Characterizing downwards closed, strongly first order, relativizable dependencies

TL;DR

This paper characterizes a class of dependencies in team semantics that are strongly first order, downwards closed, and relativizable, showing they are definable via constancy atoms and exploring their safety properties.

Contribution

It proves that all nontrivial dependencies with these properties are definable in terms of constancy atoms and establishes their safety for downwards closed dependencies.

Findings

Nontrivial strongly first order dependencies are definable via constancy atoms.

Such dependencies are safe when combined with downwards closed dependencies.

The results unify and extend understanding of dependency atoms in team semantics.

Abstract

In Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First Order Logic is equivalent to some first order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms. Additionally, it is shown that any strongly first order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First Order Logic both the strongly first order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies.