Show description

10 • Hernandez et al. (1995) considers a variety ofrestrictions on the distance scales, such as monotonically increasing interval length along the scale, and thus obtains several composition tables. Distance addition in the general case can only be approximated by a disjunction of values. But even if triangle inequality is respected, its limit case, equality, cannot imply the collinearity of the three points. This is so because in a discrete scale, extreme values may absorb the others - for instance, d(a, b) = very close, d(b, c) = very close and d( a, c) = very close are possible simultaneously, whether the three points are aligned or not.

This is necessary in local spaces, where not all information is explicitly encoded, but not in global spaces in which spatial relations can be directly "read" or at least calculated numerically. The paradigmatic case of this kind of deduction, if not the most simple case, is theorem proving in axiomatic theories of space. The efficiency of such a task lies mainly on the choice of an adequate ontology. From this standpoint, higher order representations are to be avoided. But first-order theories do not necessarily guarantee tractability.

3. INTERVAL-BASED SPACES After the great impact of Allen's interval calculus (Allen, 1983) in AI (see Chapter 7), there have been several {similar} intents of extending it to a multidimensional domain (Giisgen, 1989; Mukerjee, 1989; Mukerjee and Joe, 1990). In this approach, spatial regions are represented by tuples of intervals that are the projections of the regions on the axes of a given absolute reference frame. Spatial relations between rectangular regions are then expressed by a tuple of relations between two intervals for each axis, in the relation algebra formalism.