Expressive Power and Succinctness of the Positive Calculus of Relations

Abstract

In this paper, we study the expressive power and succinctness of the positive calculus of relations. We show that (1) the calculus has the same expressive power as that of three-variable existential positive (first-order) logic in terms of binary relations, and (2) the calculus is exponentially less succinct than three-variable existential positive logic, namely, there is no polynomial-size translation from three-variable existential positive logic to the calculus, whereas there is a linear-size translation in the converse direction. Additionally, we give a more fine-grained expressive power equivalence between the (full) calculus of relations and three-variable first-order logic in terms of the quantifier alternation hierarchy. It remains open whether the calculus of relations is also exponentially less succinct than three-variable first-order logic.