J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163-174. Springer-Verlag, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....introduced intuitionistic logic Int; later he became also famous in topology. Orlov and Godel defined S4 in order to interpret intuitionistic logic in classical one. Open sets in a topological space form a complete Heyting algebra, which is a model of Int, and can be used as a model of RCC [ Stell and Worboys, 1979; Stell, 2000] 11 S4 u is expressive enough to encode the topological meaning of the RCC 8 predicates and that of Boolean region terms. 6 Indeed, let us denote the box and the diamond of S4 by, respectively, I and C (to emphasize their topological interpretation as the interior and closure ....

J.G. Stell and M.F. Worboys. The algebraic structure of sets of regions. In Stephen C. Hirtle and Andrew U. Frank, editors, Spatial Information Theory: a theoretical basis for GIS, Proceedings of COSIT'97, volume 1329 of LNCS, pages 163--174, Berlin, October 1979. Springer.

....of location caused by the vagueness of the object de nition. 4 Modeling Location 4.1 Modeling Exact Location Spatial objects and regions of space have compositional structure. We model 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 ....

Stell, J. & Worboys, M. (1997), The algebraic structure of sets of regions, in S. Hirtle, A. Frank & K. Kuhn, eds, `Conference on Spatial Information Theory, COSIT'97', Springer.

....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 additional constraints expressing relationships between regions are added, these results do not say anything about models anymore. 8 They 8 Consider ....

John Stell and Mike Worboys. The algebraic structure of sets of regions. In Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT'97), volume 1329 of Lecture Notes in Computer Science, pages 163--174, 1997.

.... (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 ....

....the distinction between being a firm and non firm tangential part (FTPP) i.e. whether the tangential connection is point like or not. Fig.3 illustrates another better interpretation, given some suitable distance metric, would be that C(x;y) means that the distance between x and y is zero, c.f. [121]. 4 And thus C(lefthalf,righthalf) holds too. Two doughnuts with degenerate holes Block minus block Cylinder surface Loop Double doughnut (topologically, a solid block) Doughnut with gap Torus Doughnut (or Solid Torus) degenerate hole surround A doughnut with a Fig. 2. It is possible to ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Proc COSIT97, LNCS. Springer Verlag, 1997.

....using an explicit construction of models (in terms of Kripke models) Along the same line, Renz [10] showed that any path consistent atomic network has a canonical model in the Euclidean n space, for any n 1, where the regions can be assumed to be connected if n 2. Finally, Stell and Worboys [12], extending a previous result of Gotts [5] identified a class of models for the RCC axioms. So, what more could be said about models of RCC 8 This paper is an attempt to put the results about the models of RCC 8 in a simple perspective. This will be accomplished by: Using the simple notion ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Proc. of COSIT'97, LNCS, pages 163--174, 1997.

....canonical models for the RCC axioms, i.e. determining what mathematical structures fulfill all the RCC axioms, as, e.g. every region has a non tangential proper part (Randell et al. 1992) Gotts (1996) found that every connected and regular topological space is a model for the RCC axioms. Stell and Worboys (1997) identified a whole class of models base on Heyting structures. Both approaches only describe models for the RCC axioms, i.e. what kind of regions can be used at all. When additional constraints expressing relationships between regions are added, these results do not say anything about models ....

J. Stell and M. Worboys (1997). The algebraic structure of sets of regions. In Proc. COSIT'97, LNCS 1329.

No context found.

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163-174. Springer-Verlag, 1997.

No context found.

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Stephen C. Hirtle and Andrew U. Frank, editors, Proceedings of COSIT'97, volume 1329 of LNCS, pages 163-174. Springer, 1997.

.... relations between abstract graphs in the context of work on discrete representations of space [Ste00b] A data model for graphs in spatial databases has been investigated by Erwig and Guting [EG94] and a discussion of notions of part and complement for graphs appears in work by Stell and Worboys [SW97] As there is more than one notion of complement for graphs, some work is needed to investigate which of these is most appropriate to produce a scheme for relations between graphs. The ideas on vague regions in this paper could then be applied to handle vague graphs using one of the notions of ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163--174. Springer-Verlag, 1997.

....1992 paper was followed by a series of papers in which he and colleagues developed the theme of incorporating graph handling capabilities in database systems [Gut94,EG94,BG95] Erwig and Schneider [ES97] pose the question of the meaning of vagueness with reference to a graph. Stell and Worboys [SW97] have discussed the algebraic structure of the set of subgraphs of a graph. 3 Selection and Amalgamation A major motivation of this work is to clarify the distinction between selection and amalgamation generalization operations. In this section we explore the foundations of the concept less ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Hirtle and Frank [HF97], pages 163--174.

....below in section 5.3. The alternative is to relax the requirement that the union of all the subgraphs be G itself. Granulation by Node Partitioning The set of all subgraphs of a graph has a much richer algebraic structure than the set of all subsets of a set, as explained by Stell and Worboys [SW97] This allows us to define a node partition of G to be a set of subgraphs A i , i 2 I, such that A i and A j do not share nodes or edges when i 6= j, and such that every node, but not necessarily every edge, of G appears in some A i . This is equivalent to giving a partition of the set of nodes, ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Hirtle and Frank [HF97], pages 163--174.

....particularly interesting area for further work would be to consider relations between abstract graphs. A data model for graphs in spatial databases has been investigated by Erwig and Guting [EG94] and a discussion of notions of part and complement for graphs appears in work by Stell and Worboys [SW97] As there is more than one notion of complement for graphs, some work is needed to investigate which of these is most appropriate to produce a scheme for relations between graphs. The ideas on vague regions in this paper could then be applied to handle vague graphs using one of the notions of ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163--174. Springer-Verlag, 1997.

....intersection, join or union, and complement or negation) correspond to meaningful operations on crisp regions. The fact that models of RCC have the form of a Boolean algebra together with a connection relation satisfying certain axioms was demonstrated independently by D untsch [10] and by Stell [25, 23]. There is no corresponding agreement about the appropriate algebraic structure for indeterminate regions. One reason for this is the variety of di erent notions of indeterminate region, and the variety of closely related concepts such as vague regions. It is not our intention to deal with all ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163-174. Springer-Verlag, 1997.

....the open sets alone. Pointless topology has been well developed by mathematicians [27, 28, 55] but the subject does not seem to have received much attention in the spatial reasoning community. The possibility of using this kind of approach to studying models of RCC was raised by Stell and Worboys [49], but the details presented in the present paper had not been worked out at that stage. Although the approach taken in this paper to constructing models of RCC is moti3 vated by the basic idea of pointless topology, the constructions are based on structures more general than complete Heyting ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In S. C. Hirtle and A. U. Frank, editors, Spatial Information Theory, International Conference COSIT'97, Proceedings, volume 1329 of Lecture Notes in Computer Science, pages 163--174. Springer-Verlag, 1997.

....for discrete regions. Technically, the algebraic calculus arises from the bi Heyting algebra structure on the set of subgraphs of a graph (Reyes Zolfaghari 1996, Lawvere 1986) Although the possibility of applying this algebraic structure to spatial representation has been suggested already (Stell Worboys 1997, Stell 1999b) the details of its application to discrete spatial regions do not seem to have appeared before. One advantage of this algebraic calculus is that it promises to allow the extensive work on qualitative treatments of continuous space to be related to the developing theory for ....

....appeared before. One advantage of this algebraic calculus is that it promises to allow the extensive work on qualitative treatments of continuous space to be related to the developing theory for discrete space. This is because the point free approach to qualitative continuous space suggested in (Stell Worboys 1997) and studied in detail in (Stell 1999a) is based on closely related algebraic structures. There is evidence to suggest that an atomic version of the RCC system can be obtained using these structures. De nition 1 A discrete space (F; adj F ) consists of a set F and a re exive symmetric relation, ....

Stell, J. G. & Worboys, M. F. (1997), The algebraic structure of sets of regions, in Hirtle & Frank (1997), pp. 163-174.

No context found.

Stell, J.G. and Worboys, M.F. (1997). The algebraic structure of sets of regions, in Proceedings of the Conference on Spatial Information Theory, Pittsburgh, Hirtle, S. and Frank, A.U. (eds.), Springer Verlag, Lecture Notes in Computer Science 1329, pp. 163-174.

....general, a lattice which is both a Heyting algebra and a co Heyting algebra is called a bi Heyting algebra. Two of the examples of Heyting algebras we met earlier are in fact bi Heyting algebras. This is true of the subgraphs of a graph and of the two stage sets. Details of these cases appear in [SW97]. Some of the theory of bi Heyting algebras is developed in [RZ96] 8.2 Pseudosupplemented Distributive Lattices Definition 8.2 In a lattice, A, a pseudosupplement of a 2 A is an element m 2 A such that for all x in A, a x = iff m 6 x. Note that m is a pseudosupplement of a iff m is the ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Proceedings of Conference on Spatial Information Theory, COSIT'97, Lecture Notes in Computer Science. Springer-Verlag, to appear 1997.

....connection calculus (RCC) approach [GGC96] which develops work of Clarke [Cla81,Cla85] which itself can be traced back to Whitehead. RCC is a first order theory with a primitive relation of connection between regions. From the connection relation, other relations are derived. Stell and Worboys [SW97] have given a point free account of models of RCC. Another approach, also concerned with relationships between pairs of regions, is the work of Egenhofer and Franzosa. Given spatial regions A and B, with interiors A ffi , B ffi , and boundaries A, B respectively, we can measure how A ....

J. G. Stell and M. F. Worboys. The algebraic structure of sets of regions. In Hirtle and Frank [HF97], pages 163--174.

No context found.

Stell, J., and Worboys, M. F. The algebraic structure of sets of regions. In Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT 97) (1997), S. C. Hirtle and A. Frank, Eds., vol. 1329 of Lecture Notes in Computer Science, Springer{Verlag, pp. 163-174.

No context found.

STELL, J., AND WORBOYS, M. F. The algebraic structure of sets of regions. In Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT 97) (1997), S. C. Hirtle and A. Frank, Eds., vol. 1329 of Lecture Notes in Computer Science, Springer--Verlag, pp. 163--174.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC