# Two variable logic with ultimately periodic counting

arXiv: https://arxiv.org/abs/2006.01193 · HTML: https://mlsanity.com/paper/2006.01193/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## 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)

We consider the extension of two variable logic with quantifiers that state that the number of elements where a formula holds should belong to a given ultimately periodic set. We show that both satisfiability and finite satisfiability of the logic are decidable. We also show that the spectrum of any sentence is definable in Presburger arithmetic. In the process we present several refinements to the ``biregular graph method''. In this method, decidability issues concerning two-variable logics are reduced to questions about Presburger definability of integer vectors associated with partitioned graphs, where nodes in a partition satisfy certain constraints on their in- and out-degrees.