# FO = FO3 for linear orders with monotone binary relations

## TL;DR

This paper proves that monadic first-order logic over linear orders with certain monotone binary relations has the three-variable property, extending known results and providing a new proof via translation to star-free PDL.

## Contribution

It generalizes the three-variable property to a broader class of linear orders with monotone relations and introduces a novel proof method using translation to star-free PDL.

## Key findings

- Monadic first-order logic has the three-variable property over these classes.
- The proof involves translating formulas into star-free PDL.
- The results extend known cases and answer open questions.

## Abstract

We show that over the class of linear orders with additional binary relations satisfying some monotonicity conditions, monadic first-order logic has the three-variable property. This generalizes (and gives a new proof of) several known results, including the fact that monadic first-order logic has the three-variable property over linear orders, as well as over (R,<,+1), and answers some open questions mentioned in a paper from Antonopoulos, Hunter, Raza and Worrell [FoSSaCS 2015]. Our proof is based on a translation of monadic first-order logic formulas into formulas of a star-free variant of Propositional Dynamic Logic, which are in turn easily expressible in monadic first-order logic with three variables.