Some classical model theoretic aspects of bounded shrub-depth classes
Abstract
We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the class $\mathrm{TM}_{r, p}(d)$ of $p$-labeled arbitrary graphs whose underlying unlabeled graphs have tree models of height $d$ and $r$ labels. We show that this class satisfies an extension of the classical Löwenheim-Skolem property into the finite and for $\mathrm{MSO}$. This extension being a generalization of the small model property, we obtain that the graphs of $\mathrm{TM}_{r, p}(d)$ are pseudo-finite. In addition, we obtain as consequences entirely new proofs of a number of known results concerning bounded shrub-depth classes (of finite graphs) and $\mathrm{TM}_{r, p}(d)$. These include the small model property for $\mathrm{MSO}$ with elementary bounds, the classical compactness theorem from model theory over $\mathrm{TM}_{r, p}(d)$, and the equivalence of $\mathrm{MSO}$ and $\mathrm{FO}$ over $\mathrm{TM}_{r, p}(d)$ and hence over bounded shrub-depth classes. The proof for the last of these is via an adaptation of the proof of the classical Lindström's theorem characterizing $\mathrm{FO}$ over arbitrary structures.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.05799
- arXiv:
- arXiv:2010.05799
- Bibcode:
- 2020arXiv201005799S
- Keywords:
-
- Computer Science - Logic in Computer Science;
- 03C40;
- 03C52;
- 03C75;
- 03C13;
- 05C62;
- 05C38;
- 05C76
- E-Print:
- 26 pages