{"title":"There is a deep 1-generic set","authors":"Ang Li","doi":"arxiv-2409.00631","DOIUrl":null,"url":null,"abstract":"An infinite binary sequence is Bennett deep if, for any computable time\nbound, the difference between the time-bounded prefix-free Kolmogorov\ncomplexity and the prefix-free Kolmogorov complexity of its initial segments is\neventually unbounded. It is known that weakly 2-generic sets are shallow, i.e.\nnot deep. In this paper, we show that there is a deep 1-generic set.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00631","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
An infinite binary sequence is Bennett deep if, for any computable time
bound, the difference between the time-bounded prefix-free Kolmogorov
complexity and the prefix-free Kolmogorov complexity of its initial segments is
eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e.
not deep. In this paper, we show that there is a deep 1-generic set.
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