Regular graphs and the spectra of two-variable logic with counting

TL;DR

This paper demonstrates that the spectra of two-variable first-order logic with counting quantifiers are semilinear and closed under complement, using graph-theoretic characterizations of regular and biregular graphs.

Contribution

It provides the first semilinear characterization of spectra for two-variable logic with counting and links model structures to regular graph properties.

Findings

Spectra are semilinear sets.

Spectra are closed under complement.

Models correspond to collections of regular and biregular graphs.

Abstract

The {\em spectrum} of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we show that when restricted to using only two variables, but allowing counting quantifiers, the spectra of first-order logic sentences are semilinear and hence, closed under complement. At the heart of our proof are semilinear characterisations for the existence of regular and biregular graphs, the class of graphs in which there are a priori bounds on the degrees of the vertices. Our proof also provides a simple characterisation of models of two-variable logic with counting -- that is, up to renaming and extending the relation names, they are simply a collection of regular and biregular graphs.