Abstract
Space, like time, is one of the most fundamental categories of human cognition. It structures all our activities and relationships with the external world. It also structures many of our reasoning capabilities: it serves as the basis for many metaphors, including temporal, and gave rise to mathematics itself, geometry being the first formal system known.
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Notes
The term relational is also found.
The domain can hold three types of entities — points, lines, and planes — with an incidence relation (not membership) between them. In Tarski (1959), an axiomatic system of Euclidean geometry based only on points is proposed; in this same system, continuity axioms are freed from any reference to arithmetic.
When transposed to AI, one can see reasoning in global spaces as model-based and reasoning in local spaces as deductive. However, model-based reasoning in global spaces is in general replaced by more efficient numerical algorithmic methods.
Defining “good” identity criteria is very difficult. Spatiotemporal continuity is, however, often adopted as one of these criteria, which in this case would make the definition of continuity of motion circular.
For a review of possible ways of defining points in terms of regions, see Gerla (1994).
When there are no boundaries (as, for instance, for the whole space), the subset is both open and closed.
For instance, in the standard topology of IR, IR2, or IR3, all open sets but the empty set are of the same dimensionality as the whole topological space.
In the TACITUS project (Hobbs et al., 1987; Hobbs et al., 1988), a point-based global space is proposed, with independent “scales” or granular partial order relations (one for each axis).
In all generality, the case A = 7r should be distinguished.
This means that these distance calculi require some underlying orientation system.
To restrict memory size, they are sometimes compacted and hierarchically organized, as in Samet (1984; 1989).
Regions extended throughout or, formally, regions x such that x = x and x=æ.
This way, a distinction can be made between jointing along a “fiat boundary” (Smith, 1995) (for example, the relation between two halves of a ball) and touching along real, objective, boundaries (for example, the relation between the ball and the ground).
Note that Clarke’s theory and RCC do not imply this last restriction.
One may question the cognitive or physical plausibility of these particular regions. Indeed, perfect spheres may be seen as entities as abstract as points. As a consequence, a region-based geometry relying on the existence of spheres may be no more attractive than a point-based geometry.
Except when there is a NTP relation between them or when they are equal, in which case orientation has no meaning. Notice that Hernandez assumes that a TPP relation can be combined with orientation, considering the position of the common boundary, thus forbidding this shared boundary to be very long or to be scattered around the regions.
This includes the case of the relation between one history and a static object, since immobility can be seen as being relative to a point of view.
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© 1997 Springer Science+Business Media Dordrecht
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Vieu, L. (1997). Spatial Representation and Reasoning in Artificial Intelligence. In: Stock, O. (eds) Spatial and Temporal Reasoning. Springer, Dordrecht. https://doi.org/10.1007/978-0-585-28322-7_1
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DOI: https://doi.org/10.1007/978-0-585-28322-7_1
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