Upwards Closed Dependencies in Team Semantics

TL;DR

This paper proves that adding certain upwards closed dependency atoms to first-order logic with team semantics does not increase its expressive power, and introduces a class of bounded dependencies with limitations on definability.

Contribution

It establishes the invariance of expressive power when adding upwards closed dependencies and defines bounded dependencies with non-definability results.

Findings

Adding upwards closed dependency atoms does not increase expressive power.

Negations of key dependency atoms can be added without increasing expressiveness.

Unbounded dependencies cannot be defined by bounded ones.

Abstract

We prove that adding upwards closed first-order dependency atoms to first-order logic with team semantics does not increase its expressive power (with respect to sentences), and that the same remains true if we also add constancy atoms. As a consequence, the negations of functional dependence, conditional independence, inclusion and exclusion atoms can all be added to first-order logic without increasing its expressive power. Furthermore, we define a class of bounded upwards closed dependencies and we prove that unbounded dependencies cannot be defined in terms of bounded ones.