What Is a Qualitative Calculus? A General Framework

What is a qualitative calculus? Many qualitative spatial and temporal calculi arise from a set of JEPD (jointly exhaustive and pairwise disjoint) relations: a stock example is Allen’s calculus, which is based on thirteen basic relations between intervals on the time line. This paper examines the construction of such a formalism from a general point of view, in order to make apparent the formal algebraic properties of all formalisms of that type. We show that the natural algebraic object governing this kind of calculus is a non-associative algebra (in the sense of Maddux), and that the notion of weak representation is the right notion for describing most basic properties. We discuss the ubiquity of weak representations in various guises, and argue that the fundamental notion of consistency itself can best be understood in terms of consistency of one weak representation with respect to another.