Hierarchies in Dependence Logic

Abstract

We study fragments D ( k ∀) and D ( k -dep) of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that, for any fixed signature, the fragments D ( k ∀) give rise to an infinite hierarchy with respect to expressive power. On the other hand, for the fragments D ( k -dep), a hierarchy theorem is otained only in the case the signature is also allowed to vary. For any fixed signature, this question is open and is related to the so-called Spectrum Arity Hierarchy Conjecture.