Axiomatizing the Cyclic Interval Calculus

A model for qualitative reasoning about intervals on a cyclic time has been recently proposed by Balbiani and Osmani (Balbiani & Osmani 2000). In this formalism, the basic entities are intervals on a circle, and using considerations similar to Allen’s calculus, sixteen basic relations are obtained, which form a jointly disjunctive and pairwise distinct (JEPD) set of relations. The purpose of this paper is to give an axiomatic description of the calculus, based on the properties of the meets relation, from which all other fifteen relations can be deduced. We show how the corresponding theory is related to cyclic orderings, and use the results to prove that any countable model of this theory is isomorphic to the cyclic interval structure based on the rational numbers. Our approach is similar to Ladkin’s axiomatization of Allen’s calculus, although the cyclic structures introduce specific difficulties.