# Uniform One-Dimensional Fragment over Ordered Structures

## TL;DR

This paper investigates the computational complexity of the uniform one-dimensional fragment U1 over ordered structures, showing it remains NExpTime-complete, similar to FO2, but becomes undecidable with multiple orders.

## Contribution

It establishes the complexity of satisfiability for U1 over linearly ordered models and contrasts it with the undecidability when multiple orders are used.

## Key findings

- U1 satisfiability over linear orders is NExpTime-complete
- Transition from FO2 to U1 does not increase complexity in ordered models
- U1 becomes undecidable with multiple linear orders

## Abstract

The uniform one-dimensional fragment U1 is a recently introduced extension of the two-variable fragment FO2. The logic U1 enables the use of relation symbols of all arities and thereby extends the scope of applications of FO2. In this article we show that the satisfiability and finite satisfiability problems of U1 over linearly ordered models are NExpTime-complete. The corresponding problems for FO2 are likewise NextIime-complete, so the transition from FO2 to U1 in the ordered realm causes no increase in complexity. To contrast our results, we also establish that U1 with unrestricted use of two built-in linear orders is undecidable.