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Multidimensional Mereotopology

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Multidimensional Mereotopology Multidimensional Mereotopology Antony Galton School of Engineering, Computer Science and Mathematics University of Exeter Exeter EX4 4QF, UK Abstract are line, area, volume, edge, corner, and surface. Lower- To support commonsense reasoning about space, we require dimensional entities may arise as idealisations under coarse a qualitative calculus of spatial entities and their relations. granularity of what are really higher-dimensional, for ex- One requirement for such a calculus, which has not so far ample the conceptualisation of roads and rivers as line ob- been satisfactorily addressed in the mereotopological litera- jects in GIS. But sometimes we seem to need the notion of ture, is that it should be able to handle regions of different a strictly one- or two-dimensional entity inhabiting three- dimensions. Regions of the same dimension should admit dimensional space, for example the portal of Hayes’ Ontol- Boolean sum and product operations, but regions of differ- ogy of Liquids (Hayes 1985), which is defined as ‘a piece of ent dimensions should not. In this paper we propose a topo- surface which links two pieces of space and through which logical model for regions of different dimensions, based on objects and material can pass’. the idea that a region of positive codimension is a regular For the mereological component of spatial reasoning it is closed subset of the boundary of a region of the next higher important to have operations by which ‘new’ spatial enti- dimension. To satisfy the requirements of the commonsense ties can be derived from old, notably Boolean-like opera- theory, it is required that regions of the same dimension in tions of sum, product, and complement. In set theory these the model can be summed, and we show that this is always are modelled by union, intersection and complementation, the case. We conclude with a discussion of the possible ap- which form a true Boolean algebra, but in mereology it is plicability of the technical results to commonsense spatial often felt that there should be no ‘null’ entity correspond- reasoning. ing to the empty set, leading to calculi that are analogous to, but in many ways more complicated than, true Boolean Keywords algebras. An example of mereological sum applied to lower- dimensional entities would be the bringing together of a col- Qualitative spatial reasoning, mereotopology, dimension, lection of linear river stretches to form a branching river- boundary system. Introduction If we are to handle entities of different dimensions within a unified theoretical framework then we need to determine It is generally agreed that commonsense reasoning about how entities of different dimensions are related. We may space should be supported by a qualitative calculus of spa- broadly distinguish between bottom-up and top-down ap- tial entities and their relations. There is also a broad consen- proaches, according as higher-dimensional entities are de- sus that the most basic qualitative attributes and relations of rived from lower, or vice versa. The standard mathematical spatial entities are mereotopological, i.e., concerned with the point-set approach illustrates the bottom-up method: zero- relations of parthood and contact and other relations and at- dimensional points are taken as primitive, and lines, sur- tributes that may be derived from these, such as overlap, ex- faces and solids are constructed as sets of points. If arbi- ternal connection, and the distinction between tangential and trary point-sets are allowed to count as spatial entities, then non-tangential parts. But mereotopology alone is not suffi- we end up with an ontology far too rich for the purposes of cient to handle the full range of qualitative concepts impor- commonsense reasoning: arbitrary point-sets can exhibit all tant for commonsense spatial reasoning. One such concept manner of pathological behaviours, including extreme dis- which I shall not discuss here is convexity; another, which connection, fractal-type convolutions, and bizarre ‘mixed di- forms the central topic of this paper, is dimension. mension’ entities. Thus in the bottom-up approach, the pro- Qualitative spatial reasoning must engage with the con- cess of construction must be constrained in some way, for cept of dimension if it is to do justice to our common- example by allowing only simplicial complexes. sense apprehension of space. Many everyday spatial con- The top-down procedure is complementary to the bottom- cepts carry information about dimension: some examples up: starting with solids—three-dimensional chunks—as Copyright c 2004, American Association for Artificial Intelli- primitive entities, we define surfaces, lines and points