{"title":"A quest for algorithmically random infinite structures","authors":"B. Khoussainov","doi":"10.1145/2603088.2603114","DOIUrl":null,"url":null,"abstract":"The last two decades have witnessed significant advances in the investigation of algorithmic randomness, such as Martin-Löf randomness, of infinite strings. In spite of much work, research on randomness of infinite strings has excluded the investigation of algorithmic randomness for infinite algebraic structures. The main obstacle in introducing algorithmic randomness for infinite structures is that many classes of infinite structures lack measure. More precisely, it is unclear how one would define a meaningful measure through which it would be possible to introduce algorithmic randomness for infinite structures. In this paper, we overcome this obstacle by proposing a limited amount of finiteness conditions on various classes of infinite structures. These conditions will enable us to introduce measure and, as a consequence, reason about algorithmic randomness. Our classes include finitely generated universal algebras, connected graphs and tress of bounded degree, and monoids. For all these classes one can introduce algorithmic randomness concepts and prove existence of random structures. In particular, we prove that Martin-Lóf random universal algebras, graphs, trees, and monoids exist. In the case of trees we show a stronger result that Martin-Löf random computably enumerable trees exist.","PeriodicalId":20649,"journal":{"name":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2014-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"14","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2603088.2603114","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 14

Abstract

The last two decades have witnessed significant advances in the investigation of algorithmic randomness, such as Martin-Löf randomness, of infinite strings. In spite of much work, research on randomness of infinite strings has excluded the investigation of algorithmic randomness for infinite algebraic structures. The main obstacle in introducing algorithmic randomness for infinite structures is that many classes of infinite structures lack measure. More precisely, it is unclear how one would define a meaningful measure through which it would be possible to introduce algorithmic randomness for infinite structures. In this paper, we overcome this obstacle by proposing a limited amount of finiteness conditions on various classes of infinite structures. These conditions will enable us to introduce measure and, as a consequence, reason about algorithmic randomness. Our classes include finitely generated universal algebras, connected graphs and tress of bounded degree, and monoids. For all these classes one can introduce algorithmic randomness concepts and prove existence of random structures. In particular, we prove that Martin-Lóf random universal algebras, graphs, trees, and monoids exist. In the case of trees we show a stronger result that Martin-Löf random computably enumerable trees exist.
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对算法随机无限结构的探索
在过去的二十年中,在算法随机性的研究方面取得了重大进展,例如无限字符串的Martin-Löf随机性。尽管已经做了大量的工作,但对无限弦的随机性的研究却排除了对无限代数结构的算法随机性的研究。在无限结构中引入算法随机性的主要障碍是许多类无限结构缺乏测度。更确切地说,目前还不清楚如何定义一个有意义的度量,通过这个度量,可以为无限结构引入算法随机性。在本文中,我们通过在各种无限结构上提出有限数量的有限条件来克服这一障碍。这些条件将使我们能够引入度量,并因此对算法随机性进行推理。我们的课程包括有限生成的全称代数,有界度的连通图和连通树,以及一元群。对于所有这些类,可以引入算法随机性概念并证明随机结构的存在性。特别地,我们证明了Martin-Lóf随机泛代数、图、树和monoids的存在。在树的情况下,我们给出了一个更强的结果Martin-Löf随机可计算枚举树的存在。
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