On the Monadic Second-Order Transduction Hierarchy

TL;DR

This paper analyzes the hierarchy of classes of finite relational structures based on monadic second-order transductions, providing a complete description of the hierarchy and characterizing each level via tree-width variants.

Contribution

It offers a complete characterization of the monadic second-order transduction hierarchy for classes of incidence structures, including canonical representatives for each level.

Findings

Hierarchy is linear of order type ω+3

Each level characterized by a variant of tree-width

Canonical representatives include trees, paths, and grids

Abstract

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids.