N.M. Gotts. An axiomatic approach to topology for spatial information systems. Technical Report 96.25, School of Computer Studies, University of Leeds, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... based the universes R n for n 1 (in the definition of interior points x one should take n dimensional ffl neighborhoods of x) There can be different views on what sets of a topological space T = hU; Ii can be taken as interpretations of spatial regions (for a discussion consult [ Vieu, 1997; Gotts, 1996a ] 3 Following [ Gotts, 1996a ] we interpret regions only as Who was the first regular closed sets, i.e. sets X such that X = C IX (an alternative would be to take regular open sets X for which X = ICX) For example, the circle C(a; r) f(x 1 ; x 2 ) 2 R 2 : p (x 1 Gamma a 1 ) 2 ....

.... 1 (in the definition of interior points x one should take n dimensional ffl neighborhoods of x) There can be different views on what sets of a topological space T = hU; Ii can be taken as interpretations of spatial regions (for a discussion consult [ Vieu, 1997; Gotts, 1996a ] 3 Following [ Gotts, 1996a ] we interpret regions only as Who was the first regular closed sets, i.e. sets X such that X = C IX (an alternative would be to take regular open sets X for which X = ICX) For example, the circle C(a; r) f(x 1 ; x 2 ) 2 R 2 : p (x 1 Gamma a 1 ) 2 (x 2 Gamma a 2 ) 2 rg with ....

N.M. Gotts. An axiomatic approach to topology for spatial information systems. Technical Report 96.25, School of Computer Studies, University of Leeds, 1996.

....and functions which capture interesting and useful topological distinctions. The set of eight jointly exhaustive and pairwise disjoint (JEPD) relations illustrated in Figure 1 are one particularly useful set (often known as the RCC 8 calculus) A formal semantics for RCC has been given by [99, 52, 170]. Furthermore, a canonical 8 Cohn Hazarika QSR: An Overview model for arbitrary ground Boolean w s over RCC 8 atoms has been proposed [154] which is then utilised in a procedure to generate an actual 2D or 3D interpretation. Moreover this ensures (contrary to [3, 176] that an individual is ....

Gotts, N. M.: \An axiomatic approach to topology for spatial information systsems", Technical Report No. - 96.25, School of Computer Studies, University of Leeds, 1996.

....A B C D E F G H I J K L M N O P Fig. 3. Rough Location in a Regular Raster. 3.3 Location Exact Location. Every spatial object is exactly located at a single region of space in each moment of time (Casati Varzi 1995) This region may be a simple 7 For an extended discussion see (Cohn Gotts 1996). region of three dimensional space, for example, think of your body and the region of space it carves out of the air. The exact region of a spatial object may be a complex region, consisting of multiple regions of three dimensional space, as in the case of the exact region of the Hawaiian ....

....spatial objects, regions of space, and their compositional structure using the Boolean algebra (Halmos 1963, Stell Worboys 1997) of regular closed sets (Requicha 1977) Regular closed sets are sets which are equal to the closure of their interior. Regular closed point sets model regions of space (Gotts 1996). They are topologically well formed in that sense that they do not contain isolated points or in nitely thin spikes . In a Boolean algebra join, and meet, operations are de ned, which are interpreted as (regularized 9 ) union and intersection of regular closed sets. Let O be the Boolean ....

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Gotts, N. M. (1996), An axiomatic approach to topology for spatial information systems, Technical Report 96.25, School of Computer Studies.

....sum and product, a unary operation, complement, and a binary relation, connection. Sums and products of regions correspond to unions and intersections of regions. The complement of a region is that region outside it, and two regions are connected if they overlap or touch. In a recent report [22], Gotts considers the question of what particular structures are models for the RCC axioms. Gotts shows that certain topological spaces, the regular connected ones, provide models of the RCC axioms by taking a region to mean a nonempty regular closed set. Taking non empty regular open sets also ....

....them, reference is made to points within regions. Gotts observes that Using an interpretation expressed in terms of point sets might seem inconsistent with the spirit underlying the RCC approach. However no alternative has been worked out in any detail, The detailed proofs provided in [22] are needed in that they justify that certain structures are models of RCC. The disadvantage of the approach taken in [22] is not that the regions have points, but that these points are referred to in the justification. If the only known models of RCC required points to justify them, it would ....

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N. M. Gotts. An axiomatic approach to topology for spatial information systems. Research Report 96.25, University of Leeds, School of Computer Studies, 1996.

