The random members of a $\Pi^0_1$ class

TL;DR

This paper investigates various notions of randomness for elements within a given $oldsymbol{ m f ext{ extdollar} ext{ extdollar}^0_1 ext{ extdollar} ext{ extdollar}}$ class, exploring their equivalences under homogeneity conditions and discussing implications for classes of positive measure.

Contribution

It introduces and compares multiple definitions of randomness for members of $oldsymbol{ m f ext{ extdollar} ext{ extdollar}^0_1 ext{ extdollar} ext{ extdollar}}$ classes, establishing their equivalence in homogeneous cases.

Findings

Different notions of randomness coincide in certain homogeneous $oldsymbol{ m f ext{ extdollar} ext{ extdollar}^0_1 ext{ extdollar} ext{ extdollar}}$ classes.

Homogeneity conditions influence the equivalence of randomness notions.

Discussion on random members of classes with positive measure.

Abstract

We examine several notions of randomness for elements in a given $\Pi^0_1$ class $\mathcal{P}$. Such an effectively closed subset $\mathcal{P}$ of $2^\omega$ may be viewed as the set of infinite paths through the tree $T_{\mathcal{P}}$ of extendible nodes of $\mathcal{P}$, i.e., those finite strings that extend to a member of $\mathcal{P}$, so one approach to defining a random member of $\mathcal{P}$ is to randomly produce a path through $T_{\mathcal{P}}$ using a sufficiently random oracle for advice. In addition, this notion of randomness for elements of $\mathcal{P}$ may be induced by a map from $2^\omega$ onto $\mathcal{P}$ that is computable relative to $T_{\mathcal{P}}$, and the notion even has a characterization in term of Kolmogorov complexity. Another approach is to define a relative measure on $\mathcal{P}$ by conditionalizing the Lebesgue measure on $\mathcal{P}$, which becomes interesting if $\mathcal{P}$ has Lebesgue measure 0. Lastly, one can alternatively define a notion of incompressibility for members of $\mathcal{P}$ in terms of the amount of branching at levels of $T_{\mathcal{P}}$. We explore some notions of homogeneity for $\Pi^0_1$ classes, inspired by work of van Lambalgen. A key finding is that in a specific class of sufficiently homogeneous $\Pi^0_1$ classes $\mathcal{P}$, each of these approaches coincides. We conclude with a discussion of random members of $\Pi^0_1$ classes of positive measure.