Connection Matrices and the Definability of Graph Parameters

TL;DR

This paper extends the Finite Rank Theorem for connection matrices of graph parameters definable in CMSOL, demonstrating its broad applicability in establishing non-definability results and proving a Feferman-Vaught Theorem for CFOL.

Contribution

It provides a detailed proof of the Finite Rank Theorem for CMSOL-definable graph parameters and applies it to derive new non-definability results and a Feferman-Vaught Theorem for CFOL.

Findings

Finite Rank Theorem extended and proved in detail.

New non-definability results for graph properties and parameters.

Feferman-Vaught Theorem established for CFOL.

Abstract

In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers.