# Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

## TL;DR

This paper investigates the conditions under which dependency atoms in Team Semantics can be added to First Order Logic without increasing its expressive power, focusing on relativizable and non-jumping dependencies.

## Contribution

It characterizes strongly first order dependencies in Team Semantics under the assumptions of relativizability and non-jumping, and analyzes the impact of global disjunction.

## Key findings

- Global disjunction preserves expressiveness for strongly first order dependencies.
- Non-strongly first order dependencies cannot generally include global disjunction without increasing expressiveness.
- Provides a classification of dependency atoms based on their properties and impact on logic expressiveness.

## Abstract

Team Semantics generalizes Tarski's Semantics for First Order Logic by allowing formulas to be satisfied or not satisfied by sets of assignments rather than by single assignments. Because of this, in Team Semantics it is possible to extend the language of First Order Logic via new types of atomic formulas that express dependencies between different assignments. Some of these extensions are much more expressive than First Order Logic proper; but the problem of which atoms can instead be added to First Order Logic without increasing its expressive power is still unsolved. In this work, I provide an answer to this question under the additional assumptions (true of most atoms studied so far) that the dependency atoms are relativizable and non-jumping. Furthermore, I show that the global (or Boolean) disjunction connective can be added to any strongly first order family of dependencies without increasing the expressive power, but that the same is not true in general for non strongly first order dependencies.