Hierarchies in Inclusion Logic with Lax Semantics

We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers, the number of inclusion atoms, or the arity of inclusion atoms. We show that the whole expressive power of inclusion logic can be captured using only five inclusion atoms in finite ordered models or, alternatively, only one universal quantifier in general. The arity hierarchy is shown to be strict by relating the question to the study of arity hierarchies in fixed point logics.