T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in the examples of optical, cartographic and semantic projection considered above. The grids of the latter satisfy a requirement to the effect that the cells (objects) within the target fit exactly to the corresponding cells of the user s grid. This condition can in various ways be weakened. Bittner and Stell (1998) offer an approach to spatial grids otherwise similar to the one advanced here but within which the restriction on cell object fit is relaxed through the notion of rough location. Smith and Brogaard (2000, 2001) show how the theory of true grids can be used to develop a theory of vagueness. ....

Bittner, T. and Stell, J. G. 1998 "A Boundary-Sensitive Approach to Qualitative Location", Annals of Mathematics and Artificial Intelligence, 24, 93--114.

....a cell to a guest in a hotel room) Partitions in our sense arise also when we make certain sorts of observations or experiments (compare an object in a cell to a bacterium in a petri dish) The requirement that an object must fit entirely within its corresponding cell can of course be weakened. (Bittner and Stell (1998) offer an approach to spatial partitions otherwise similar to the one advanced here but within which the restriction on cell object fit is relaxed through the notion of rough location. In Smith and Brogaard (2001) we argue that already the departure from the requirement of exact location allows ....

Bittner, T. and Stell, J. G. 1998 "A Boundary-Sensitive Approach to Qualitative Location," Annals of Mathematics and Artificial Intelligence, 24, 93--114.

....can be found in figure 1. Crisping Yolk Egg Figure 1: An Egg Yolk structure However, there is another notion of indefiniteness which may apply to a region: its location. 8] have proposed a qualitative calculus for representing and reasoning about the position of point like spatial entities. [27, 26] has proposed a technique for representing and reasoning about the location of spatial regions by locating them with respect to a background region partition (e.g. the States of the USA) using binary relations from a prexisting qualitative topological calculus (the RCC 8 calculus [9] In this ....

Bittner T. and Stell J. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, to appear.

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T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....(resolution) of a given partition and the number of sets we can represent exactly and the roughness of the approximation of the other sets. This will be discussed in Section 9. 8 T. Bittner Approximate Qualitative Temporal Reasoning 2.3.2. Approximation and temporal granularities In [9] and [43] the technique of approximation of subsets of a set was applied to regions of space by interpreting the three values fo, po, and no as different degree of spatial overlap. On the spatial interpretation these values measure the extent to which a region ( overlaps with the cells of the ....

....the interval and dashed lines signify the interval . imations. 4.1. Approximating regions 4.1.1. Boundary insensitive approximation Consider the set of regions, of a one dimensional space. By imposing a partition, on we can approximate elements of by elements of x [9]. That is, we approximate regions in by functions from to the set fo po no . The function which assigns to each region D its approximation will be denoted x c1 x . The value of x U is fo if covers all of the cell , it is po if covers some but not all of the ....

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T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....negation of somewhere i.e. nowhere, and ; as the conjunction somewhere and not everywhere. This threefold classi cation has been used by Worboys [Wor98b] as the basis of a theory of nite resolution spatial data. The system of three extents is not the only one, and Bittner and Stell [Bit97, BS98] have shown how it can be made more detailed by taking account of how the region being described interacts with the boundaries between each pair of cells in the granulation. While applications of the system of three extents have been studied, and more elaborate systems of purely spatial extents ....

....It should be possible to extend the system of eleven spatio temporal extents to a more detailed classi cation by taking into account the boundaries of the cells in the granulation. This would form a natural generalization to the spatiotemporal context of the researches of Bittner and Stell [Bit97, BS98] A related question is how relations between granularly represented spatio temporal regions can be described. What, for example, are the appropriate analogues of systems such as RCC5, RCC8, and the 4 and 9 intersection models One direction to investigate this would be by using the systems of ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Articial Intelligence, 24:93-114, 1998.

....partition) might overlap on their boundaries, but not their interiors. Because of the additional topological structure, it is possible to make a more detailed classification of overlaps between subsets and cells in the partition. An account of how this can be done was given in our earlier paper [BS98]. This is, however, only one of the directions in which the basic rough sets approach to approximation can be generalized to spatial approximation. Our concern in the present paper is relationships between spatial regions when these regions have been given approximate descriptions. The study of ....

....The domain of regions is equipped with a meet operation interpreted as the intersection of regions. In the domain of approximation functions the meet operation between regions is approximated by pairs of greatest minimal, min , and least maximal, max , meet operations on approximation mappings [BS98]. Consider the operations min and max on the set 3 = ffo; po; nog that are defined as follows. min no po fo no no no no po no no po fo no po fo max no po fo no no no no po no po po fo no po fo These operations extend to elements of G 3 (i.e. the set of functions from G to 3 ) by ....

