We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison’s theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms.
{"title":"Primitive recursive ordered fields and some applications","authors":"Victor Selivanov, Svetlana Selivanova","doi":"10.3233/com-210386","DOIUrl":"https://doi.org/10.3233/com-210386","url":null,"abstract":"We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison’s theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135251560","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}
Pub Date : 2022-02-01DOI: 10.1007/978-3-030-83202-5
G. Tourlakis
{"title":"Computability","authors":"G. Tourlakis","doi":"10.1007/978-3-030-83202-5","DOIUrl":"https://doi.org/10.1007/978-3-030-83202-5","url":null,"abstract":"","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":"116 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86803719","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}
This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.
{"title":"Dimension spectra of lines1","authors":"Neil Lutz, D. M. Stull","doi":"10.3233/com-190292","DOIUrl":"https://doi.org/10.3233/com-190292","url":null,"abstract":"This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.","PeriodicalId":42452,"journal":{"name":"Computability-The Journal of the Association CiE","volume":"29 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81551859","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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