Qualitative Reasoning about Relative Directions : Computational Complexity and Practical Algorithm

Abstract

Qualitative spatial reasoning (QSR) enables cognitive agents to reason about space using abstract symbols. Among several aspects of space (e.g., topology, direction, distance) directional information is useful for agents navigating in space. Observers typically describe their environment by specifying the relative directions in which they see other objects or other people from their point of view. As such, qualitative reasoning about relative directions, i.e., determining whether a given statement involving relative directions is true, can be advantageously used for applications, for example, robot navigation, computer-aided design and geographical information systems. Unfortunately, despite the apparent importance of reasoning about relative directions, QSR-research so far could not provide efficient decision procedures for qualitative reasoning about relative directions. Accordingly, the question about how to devise an efficient decision procedure for qualitative reasoning about relative directions has meanwhile turned to the question about whether an efficient decision procedure exists at all. Answering the latter existential question, which requires a formal analysis of relative directions from a computational complexity point of view, has remained an open problem in the field of QSR. The present thesis solves the open problem by proving that there is no efficient decision procedure for qualitative reasoning about relative directions, even if only left or right relations are involved. This is surprising as it contradicts the early premise of QSR believed by many researchers in and outside the field, that is, abstracting from an infinite domain to a finite set of relations naturally leads to efficient reasoning. As a consequence of this rather negative result, efficient reasoning with any of the well-known relative direction calculi (OPRAm, DCC, DRA, LR) is impossible. Indeed, the present thesis shows that all the relative direction calculi belong to one and the same class of ∃R-complete problems, which are the problems that can be reduced to the NP-hard decision problem of the existential theory of the reals, and vice versa. Nevertheless, in practice, many interesting computationally hard AI problems can be tackled by means of approximative algorithms and heuristics. In the same vein, the present thesis shows that qualitative reasoning about relative directions can also be tackled with approximative algorithms. In the thesis we develop the qualitative calculus SVm which allows for a practical algorithm for qualitative reasoning about relative directions. SVm also provides an effective semi-decision procedure for the OPRAm calculus, the most versatile one among the relative direction calculi. In this thesis we substantiate the usefulness of SVm by applying it in the marine navigation domain.