The Random Members of a ${\Pi }_{1}^{0}$ Class

Abstract

We examine several notions of randomness for elements in a given ${\Pi }_{1}^{0}$ class $\mathcal {P}$. Such an effectively closed subset $\mathcal {P}$ of 2^ωmay 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^ωonto $\mathcal {P}$ 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 ${\Pi }_{1}^{0}$ classes, inspired by work of van Lambalgen. A key finding is that in a specific class of sufficiently homogeneous ${\Pi }_{1}^{0}$ classes $\mathcal {P}$, all of these approaches coincide. We conclude with a discussion of random members of ${\Pi }_{1}^{0}$ classes of positive measure.