Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs (...) where possible, followed by a survey of recent advances. (shrink)

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we (...) describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work. (shrink)

Computable structures of Scott rank ${\omega_1^{CK}}$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of ${\mathcal{L}_{\omega_1 \omega}}$ , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank ${\omega_1^{CK}}$ whose computable infinitary theories are each ${\aleph_0}$ -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank ${\omega_1^{CK}}$ , which guarantee that the (...) resulting structure is a model of an ${\aleph_0}$ -categorical computable infinitary theory. (shrink)

Let L be a countable language, let I be an isomorphism-type of countable L-structures, and let a∈2ω. We say that I is a-strange if it contains a computable-from-a structure and its Scott rank is exactly ω1a. For all a, a-strange structures exist. Theorem : If C is a collection of ℵ1 isomorphism-types of countable structures, then for a Turing cone of a’s, no member of C is a-strange.