Measuring the Complexity of Countable Presburger Models

TL;DR

This paper explores the complexity classification of Presburger models using Scott analysis and degree spectra, constructing Presburger groups from linear orders to analyze their structural properties.

Contribution

It introduces methods to analyze Presburger models' complexity via Scott sentences and degree spectra, linking linear orders to Presburger groups.

Findings

Characterization of Scott sentence complexities for Presburger models

Construction of Presburger groups from linear orders preserving structural features

Insights into the degree spectra of Presburger models

Abstract

We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger arithmetic. Many of our results will be achieved by showing how given a linear order $\mathcal{L}$, we can construct a Presburger group $P_\mathcal{L}$ that maintains much of the structure of $\mathcal{L}$.