We give an overview of an approach to qualitative spatial reasoning based on directional orientation information as available through perception processes or natural language descriptions. Qualitative orientations in 2-dimensional space are given by the relation between a point and a vector. The paper presents our basic iconic notation for spatial orientation relations that exploits the spatial structure of the domain and explores a variety of ways in which these relations can be manipulated and combined for spatial reasoning. Using this notation, we explore a method for exploiting interactions between space and movement in this space for enhancing the inferential power. Finally, the orientation-based approach is augmented by distance information, which can be mapped into position constraints and vice versa.

Abstract. In AI a large number of calculi for efficient reasoning about spatial and temporal entities have been developed. The most prominent temporal calculi are the point algebra of linear time and Allen’s interval calculus. Examples of spatial calculi include mereotopological calculi, Frank’s cardinal direction calculus, Freksa’s double cross calculus, Egenhofer and Franzosa’s intersection calculi, and Randell, Cui, and Cohn’s region connection calculi. These calculi are designed for modeling specific aspects of space or time, respectively, to the effect that the class of intended models may vary widely with the calculus at hand. But from a formal point of view these calculi are often closely related to each other. For example, the spatial region connection calculus RCC5 may be considered a coarsening of Allen’s (temporal) interval calculus. And vice versa, intervals can be used to represent spatial objects that feature an internal direction. The central question of this paper is how these calculi as well as their mutual dependencies can be axiomatized by algebraic specifications. This question will be investigated within the framework of the Common Algebraic Specification Language (CASL), a specification language developed by the Common Framework Initiative for algebraic specification and development (COFI). We explain scope and expressiveness of CASL by discussing the specifications of some of the calculi mentioned before. 1

We develop a theory for ternary point-based calculi such that the relations are invariant when all points are mapped by rotations, scalings or translations and propose methods to determine arbitrary transformations and compositions of such relations. We argue that calculi based on such relation systems should satisfy two criteria. First, the relation system should be closed under transformations, compositions and intersections and have a finite base that is jointly exhaustive and pairwise. This implies that the well-known path consistency algorithm can be used to conclude implicit knowledge without any loss of information. If this is the case, we call the calculus practical. Second, we say that a relation system is natural if all relations and their complements give rise to sets of points that are connected. The main result of the paper is then the identification of a maximally refined calculus amongst the practical natural RST calculi, which turns out to be very similar to Ligozat’s flip-flop calculus. From that it follows, e.g., that there is no finite refinement of the TPCC calculus by Moratz et al that is closed under transformations, composition, and intersection.

Abstract. A multitude of calculi for qualitative spatial reasoning (QSR) have been proposed during the last two decades. The number of practical applications that make use of QSR techniques is, however, comparatively small. One reason for this may be seen in the difficulty for people from outside the field to incorporate the required reasoning techniques into their software. Sometimes, proposed calculi are only partially specified and implementations are rarely available. With the SparQ toolbox presented in this text, we seek to improve this situation by making common calculi and standard reasoning techniques accessible in a way that allows for easy integration into applications. We hope to turn this into a community effort and encourage researchers to incorporate their calculi into SparQ. This text is intended to present SparQ to potential users and contributors and to provide an overview on its features and utilization. 1

Abstract. In the domain of qualitative constraint reasoning, a subfield of AI which has evolved in the last 25 years, a large number of calculi for efficient reasoning about space and time has been developed. Reasoning problems in such calculi are usually formulated as constraint satisfaction problems. For temporal and spatial reasoning, these problems often have infinite domains, which need to be abstracted to (finite) algebras in order to become computationally feasible. Ligozat [13] has argued that the notion of weak representation plays a crucial rôle: it not only captures the correspondence between abstract relations (in a relation algebra or non-associative algebra) and relations in a concrete domain, but also corresponds to algebraically closed constraint networks. In this work, we examine properties of the category of weak representations and treat the relations between partition schemes, non-associative algebras and concrete domains in a systematic way. This leads to the notion of semi-strong representation, which captures the correspondence between abstract and concrete relations better than the notion of weak representation does. The slogan is that semi-strong representations avoid unnecessary loss of information. Furthermore, we hope that the categorical perspective will help in the future to provide new insights on the important problem of determining whether algebraic closedness decides consistency of constraint networks. 1

Abstract Various calculi have been designed for qualitative constraintbased representation and reasoning. Especially for orientation calculi, it happens that the well-known method of algebraic closure cannot decide consistency of constraint networks, even when considering networks over base relations ( = scenarios) only. We show that this is the case for all relative orientation calculi capable of distinguishing between “left of” and “right of”. Indeed, for these calculi, it is not clear whether efficient (i.e. polynomial) algorithms deciding scenario-consistency exist. As a partial solution of this problem, we present a technique to decide global consistency in qualitative calculi. It is applicable to all calculi that employ convex base relations over the real-valued space R n and it can be performed in polynomial time when dealing with convex relations only. Since global consistency implies consistency, this can be an efficient aid for identifying consistent scenarios. This complements the method of algebraic closure which can identify a subset of inconsistent scenarios. Keywords: Qualitative Spatio-Temporal Reasoning 1

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