Expressive power and succinctness of the positive calculus of binary relations

Abstract

In this paper, we study the expressive power and succinctness of the positive calculus of (binary) relations. We show that, for binary relations, (1) the calculus has the same expressive power as that of three-variable existential positive (first-order) logic, and (2) the calculus is exponentially less succinct than three-variable existential positive logic, namely, there is no subexponential-size translation from three-variable existential positive logic to the calculus. 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.