Sequences with Trivial Initial Segment Complexity
Tom Florian Sterkenburg
Abstract:
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 diļ¬erent
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.