[Article contains additional citation context not shown here]

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....the mapping, r, exists. Consequently, we can use the notion r(o) to refer on a formal level to the exact region of the object o 2 O. 9 See (Gotts 1996) or (Requicha 1977) 4. 2 Approximating Regions of Space We now review the notion of relationship mappings which was originally introduced by Bittner Stell (1998). In this paper we use relationship mappings in order to model rough location of analysis created at objects within the regional partition created by the underlying measurement process. Overlap Sensitive Approximation. Let the set 9 = f3; 0 3 g be a set of ways elements g 2 G can relate to an ....

....function p 3 applied to region r and partition region g returns (2; 3) if and only if g is a part of r. It yields (0 2 ; 3) if g and r overlap, without g being a part of r, and (0 2 ; 0 3 ) otherwise. Assuming 8g 2 G(g 6= the case p 8 (g; r) 2 and p 9 (g; r) 0 2 cannot occur (Bittner Stell 1998). The relationship function, p 3 , induces an approximation function 3 as discussed above. Consider gure 3. The graph of the approximation function ( 3 r) G 3 is: G A B . K L : 3 (0 2 ; 0 3 ) 0 2 ; 3) 2; 3) 0 2 ; 3) 4.3 Modeling Rough Location In this ....

Bittner, T. & Stell, J. G. (1998), `A boundary-sensitive approach to qualitative location ', Annals of Mathematics and Articial Intelligence 24, 93-114.

....For example, merging of two adjacent areas in an areal decomposition is equivalent in the dual graph to amalgamation of two nodes and elimination of the edge between them. One direction for further work using dual graphs would be to develop the boundary sensitive approach to qualitative location [BS98,Ste99] in a multi resolution context. A detailed exploration of generalizations of areal decompositions of the plane in terms of simplification of the dual graph will be reported in later publications. ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....Relations characterizing part location can be de ned, for example, by taking boundary parts into account or ignoring them. This results in boundary sensitive and boundary insensitive relations. In this paper we discuss boundary insensitive relations. Boundary sensitive relations were discussed by Bittner Stell (1998) or Bittner (1999) 2. There are multiple ways how (object) parts of spatial objects can be related to regional parts of spatial regions. Above we distinguished, for example, the relations WL(x; y) P L(x; y) and GL(x; y) Partitioning Sets of Part Location Relations. The de nitions of the part ....

....partition the domain of pairs of objects. Jointly exhaustive and pair wise disjoint sets of relations are particularly important for the formalization of rough location. They provide the basis for the formalization of rough location by means of rough sets (Pawlak 1982) and location mappings (Bittner Stell 1998, Bittner 1999) Based on the de nitions of Casati Varzi (1995) the following sets of JEPD part location predicates can be (trivially) de ned: Name Intended Meaning Relation Set Contained Sensitive the region of x is either fWL(x; y) WL(x; y)g a part of y or not Containment Sensitive y is ....

Bittner, T. & Stell, J. G. (1998), `A boundary-sensitive approach to qualitative location ', Annals of Mathematics and Articial Intelligence 24, 93-114.

....is explained. Two kinds of vague graphs are identified in section 4. In section 5 four notions of granularity for graphs are presented. Some of the structures discussed in section 5 have connections to the boundary sensitive approach to qualitative location developed by Bittner and Stell [Bit97,BS98] These connections are the subject of section 6. Conclusions and further work are outlined in section 7. 2 Informal Examples 2.1 Vague Subgraphs A network of roads can be represented as a graph. We might then want to select the part of the road network which is currently flooded. Such a query ....

....relevant local area networks in detail. 6 Boundary Sensitive Location The regular granular subgraphs for graph equivalence granulation, described in section 5. 2, provide an alternative way of presenting the boundary sensitive approach to qualitative location developed by Bittner and Stell [Bit97,BS98] Suppose we have a partition of the plane, or of some bounded portion of it, where the cells of the partition are regular closed regions which may intersect only in their boundaries. If we want to describe a region r, which we will assume to be regular open, we can do this by describing the ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....partition) might overlap on their boundaries, but not their interiors. Because of the additional topological structure, it is possible to make a more detailed classification of overlaps between subsets and cells in the partition. An account of how this can be done was given in our earlier paper [BS98]. This is, however, only one of the directions in which the basic rough sets approach to approximation can be generalized to spatial approximation. Our concern in the present paper is relationships between spatial regions when these regions have been given approximate descriptions. The study of ....

