An Introduction to the Scott Complexity of Countable Structures and a Survey of Recent Results

Abstract

Every countable structure has a sentence of the infinitary logic 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.

Computable scott sentences for quasi–Hopfian finitely presented structures

“…In [21] Scott proved that every countable structure is characterized up to isomorphism by a sentence in L ω1,ω . Recently, there has been considerable interest [2,6,8,9,10,11,15] in understanding how simple such a sentence can been chosen, in the sense of the following stratification of L ω1,ω from computable model theory:…”

“…computable Σ β -formula) for some β < α; (2.3) ϕ(x) is d-Σ α if it is a conjunction of a Σ α -formula and a Π α -formula. For a general overview of the motivation, history and literature surrounding the topic of optimal Scott sentences we refer the reader to the excellent survey [8]. In this paper we deal exclusively with optimal (computable) Scott sentences for finitely generated structures, and in particular for finitely presented groups; this topic has been central to the area.…”

“…that right-angled Coxeter groups of finite rank have a computable d-Σ 2 Scott sentence. Many questions on optimal Scott sentences for finitely generated structures are still open (see in particular [8,9] for an overview). The following two questions appear to be the most important in the area (cf.…”