In mathematics, we know there are some concepts-objects, constructions, structures, proofs-
that are more complex and difficult to describe than others. Computable structure theory
quantifies and studies the complexity of mathematical structures, structures such as graphs,
groups, and orderings. Written by a contemporary expert in the subject, this is the first full
monograph on computable structure theory in 20 years. Aimed at graduate students and
researchers in mathematical logic, it brings new results of the author together with many …

We exploit properties of certain directed graphs, obtained from the families of sets with
special effective enumeration properties, to generalize several results in computable model
theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such
enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a
computable successor ordinal α, we transform a countable directed graph G into a structure
G∗ such that G has a Δα0 isomorphic copy if and only if G∗ has a computable isomorphic …