Dependency calculus: Reasoning in a general point relation algebra

Abstract

The point algebra is a fundamental formal calculus for spatial and temporal reasoning. We present a new generalization that meets all requirements to describe dependencies on networks. Applications range from traffic networks to medical diagnostics. We investigate satisfaction problems, tractable subclassses, embeddings into other relation algebras, and the associated interval algebra. 1 The Dependency Calculus Reasoning about complex dependencies between events is a crucial task. However, qualitative reasoning has so far concentrated on spatial and temporal issues. In contrast, we present a calculus [Ragni and Scivos, 2005], a proper generalization of the nonlinear relation algebra, created for specific questions of reasoning about consequences. This algebra, called dependency calculus (DC), meets all requirements to describe dependencies in networks. There are two aspects: dependencies of points are described by the point algebra PAdc, and of intervals by the associated interval algebra IAdc. For these we analyze questions concerning the satisfaction problems, and show the correspondence to other relation algebras. For this, we use an isomorphism preserving the tractability of subclasses. This method promises to structure the field of relation algebras and to transfer algebraic aspects and complexity results from one algebra to another. If we observe pollution in an ecosystem of flowing water, we can draw conclusions about pollution at other points (cf. Fig. 1). If pollution is found at D, F is polluted as well. It