Date

2019

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mathematics

First Adviser

Stanley, Lee J.

Abstract

Algorithmic randomness is primarily concerned with quantifying the degree of randomness of infinite binary strings, and is usually carried out in the setting of Cantor space. One characterization of randomness involves prefixes ``being as hard as possible to describe". Also of interest are the infinite binary strings whose prefixes are as easy as possible to describe i.e., the $K$-trivial strings. We will study these strings in the setting of computable metric spaces, and investigate several definitions which attempt to correctly generalize $K$-triviality. We describe some of the difficulties inherent in a natural-seeming approach, and offer partial results where new definitions relate to a more established definition of $K$-triviality under the right conditions.

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