On the uniform one-dimensional fragment

TL;DR

This paper surveys the properties of the uniform one-dimensional fragment of first-order logic, explores its relation to higher-arity description logics, and introduces a new description logic variant with new expressivity results.

Contribution

It provides a comprehensive survey of U1, analyzes its relationship with description logics like DLR_reg, and introduces a new description logic variant with novel expressivity insights.

Findings

U1 extends two-variable logic to all arities

Established connections between U1 and description logics like DLR_reg

Proved new results on the expressivity of U1 and related logics

Abstract

The uniform one-dimensional fragment of first-order logic, U1, is a formalism that extends two-variable logic in a natural way to contexts with relations of all arities. We survey properties of U1 and investigate its relationship to description logics designed to accommodate higher arity relations, with particular attention given to DLR_reg. We also define a description logic version of a variant of U1 and prove a range of new results concerning the expressivity of U1 and related logics.