One-Variable Logic Meets Presburger Arithmetic

TL;DR

This paper introduces a new one-variable logic extended with Presburger constraints, demonstrating its NP-completeness and unifying various counting-based logical fragments.

Contribution

It presents the first NP-complete decision procedure for this extended one-variable logic, unifying multiple counting and cardinality fragments.

Findings

Proves NP-completeness of the logic

Provides an optimal algorithm for finite satisfiability

Unifies multiple counting-based logical fragments

Abstract

We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove NP-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.