The Geometry of L^k-Canonization I: Rosiness from Efficient Constructibility

TL;DR

This paper links the efficient reconstruction of finite models in certain theories to their rosiness and super-rosiness, revealing structural properties and amalgamation conditions.

Contribution

It establishes that theories of finite structures with efficient model recovery are necessarily rosy and super-rosy, connecting model-theoretic properties with computational efficiency.

Findings

Finite models can be recovered efficiently from diagrams of finite subsets.

Such theories are necessarily rosy and have super-rosy completions.

Efficient solutions to the Strong $L^k$-Canonization Problem imply certain amalgamation properties.

Abstract

We demonstrate that for the $k$-variable theory $T$ of a finite structure (satisfying certain amalgamation conditions), if finite models of $T$ can be recovered from diagrams of finite {\em subsets} of model of $T$ in a certain "efficient" way, then $T$ is rosy -- in fact, a certain natural $\aleph_0$-categorical completion $T^{\lim}$ of $T$ is super-rosy of finite $U^\thorn$-rank. In an appendix, we also show that any $k$-variable theory $T$ of a finite structure for which the Strong $L^k$-Canonization Problem is efficient soluble has the necessary amalgamation properties up to taking an appropriate reduct.