \\\\} \\\\)</span>. It’s obvious that it is possible to define such relations between the elements of the set of mentioned strings and between the elements of the set of nonnegative integers that these two sets will be isomorphic (with respect to the relations in question). The article shows that it is possible to define such relations between the elements of <span>\\\\({\\\\mathbf{P}}\\\\)</span> and between the elements of <span>\\\\(\\\\hat {{\\\\mathbf{P}}}\\\\)</span> that there will be homomorphic mappings from <span>\\\\({\\\\mathbf{P}}\\\\)</span> to <span>\\\\(\\\\hat {{\\\\mathbf{P}}}\\\\)</span> and vice versa, from <span>\\\\(\\\\hat {{\\\\mathbf{P}}}\\\\)</span> to <span>\\\\({\\\\mathbf{P}}\\\\)</span> (with respect to the relations in question). Based on the notions of <i>T</i>-mitoticity and <i>T</i>-autoreducibility, Ambos-Spies introduced the notions of <i>P‑T</i>-mitoticity, weakly <i>P</i>-<i>T</i>-mitoticity and <i>P</i>-<i>T</i>-autoreducibility. By analogy with the mentioned notions we introduce the notions of <span>\\\\(\\\\hat {P}\\\\)</span>-<i>T</i>-mitoticity, weakly <span>\\\\(\\\\hat {P}\\\\)</span>-<i>T</i>-mitoticity and <span>\\\\(\\\\hat {P}\\\\)</span>-<i>T</i>-autoreducibility. It is proved in the article that the index sets {<span>\\\\({\\\\text{z}}\\\\,|\\\\,{{{\\\\text{W}}}_{{\\\\text{z}}}}\\\\)</span> is <span>\\\\({{\\\\hat {P}}}\\\\)</span><i>-T-</i>mitotic}, <span>\\\\({\\\\text{\\\\{ z}}\\\\,|\\\\,{{{\\\\text{W}}}_{{\\\\text{z}}}}\\\\)</span> is weakly <span>\\\\({{\\\\hat {P}}}\\\\)</span><i>-T-</i>mitotic}, <span>\\\\({\\\\text{\\\\{ }}~{\\\\text{z}}\\\\,|\\\\,{{{\\\\text{W}}}_{{\\\\text{z}}}}\\\\)</span> is <span>\\\\({{\\\\hat {P}}}\\\\)</span>-<i>T-</i>autoreducible} and <span>\\\\({\\\\text{\\\\{ z}}\\\\,|\\\\,{{{\\\\text{W}}}_{{\\\\text{z}}}} \\\\in {\\\\mathbf{\\\\hat {P}}}\\\\} \\\\)</span> are <span>\\\\({{{\\\\mathbf{\\\\Sigma }}}_{3}}\\\\)</span>-complete.</p>\",\"PeriodicalId\":35400,\"journal\":{\"name\":\"PATTERN RECOGNITION AND IMAGE ANALYSIS\",\"volume\":\"27 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-04-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"PATTERN RECOGNITION AND IMAGE ANALYSIS\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1134/s1054661824010127\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1054661824010127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Polynomial Time Turing Mitoticity and Arithmetical Hierarchy
Abstract
Let \(\omega \) be the set of all nonnegative integers. Let P be a class of problems recognized by deterministic Turing machines, which run in polynomial time. It is known that effective enumeration of the sets of the class P (namely, \({{P}_{0}},{{P}_{1}}\), …, \({{P}_{i}}\), …) exists and thus \({\mathbf{P}} = \{ {{P}_{i}}\,|\,i \in \omega \} .\) Note that for each \(i\), \({{P}_{i}}\) is a set of strings that are sequences of 0s and 1s. Based on available numbering of computably enumerable (c.e.) sets \({{\{ {{W}_{i}}\} }_{{i \in \omega }}}\), a sequence of sets of non-negative numbers \({{\hat {P}}_{i}}\) is constructed such that there is an effective enumeration of them. Let us define \({\mathbf{\hat {P}}}\) as follows: \(~{\mathbf{\hat {P}}} = \{ {{\hat {P}}_{i}}\,|\,i \in \omega \} \). It’s obvious that it is possible to define such relations between the elements of the set of mentioned strings and between the elements of the set of nonnegative integers that these two sets will be isomorphic (with respect to the relations in question). The article shows that it is possible to define such relations between the elements of \({\mathbf{P}}\) and between the elements of \(\hat {{\mathbf{P}}}\) that