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

arXiv: https://arxiv.org/abs/1304.0829 · HTML: https://mlsanity.com/paper/1304.0829/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2013.

## Built on by (most cited first, 1 of 1)

- [A Fast Model Counting Algorithm for Two-Variable Logic with Counting and Modulo Counting Quantifiers](https://mlsanity.com/paper/2605.03391/) (2026)

## Abstract (arXiv)

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.