Order-Invariant Types and Their Applications

TL;DR

This paper applies model-theoretic types to analyze order-invariant properties, demonstrating their expressibility and providing new theorems that connect logical definability with combinatorial properties in structures like trees.

Contribution

It introduces the novel application of types to order-invariant properties and proves key theorems linking logic and combinatorics in this context.

Findings

Order-invariant MSO properties are expressible in MSO with counting quantifiers over trees.

An analog of the Feferman-Vaught theorem for order-invariant properties is established.

The approach bridges model theory and combinatorics for order-invariant properties.

Abstract

Our goal is to show that the standard model-theoretic concept of types can be applied in the study of order-invariant properties, i.e., properties definable in a logic in the presence of an auxiliary order relation, but not actually dependent on that order relation. This is somewhat surprising since order-invariant properties are more of a combinatorial rather than a logical object. We provide two applications of this notion. One is a proof, from the basic principles, of a theorem by Courcelle stating that over trees, order-invariant MSO properties are expressible in MSO with counting quantifiers. The other is an analog of the Feferman-Vaught theorem for order-invariant properties.