....p 2 U , the family of all neighbourhoods of p in U will be called the neighbourhood system of p in U . A neighbourhood system N has the property that every finite intersection of members of N belongs to N . It is possible to use any topological space which is a model for the RCC axioms. Gotts [10] has shown that every regular connected topological space is a model for the RCC axioms (see also Section 7) So, whenever we refer to a topological space in the remainder of the paper, we mean a regular connected topological space. 3 MODAL ENCODING CANONICAL MODELS 5 Relation Model Constraints ....

....in d dimensional space in times O(n 2 A Theta I Theta ) 7 Discussion Related Work There is some work on identifying canonical models for the RCC axioms, i.e. determining what mathematical structures fulfil all the RCC axioms, as, e.g. every region has a non tangential proper part. Gotts [10] found that every connected T 3 space, i.e. every connected and regular topological space is a model for the RCC axioms. Stell and Worboys [20] identified a whole class of models. Both approaches only describe models of the RCC axioms, i.e. what kind of regions can be used at all. When ....

Nicholas M. Gotts. An axiomatic approach to topology for spatial information systems. Technical Report 96-25, University of Leeds, School of Computer Studies, 1996.

.... (sharing a point, if one wants to think of regions as consisting of sets of points) C(x; y) In the RCC system this interpretation 2 is slightly changed to the closures of the regions sharing a point 3 this has the effect of collapsing the 2 A formal semantics for RCC has been given by [69, 37, 121]. 3 Actually, given the disdain of the RCC theory as presented in [108] for points, a distinction between a region, its closure and its interior, which it is argued has no relevance for the kinds of domain with which QSR is concerned (another reason for abandoning traditional mathematical ....

N M Gotts. An axiomatic approach to topology for spatial information systems. Technical report, Report 96.25, School of Computer Studies, University of Leeds, 1996.

....for each a # X and each closed set x not containing a, there are disjoint open sets w; v such that a # w; x # v. #X; # is called connected if the only open closed (clopen) sets are X and #. For properties of topological spaces not mentioned here, we invite the reader to consult [20] As shown in [21], a standard RCC model is the complete Boolean algebra RO#X# of regular open sets of a connected regular topological space #X; #, where for x; y # RO#X# xCy ### ## cl#x# # cl#y# ## #: 6.4) Theorem 6.3 gives the topological properties of the base relations and the building blocks of the others, ....

Gotts, N. M. (1996a). An axiomatic approach to topology for spatial information systems. Research Report 96.25, School of Computer Studies, University of Leeds.

....may hold between regions, partial binary operations sum and prod , and a unary operation compl . Eight axioms are expressed in a first order sorted logic using just the connection relation and the three operations. The axioms have intuitive meanings in which points play no role. In a recent report [Got96], Gotts considers the question of models for the RCC axioms. Gotts shows that certain topological spaces, the T 3 connected ones, provide models of the RCC axioms by taking a region to mean a nonempty regular closed set. Taking non empty regular open sets also gives a model. These models have the ....

N. M. Gotts. An axiomatic approach to topology for spatial information systems. Research Report 96.25, University of Leeds, School of Computer Studies, 1996.

....calculus for automated reasoning, thus making efficiency an important consideration. Since the present paper is concerned with certain theoretical issues, and not with implementation, I have not followed the original formulation. The description given below is slightly adapted from that given in [Got96, GGC96]. Full details are given below, but the most important ingredients in a model of the Region Connection Calculus are a set, R, of regions and a binary relation C on R. Various further binary relations on R are defined in terms of C , and these definitions are needed to state the axioms. These ....

....z) iff C (x; y) or C (x; z) R6. 8x; y; z 2 R Delta prod (y; z) 2 R implies C (x; prod (y; z) iff 9w 2 R Delta P (w; y) and P (w; z) and C (x; w) R7. 8x; y 2 R Delta prod (x; y) 2 R iff O (x; y) R8. 8x 2 R Delta 9y 2 R Delta NTPP (y; x) 2. 2 Examples of Models of RCC In a recent report [Got96], Gotts considers the question of what particular structures are models for the RCC axioms. Gotts shows that certain topological spaces, the regular connected ones, provide models of the RCC axioms by taking a region to mean a nonempty regular closed set. Taking non empty regular open sets also ....

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N. M. Gotts. An axiomatic approach to topology for spatial information systems. Research Report 96.25, University of Leeds, School of Computer Studies, 1996.

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Gotts, N. M.: 1996a, An axiomatic approach to topology for spatial information systems, Research Report 96.25, University of Leeds, School of Computer Studies.

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Gotts, N. M.: 1996a, An axiomatic approach to topology for spatial information systems, Technical report, Report 96.25, School of Computer Studies, University of Leeds.

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N.M. Gotts (1996). An axiomatic approach to topology for spatial information systems. Technical Report 96-25, Univ. of Leeds, School of Computer Studies.

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