there will be homomorphic mappings from \({\mathbf{P}}\) to \(\hat {{\mathbf{P}}}\) and vice versa, from \(\hat {{\mathbf{P}}}\) to \({\mathbf{P}}\) (with respect to the relations in question). Based on the notions of T-mitoticity and T-autoreducibility, Ambos-Spies introduced the notions of P‑T-mitoticity, weakly P-T-mitoticity and P-T-autoreducibility. By analogy with the mentioned notions we introduce the notions of \(\hat {P}\)-T-mitoticity, weakly \(\hat {P}\)-T-mitoticity and \(\hat {P}\)-T-autoreducibility. It is proved in the article that the index sets {\({\text{z}}\,|\,{{{\text{W}}}_{{\text{z}}}}\) is \({{\hat {P}}}\)-T-mitotic}, \({\text{\{ z}}\,|\,{{{\text{W}}}_{{\text{z}}}}\) is weakly \({{\hat {P}}}\)-T-mitotic}, \({\text{\{ }}~{\text{z}}\,|\,{{{\text{W}}}_{{\text{z}}}}\) is \({{\hat {P}}}\)-T-autoreducible} and \({\text{\{ z}}\,|\,{{{\text{W}}}_{{\text{z}}}} \in {\mathbf{\hat {P}}}\} \) are \({{{\mathbf{\Sigma }}}_{3}}\)-complete.
期刊介绍:
The purpose of the journal is to publish high-quality peer-reviewed scientific and technical materials that present the results of fundamental and applied scientific research in the field of image processing, recognition, analysis and understanding, pattern recognition, artificial intelligence, and related fields of theoretical and applied computer science and applied mathematics. The policy of the journal provides for the rapid publication of original scientific articles, analytical reviews, articles of the world''s leading scientists and specialists on the subject of the journal solicited by the editorial board, special thematic issues, proceedings of the world''s leading scientific conferences and seminars, as well as short reports containing new results of fundamental and applied research in the field of mathematical theory and methodology of image analysis, mathematical theory and methodology of image recognition, and mathematical foundations and methodology of artificial intelligence. The journal also publishes articles on the use of the apparatus and methods of the mathematical theory of image analysis and the mathematical theory of image recognition for the development of new information technologies and their supporting software and algorithmic complexes and systems for solving complex and particularly important applied problems. The main scientific areas are the mathematical theory of image analysis and the mathematical theory of pattern recognition. The journal also embraces the problems of analyzing and evaluating poorly formalized, poorly structured, incomplete, contradictory and noisy information, including artificial intelligence, bioinformatics, medical informatics, data mining, big data analysis, machine vision, data representation and modeling, data and knowledge extraction from images, machine learning, forecasting, machine graphics, databases, knowledge bases, medical and technical diagnostics, neural networks, specialized software, specialized computational architectures for information analysis and evaluation, linguistic, psychological, psychophysical, and physiological aspects of image analysis and pattern recognition, applied problems, and related problems. Articles can be submitted either in English or Russian. The English language is preferable. Pattern Recognition and Image Analysis is a hybrid journal that publishes mostly subscription articles that are free of charge for the authors, but also accepts Open Access articles with article processing charges. The journal is one of the top 10 global periodicals on image analysis and pattern recognition and is the only publication on this topic in the Russian Federation, Central and Eastern Europe.