Boolean Dependence Logic and Partially-Ordered Connectives

TL;DR

This paper introduces Boolean dependence logic, a new variant of dependence logic that uses Boolean dependence atoms to express quantification of relations, and compares its expressive power to other logical systems.

Contribution

It defines Boolean dependence logic, shows its expressive power matches dependence logic, and establishes a hierarchy among its syntactic fragments and related logics.

Findings

Boolean dependence logic's expressive power coincides with dependence logic.

Fragments of Boolean dependence logic match those of first-order logic with partially-ordered connectives.

A strict hierarchy exists among the syntactic fragments of Boolean dependence logic.

Abstract

We introduce a new variant of dependence logic called Boolean dependence logic. In Boolean dependence logic dependence atoms are of the type =(x_1,...,x_n,\alpha), where \alpha is a Boolean variable. Intuitively, with Boolean dependence atoms one can express quantification of relations, while standard dependence atoms express quantification over functions. We compare the expressive power of Boolean dependence logic to dependence logic and first-order logic enriched by partially-ordered connectives. We show that the expressive power of Boolean dependence logic and dependence logic coincide. We define natural syntactic fragments of Boolean dependence logic and show that they coincide with the corresponding fragments of first-order logic enriched by partially-ordered connectives with respect to expressive power. We then show that the fragments form a strict hierarchy.