Gérard Ligozat

Reasoning about Cardinal Directions

THIS PAPER is about the complexity of reasoning about cardinal directions in 2D space. By cardinal directions, we mean North, South, East, West and the intermediate directions Northeast, Northwest, Southeast, and Southwest. To make things more concrete, consider the problem of picture description as presented by Chang and Jungert [2]: Figure 1 is a symbolic picture representing four cities, e.g. Paris (P ), Cologne (C ), Brussels (B ), and The Hague (H ). In terms of the language of 2D strings described by Chang and Jungert [2], the information about the locations of the four cities might be represented by the following 2D string:

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.

Qualitative triangulation for spatial reasoning

This paper presents a systematic way of defining qualitative calculi for spatial reasoning. These calculi, which derive from the concept of qualitative triangulation, allow inference about the relative relationships of punctual objects in two-dimensional space. After introducing the general concept of qualitative triangulation, we discuss the main aspects of some important members of this family of calculi, including the so-called flipflop calculus, which subsumes the relative calculus in dimension one, and the calculus introduced by Freksa (orientation-based spatial inference). This allows us to present in a general setting the notions of coarse and fine inference, as well as the conceptual neighborhood properties of sets of spatial relations. We also show how these calculi can be used for actual inference, and how switching from a particular calculus to a refinement of it can be used to strengthen the inference.

A Generic Toolkit for n-ary Qualitative Temporal and Spatial Calculi

Temporal and spatial reasoning is a central task for numerous applications in many areas of artificial intelligence. For this task, numerous formalisms using the qualitative approach have been proposed. Clearly, these formalisms share a common algebraic structure. In this paper we propose and study a general definition of such formalisms by considering calculi based on basic relations of an arbitrary arity. We also describe the QAT (the qualitative algebra toolkit), a JAVA constraint programming library allowing to handle constraint networks based on those qualitative calculi

’’Corner‘‘ Relations in Allen‘s algebra

This paper proves a key result in the maximality proof of ORD-Horn relations, namely, the fact that any subclass of Allens algebra which contains all atomic relations, is closed under conversion, intersection and composition, and contains a relation which is not ORD-Horn will contain one (in fact, two at least) of four specific relations, the “corner” relations. Our proof uses the structural properties of ORD-Horn relations, where the original proof was by machine enumeration.

Eligible and frozen constraints for solving temporal qualitative constraint networks

In this paper we consider the consistency problem for qualitative constraint networks representing temporal or spatial information. The most efficient method for solving this problem consists in a search algorithm using, on the one hand, the weak composition closure method as a local propagation method, and on the other hand, a decomposition of the constraints into subrelations of a tractable set. We extend this algorithm with the notion of eligibility and the notion of frozen constraints. The first concept allows to characterise constraints which will not be considered during the search. The second one allows to freeze constraints in order to avoid unnecessary updates.

Categorical methods in qualitative reasoning: the case for weak representations

This paper argues for considering qualitative spatial and temporal reasoning in algebraic and category-theoretic terms. A central notion in this context is that of weak representation (WR) of the algebra governing the calculus. WRs are ubiquitous in qualitative reasoning, appearing both as domains of interpretation and as constraints. Defining the category of WRs allows us to express the basic notion of satisfiability (or consistency) in a simple way, and brings clarity to the study of various variants of consistency. The WRs of many popular calculi are of interest in themselves. Moreover, the classification of WRs leads to non-trivial model-theoretic results. The paper provides a not-too-technical introduction to these topics and illustrates it with simple examples.

On the consistency problem for the INDU calculus

In this paper, we further investigate the consistency problem for the qualitative temporal calculus INDU introduced by A. K. Pujari et al. (1999). We prove the intractability of the consistency problem for the subset of preconvex relations. On the other hand, we show the tractability of strongly preconvex relations. Furthermore, we also define another interesting set of relations for which the consistency problem can be decided by a method similar to the usual path-consistency method.

When Tables Tell It All: Qualitative Spatial and Temporal Reasoning Based on Linear Orderings

In [8] Bennett, Isli and Cohn put out the following challenge to researchers working with theories based on composition tables (CT): give a general characterization of theories and relational constraint languages for which a complete proof procedure can be specified by a CT. For theories based on CTs, they make the distinction between a weak, consistency-based interpretation of the CT, and a stronger extensional definition. In this paper, we take up a limited aspect of the challenge, namely, we characterize a subclass of formalisms for which the weak interpretation can be related in a canonical way to a structure based on a total ordering, while the strong interpretations have the property of aleph-zero categoricity (all countable models are isomorphic). Our approach is based on algebraic, rather than logical, methods. It can be summarized by two keywords: relation algebra and weak representation.

From Language to Motion, and Back: Generating and Using Route Descriptions

Route descriptions are natural language productions in response to the question: How do I get from A to B?