Overview
This research focuses on classifying the complexity of countable Presburger models. The classification employs two distinct analytical approaches: Scott analysis and degree spectra. A primary objective is to investigate the range of possible Scott sentence complexities and the spectrum of degrees achievable by models of Presburger arithmetic.
Approach
The methodologies utilized for classifying the complexity of Presburger models are Scott analysis and degree spectra. A significant tactic in achieving the research's results involves constructing a Presburger group, designated $P_\mathcal{L}$, from a given linear order $\mathcal{L}$. This construction is designed such that $P_\mathcal{L}$ preserves a substantial portion of the structural properties inherent to $\mathcal{L}$.
Findings
- The study classifies the complexity of Presburger models through Scott analysis.
- The study classifies the complexity of Presburger models through degree spectra.
- It investigates the possible Scott sentence complexities associated with models of Presburger arithmetic.
- It investigates the possible degree spectra associated with models of Presburger arithmetic.
- Many results are obtained by demonstrating how a Presburger group $P_\mathcal{L}$ can be constructed from a linear order $\mathcal{L}$, maintaining much of $\mathcal{L}$'s structure.