Unary counting quantifiers do not increase the expressive power of Presburger aritmetic: an alternative shorter proof

TL;DR

This paper offers an alternative, shorter proof that unary counting quantifiers do not enhance the expressive power of Presburger arithmetic, and discusses related complexity issues not addressed in the original presentation.

Contribution

It provides a new, concise proof of a known result and explores complexity aspects of Presburger arithmetic that were previously overlooked.

Findings

Unary counting quantifiers do not increase expressive power

A shorter proof of the main result is presented

Complexity issues in Presburger arithmetic are analyzed

Abstract

This work was presented in June 5-7, 2017 at the conference "Journ\'{e}es sur les Arithm\'{e}tiques Faibles -- Weak Arithmetics Days" held in Saint-Pertersburg of which no proceeding was ever published. It was not a new result but showed that a different approach is possible. The paper presented at ICALP 2024 addresses, among other problems, the complexity issues which were ignored in my 2017 talk.