Complexity of Scott Sentences

TL;DR

This paper develops effective versions of results on Scott sentences, linking the complexity of Scott sentences to the definability of orbits in computable structures, with applications to finitely generated groups.

Contribution

It provides new effective characterizations of Scott sentences and orbit definitions, extending prior non-effective results to the computable setting.

Findings

Effective versions of Scott sentence results are established.

A computable finitely generated group has a computable d-Σ₂ Scott sentence iff its generating tuple's orbit is Π₁-definable.

The complexity of Scott sentences correlates with orbit definability in computable structures.

Abstract

We give effective versions of some results on Scott sentences. We show that if $\mathcal{A}$ has a computable $\Pi_\alpha$ Scott sentence, then the orbits of all tuples are defined by formulas that are computable $\Sigma_\beta$ for some $\beta <\alpha$. (This is an effective version of a result of Montalb\'{a}n.) We show that if a countable structure $\mathcal{A}$ has a computable $\Sigma_\alpha$ Scott sentence and one that is computable $\Pi_\alpha$, then it has one that is computable $d$-$\Sigma_\beta$ for some $\beta < \alpha$. (This is an effective version of a result of A. Miller.) We also give an effective version of a result of D. Miller. Using the non-effective results of Montalb\'{a}n and A. Miller, we show that a finitely generated group has a $d$-$\Sigma_2$ Scott sentence iff the orbit of some (or every) generating tuple is defined by a $\Pi_1$ formula. Using our effective results, we show that for a computable finitely generated group, there is a computable $d$-$\Sigma_2$ Scott sentence iff the orbit of some (every) generating tuple is defined by a computable $\Pi_1$ formula.