{"title":"PAC和PACi的不可比度、连接和嵌入的有效概念类","authors":"D. G. M. Senadheera","doi":"10.1017/bsl.2022.39","DOIUrl":null,"url":null,"abstract":"Abstract The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples. Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees. Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them. Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesis E-mail: senadheerad@winthrop.edu URL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Effective Concept Classes of PAC and PACi Incomparable Degrees, Joins and Embedding of Degrees\",\"authors\":\"D. G. M. Senadheera\",\"doi\":\"10.1017/bsl.2022.39\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples. Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees. Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them. Abstract prepared by Dodamgodage Gihnee M. 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引用次数: 1
摘要
大概近似正确(PAC)学习是Leslie Valiant在1984年提出的一种机器学习模型。PACi可约性是指与大小和计算时间无关的PACi可约性。PAC学习中的这种可约性类似于图灵可计算性中的可约性。在PAC可约性下,概念类的排序是非线性的,即使在特定的具体例子中也是如此。由于与图灵可约性的相似性,我们怀疑在图灵不可比拟度中,PACi和PAC可约性可能存在不可比拟的PACi和PAC度。1957年Friedberg和1956年Muchnik分别用优先级构造法构造了不可比较度的可计算枚举集合A和B,解决了Post问题。我们将这一思想应用于PACi和PAC可约性,并构造了两个有效的概念类C和D,使得C不可约为D,反之亦然。在考虑PAC可约性时,有必要研究有效概念类的大小,因此我们使用Kolmogorov复杂度来获得大小。PAC学习模型中概念类的不可学习性可以用PAC不可比较度的存在来解释。与图灵跳迁类似,我们给出了零跳迁的有效概念类的跳迁操作。为了定义PACi度的跳零算子,构造了所有有效概念类的联接,并证明了它是最大元。已有的学位已经证明了许多特性。因此,我们可以探索证明这些性质的PACi和PAC度。但是如果我们证明从这些度到PACi和PAC度的嵌入,那么这些属性将对PACi和PAC度成立,而不需要明确地证明它们。摘要由Dodamgodage Gihnee M. Senadheera撰写,直接摘自论文E-mail: senadheerad@winthrop.edu URL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864
Effective Concept Classes of PAC and PACi Incomparable Degrees, Joins and Embedding of Degrees
Abstract The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples. Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees. Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them. Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesis E-mail: senadheerad@winthrop.edu URL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864