Jochen Renz

Qualitative spatial representation and reasoning

Publisher Summary Early attempts at qualitative spatial reasoning within the qualitative reasoning (QR) community led to the poverty conjecture. The need for spatial representations and spatial reasoning is ubiquitous in artificial intelligence (AI) from robot planning and navigation to interpreting visual inputs to understanding natural language. In all these cases, the need to represent and reason about spatial aspects of the world is of key importance. Related fields of research such as geographic information science (GIScience) have also driven the spatial representation and reasoning community to produce efficient, expressive, and useful calculi. There has been considerable research in spatial representations that are based on metric measurements, in particular within the vision and robotics communities, and also on raster and vector representations in GIScience. This chapter focuses on symbolic and, in particular, qualitative representations. The challenge of qualitative spatial reasoning (QSR) is to provide calculi that allow a machine to represent and reason with spatial entities without resort to the traditional quantitative techniques prevalent in, for example, computer graphics or computer vision communities.

Qualitative spatial reasoning with topological information

Background.- Qualitative Spatial Representation and Reasoning.- The Region Connection Calculus.- Cognitive Properties of Topological Spatial Relations.- Computational Properties of RCC-8.- A Complete Analysis of Tractability in RCC-8.- Empirical Evaluation of Reasoning with RCC-8.- Representational Properties of RCC-8.- Conclusions.- A. Enumeration of the Relations of the Maximal Tractable Subsets of RCC-8.

Qualitative direction calculi with arbitrary granularity

Binary direction relations between points in two-dimensional space are the basis to any qualitative direction calculus. Previous calculi are only on a very low level of granularity. In this paper we propose a generalization of previous approaches which enables qualitative calculi with an arbitrary level of granularity. The resulting calculi are so powerful that they can even emulate a quantitative representation based on a coordinate system. We also propose a less powerful, purely qualitative version of the generalized calculus. We identify tractable subsets of the generalized calculus and describe some applications for which these calculi are useful.

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 Allens 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.

Efficient methods for qualitative spatial reasoning

The theoretical properties of qualitative spatial reasoning in the RCC-8 framework have been analyzed extensively. However, no empirical investigation has been made yet. Our experiments show that the adaption of the algorithms used for qualitative temporal reasoning can solve large RCC-8 instances, even if they are in the phase transition region - provided that one uses the maximal tractable subsets of RCC-8 that have been identified by us. In particular, we demonstrate that the orthogonal combination of heuristic methods is successful in solving almost all apparently hard instances in the phase transition region up to a certain size in reasonable time.

Combining topological and size information for spatial reasoning

Information about the size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we introduce four classes of qualitative and metric size constraints, and we study their integration with the Region Connection Calculus RCC-8, a well-known approach to qualitative spatial reasoning with topological relations. We propose a new path-consistency algorithm for combining RCC-8 relations and qualitative size relations. The algorithm is complete for deciding satisfiability of an input set of topological constraints over one of the three maximal tractable subclasses of RCC-8 containing all the basic relations. Moreover, its time complexity is cubic and is the same as the complexity of the best-known method for deciding satisfiability when only these topological relations are considered. We also provide results on finding a consistent scenario in cubic time for these combined classes. Regarding metric size constraints, we first study their combination with RCC-8 and we show that deciding satisfiability for the combined sets of constraints is NP-hard, even when only the RCC-8 basic relations are used. Then we introduce RCC-7, a subalgebra of RCC-8 that can be used for applications where spatial regions cannot partially overlap. We show that reasoning with the seven RCC-7 basic relations and the universal relation is intractable, but that reasoning with the RCC-7 basic relations combined with metric size information is tractable. Finally, we give a polynomial algorithm for the latter case and a backtracking algorithm for the general case.

A Cognitive Assessment of Topological Spatial Relations: Results from an Empirical Investigation

Whether or not a formal approach to spatial relations is a cognitively adequate (the term will be explicated in this paper) model of human spatial knowledge is more often based on the intuition of the researchers than on empirical data In contrast, the research reported here is concerned with an empirical assessment of one of the three general classes of spatial relations, namely topological knowledge. In the reported empirical investigation, subjects had to group numerous spatial configurations consisting of two circles with respect to their similarity. As is well known, such tasks are solved on the basis of underlying spatial concepts. The results were compared with the RCC-theory and Egenhofers approach to topological relations and support the assumption that both theories are cognitively adequate in a number of important aspects.

A Canonical Model of the Region Connection Calculus

Although the computational properties of the Region Connection Calculus RCC-8 are well studied, reasoning with RCC-8 entails several representational problems. This includes the problem of representing arbitrary spatial regions in a computational framework, leading to the problem of generating a realization of a consistent set of RCC-8 constraints. A further problem is that RCC-8 performs reasoning about topological space, which does not have a particular dimension. Most applications of spatial reasoning, however, deal with two- or three-dimensional space. Therefore, a consistent set of RCC-8 constraints might not be realizable within the desired dimension. In this paper we address these problems and develop a canonical model of RCC-8 which allows a simple representation of regions with respect to a set of RCC-8 constraints, and, further, enables us to generate realizations in any dimension d = 1. For three-and higher-dimensional space this can also be done for internally connected regions.

Towards Cognitive Adequacy of Topological Spatial Relations

Qualitative spatial reasoning is often considered to be akin to human reasoning. This, however, is mostly based on the intuition of researchers rather than on empirical data. In this paper we continue our effort in empirically studying the cognitive adequacy of systems of topological relations. As compared to our previous empirical investigation [7], we partially lifted constraints on the shape of regions in configurations that we presented subjects in a grouping task.With a high level of agreement, subjects distinguished between different possibilities of how spatial regions can touch each other. Based on the results of our investigation, we propose to develop a new system of topological relations on a finer level of granularity than previously considered.

Combining Topological and Qualitative Size Constraints for Spatial Reasoning

Information about the relative size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we combine a simple framework for reasoning about qualitative size relations with the Region Connection Calculus RCC-8, a widely studied approach for qualitative spatial reasoning with topological relations. Reasoning about RCC-8 relations is NP-hard, but a large maximal tractable subclass of RCC-8 called H8 was identified. Interestingly, any constraint in RCC-8 - H8 can be consistently reduced to a constraint in H8, when an appropriate size constraint between the spatial regions is supplied. We propose an O(n3) time path-consistency algorithm based on a novel technique for combining RCC-8 constraints and relative size constraints, where n is the number of spatial regions. We prove its correctness and completeness for deciding consistency when the input contains topological constraints in H8. We also provide results on finding a consistent scenario in O(n3) time both for combined topological and relative size constraints, and for topological constraints alone. This is an O(n2) improvement over the known methods.