Dependence logic with a majority quantifier

We study the extension of dependence logic D by a majority quantifier M over finite structures. Dependence logic [19] extends first-order logic by dependence atomic formulas =(t1, . . . , tn) the intuitive meaning of which is that the value of the term tn is completely determined by the values of t1, . . . , tn−1. While in first-order logic the order of quantifiers solely determines the dependence relations between variables, in dependence logic more general dependencies between variables can be expressed. Historically dependence logic was preceded by partially ordered quantifiers (Henkin quantifiers) of Henkin [8] and Independence-Friendly (IF) logic of Hintikka and Sandu [9]. It is known that both IF logic and dependence logic are equivalent to existential second-order logic ESO in expressive power. From the point of view of descriptive complexity theory, this means that dependence logic captures the class NP. The framework of dependence logic has turned out be flexible to allow interesting generalizations. For example, the extensions of dependence logic in terms of so-called