MoL-2011-01: Sequences with Trivial Initial Segment Complexity

MoL-2011-01: Sterkenburg, Tom Florian (2011) Sequences with Trivial Initial Segment Complexity. [Report]

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The field of algorithmic randomness is concerned with making precise the intuitive notion of the randomness of individual objects, and is grounded on concepts from computability theory. Not only do different formalisations of irregularity, incompressibility and unpredictability lead to the same class of random binary sequences, they also allow us to compare such sequences on their randomness or their power to find regularities in other sets. Much like the Turing-degrees of computational content, we can define degrees of randomness. A triviality notion with respect to such structures is that of the K-trivial sets, the sets all of whose initial segments are trivial in the sense that they are easily compressible. This thesis provides a general discussion of algorithmic randomness, as well as original results concerning two topics related to sequences with trivial initial segment complexity. First we apply the classical notion of splitting in the c.e. Turing-degrees to the c.e. degrees of randomness given by the LR-, K- and C-reducibilities. But the main topic is a question by Downey, Miller and Yu about the arithmetical complexity of the function that computes the finite number of K-trivial sets via a given constant. Representing these sets as paths of certain trees, we find a solution to this problem by inspecting the general complexity of calculating the number of paths of trees and reducing the complexity of our particular family of K-triviality trees.

Item Type: Report
Report Nr: MoL-2011-01
Series Name: Master of Logic Thesis (MoL) Series
Year: 2011
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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