as gence (www.aaai.org). All rights reserved. sets of solids. (Roughly, a lower-dimensional entity is de- KR 2004 45 fined as the set of all solids which we want to regard as But area A in the diagram is exactly such a region, which ‘containing’ that entity.) This approach was explored by de means that P is TPP to L. The same reasoning would ap- Laguna (1922) and Tarski (1956), who were motivated by ply to any proper part of L, from which we conclude that L the thought that the spatial entities which are in some sense has no non-tangential proper parts, thereby contradicting the the most ‘real’ are precisely the solid, three-dimensional ob- axiom. jects in the world around us, and the three-dimensional re- gions that they do, or can, occupy. Lower-dimensional en- L tities are conceived of as in some sense dependent on these, and the top-down approach affirms this dependence by actu- A ally deriving them from solids—notwithstanding the rather counter-intuitive flavour of the resulting characterisations. P In this paper I review a number of mereotopological schemes from the literature, focussing on whether and how Figure 1: Proper parthood for regions of positive codimen- they handle regions of different dimensions. I then propose sion a mathematical model within which we can define spatial regions of different dimensions in a way which does jus- tice to the essential insight that lower-dimensional regions To avoid this conclusion, we need to interpret RCC so EC EC arise as parts of the boundaries of higher-dimensional re- that A is not, after all, to P . Regions are so long gions. An important concept which will facilitate much as they are connected but do not overlap, so we need A and of the discussion is that of codimension: by this is meant P to either overlap or be disconnected. In a point-set topo- the number of dimensions by which a region falls short of logical interpretation in which regions are connected if their the dimensionality of the space in which it is considered to closures are non-disjoint, A is certainly connected to P . If be embedded. Thus a one-dimensional object embedded in A is open, it does not overlap P , whereas if it is closed it three-dimensional space has a codimension of 2. By ‘lower- has a point in common with P —but this is only overlap if dimensional’ regions is meant regions of positive codimen- a point counts as a region, in which case the notion of ex- sion. ternal connection disappears entirely, so all proper parts are non-tangential, contrary to the spirit of RCC. Thus while it Current approaches is possible in principle to interpret RCC in such a way that regions of positive codimension can be accommodated, to Regional Connection Calculus do so would deprive RCC of some of its expressive power, One of the best-known approaches to the logical codifica- since several of the defined predicates become null. Thus tion of commonsense mereotopological theory is the Re- RCC is fundamentally antagonistic to regions of positive co- gional Connection Calculus (RCC) of (Randell, Cui, & dimension. Cohn 1992). In the basic RCC-8 formulation, there is a sin- gle non-logical primitive, the binary relation C, interpreted Intersection matrices as ‘connection’ or ‘contact’. Additional relations are defined in terms of C, notably the following: Independently of RCC, Egenhofer introduced a method of capturing certain mereotopological relations between re- Part gions by means of matrices which record the nature of the P x; y z C z; x C z; y ( ) =def 8 ( ( ) ! ( )) intersections between salient parts of the regions (Egenhofer Proper part 1989; 1991). An example is the 9-intersection matrix, de- PP P P (x; y) =def (x; y) ^ : (y; x) fined for the regions X and Y as Overlap O(x; y) =def 9z(P(z; x) ^ P(z; y)) b(X) \ b(Y ) b(X) \ i(Y ) b(X) \ c(Y ) External connection i(X) \ b(Y ) i(X) \ i(Y ) i(X) \ c(Y ) c b c i c c ! EC(x; y) =def C(x; y) ^ :O(x; y) (X) \ (Y ) (X) \ (Y ) (X) \ (Y ) Tangential proper part where b(X), i(X), and c(X) are respectively the boundary, TPP(x; y) = PP(x; y) ^ 9z(EC(z; x) ^ EC(z; y)) def interior, and complement of X. In this context, these notions Non-tangential proper part must be interpreted as follows. The complement is always NTPP PP TPP (x; y) =def (x; y) ^ : (x; y) understood with respect to the embedding space, but the in- A key axiom of RCC is 8x9yNTPP(y; x); which says terpretation of the other terms depends on the dimension of that every region has a non-tangential proper part. The mo- X. For example, if a line segment S in three-dimensional tivation for this axiom is to ensure that space is not discrete, space is modelled as a subset of R3 with the usual topology, but it also succeeds in ruling out regions of positive codi- then the boundary of S is @(S) = S and the interior of S is mension.
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