Structural properties of the first-order transduction quasiorder

TL;DR

This paper explores the complex structure of first-order transduction quasiorders on infinite classes of finite graphs, revealing their algebraic properties and implications for class hierarchies and logical properties.

Contribution

It establishes a local normal form for FO transductions and characterizes the quasiorder as a bounded distributive join-semilattice, advancing understanding of their structural and expressive properties.

Findings

FO transduction quasiorder is a bounded distributive join-semilattice

Classes with pathwidth at most k form a strict hierarchy in the quasiorder

FO transductions of paths are perturbations of classes with bounded bandwidth

Abstract

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of \emph{additive} classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder.