Computable Randomness, and Coding the Orbits of the Collatz Map

Graduation date:2018In this thesis I will look at a definition of computable randomness from Algorithmic Information Theory as defined by Andre Nies through the lens of Computable Analaysis as\ud defined by Klaus Weihrauch. I will show that despite the fact that these two paradigms\ud generate distinct classes of computable supermartingales, the class of sets on which no\ud computable supermartingale succeeds of either type is identical. Therefore, both theories\ud generate the same collection of computably random sets. I will then consider how one\ud might apply some of the techniques in Algorithmic Information Theory, including prefix\ud free codes and the Kraft Inequality, to the study of the Collatz Conjecture