Index sets for Π01 classes

Index sets for classes

A class is an e ectively closed set of reals We study properties of these classes determined by cardinality measure and category as well as by the complexity of the members of a class P Given an e ective enumeration fPe e g of the classes the index set I for a certain property such as having positive measure is the set of indices e such that Pe has the property For example the index set of bina...

Non-branching degrees in the Medvedev lattice of Π01 classes

Special thanks to my advisor Steffen Lempp for his guidance and conversation. Abstract Always Sometimes 2 In this talk I will present the necessary background and then describe and characterize the non-branching degrees. Significantly, I will show three distinct classes of non-branching degrees. The Medvedev lattice of Π 0 1 classes is a lattice of certain subsets of 2 ω under a natural reducti...

Π01 Classes, Peano Arithmetic, Randomness, and Computable Domination

We present an overview of the topics in the title and of some of the key results pertaining to them. These have historically been topics of interest in computability theory and continue to be a rich source of problems and ideas. In particular, we draw attention to the links and connections between these topics and explore their significance to modern research in the field.

Index Sets and Presentations of Complexity Classes ( revised

This paper draws close connections between the ease of presenting a given complexity class C and the position of the index sets IC = f i : L(Mi) 2 C g and JC = f i : Mi is total ^ L(Mi) = 2 C g in the arithmetical hierarchy. For virtually all classes C studied in the literature, the lowest levels attainable are IC 2 P03 and JC 2 Q02; the rst holds i C is 02-presentable, and the second i C is re...

On Pre-θ-Open Sets and Two Classes of Functions