Dependence logic with a majority quantifier

We study the extension of dependence logic D by a majority quantifier M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, D(M) captures the complexity class counting hierarchy.

💡 Research Summary

The paper investigates the logical and computational power of extending dependence logic (denoted D) with a majority quantifier, resulting in a new logic called D(M). Dependence logic enriches first‑order logic by adding dependence atoms of the form = (t₁,…,tₙ), which express that the value of the last term is functionally determined by the preceding ones. Its semantics is given in terms of teams—sets of assignments—rather than single assignments. It is well‑known that D is equivalent in expressive power to existential second‑order logic (ESO) and therefore captures the class NP.

The authors introduce a majority quantifier M that operates on teams. For a team X over a structure A, the formula M x φ holds if at least half of all functions F from X to the domain of A make the formula φ true when the variable x is interpreted via F. Formally, \