....The domain of regions is equiped with a meet operation interpreted as the intersection of regions. In the domain of approximation functions the meet operation between regions is approximated by pairs of greatest minimal, min , and least maximal, max , meet operations on approximation mappings [BS98]. Consider the operations min and max on the set 3 = ffo; po; nog that are defined as follows. min no po fo no no no no po no no po fo no po fo max no po fo no no no no po no po po fo no po fo These operations extend to elements of G 3 (i.e. the set of functions from G to 3 ) by ....

[Article contains additional citation context not shown here]

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....(Carnap 1966) functional) relation of (exact) location between them and concentrate on the approximation of the exact regions (of objects) with respect to an underlying regional partition. Moreover, I concentrate on temporal regions and approximations of temporal regions. This paper builds on (Bittner Stell 1998) and (Bittner Stell 2000) in which various ways of providing qualitative approximations of regions with respect to a partition of the plane as well as reasoning about those approximations were described. Bittner Stell 2000) showed that approximate qualitative reasoning is based on: 1) ....

....of the lower RCC 15 1 relations between connected intervals. Approximations Approximating regions Boundary insensitive approximation Consider the set of regions, R, of a one dimensional space. By imposing a partition, G, on R we can approximate elements of R by elements of G 3 (Bittner Stell 1998). That is, we approximate regions in R by functions from G to the set 3 = ffo; po; nog. The function which assigns to each region r 2 R its approximation will be denoted 3 : R G 3 . The value of ( 3 r)g is fo if r covers all the of the cell g, it is po if r covers some but not all of ....

[Article contains additional citation context not shown here]

Bittner, T., and Stell, J. G. 1998. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence 24:93--114.

....The domain of regions is equipped with a meet operation interpreted as the intersection of regions. In the domain of approximation functions the meet operation between regions is approximated by pairs of greatest minimal, and least maximal, meet operations on approximation mappings [BS98] Consider the operations and on the set 3 = ffo; po; nog that are defined as follows: no po fo no no no no po no no po fo no po fo no po fo no no no no po no po po fo no po fo These operations extend to elements of G 3 (i.e. the set of functions from G to 3 ) by (X Y )g = ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....the Canadian GEOID network is gratefully acknowledged. i.e. different kinds of things, qualitative relations between lines [All83] and qualitative relations between regions [CBGG97] At the formal level I will use a language based on the qualitative notion of boundary sensitive rough location [BS98] to describe built environments. I am going to show that this notion provides the basis for the formal description of built environments and for the evaluation of the complexity of navigation. This paper is structured as follows. I start with an informal analysis of the ontological makeup of ....

....problem (2) In this paper I focus on problem (1) This provides the basis for solving problem (2) 3 Approximating regions and relations between approximations In this section I shortly discuss the formal notions needed in the remainder of this paper. These notions were originally introduced in [BS98] and [BS00] For an extended discussion see also [Bit99] 3.1 Approximations Suppose we have a space R of detailed or precise regions. By imposing a partition, G, on R we can approximate elements of R by elements of bs G G . That is, we approximate regions in R by functions from G G to the ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....indeterminate regions arise is from geographic data obtained by a process of cartographic generalisation. For example, a region given crisply on a large scale map may become indeterminate when represented on a small scale map only capable of showing a much lower level of detail. Bittner and Stell [2] have provided an account of how regions can be approximated by descriptions with respect to a frame of reference. Their work is concerned with how a single region relates to the cells in a frame of reference, rather than relationships between regions themselves. 1.2.3 Main features of our ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Articial Intelligence, 24:93-114, 1998. 28

....descriptions of space are not restricted in their application to commonsense tasks such as the above examples. They are, for example, relevant to Geographic Information Systems (GIS) where qualitative descriptions of how two geographic regions are related to each other have been widely studied [7, 20]. QSR also impinges on linguistics and psychology, having application to understanding spatial expressions within natural language [19] and wayfinding both in small scale and large scale environments [42] Models of space which have been proposed as appropriate for tackling the kinds of problems ....

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

....has been studied in AI and logic, for example in van Lambalgen s logic of vision (van Lambalgen van der Does 1997) and in the theory of rough sets (Or lowska 1998) the topic is under represented in the literature on space. Some papers which do tackle the topic are (Euzenat 1994, Euzenat 1995, Bittner Stell 1998), and literature in the GIS context such as (Timpf Frank 1997, Worboys 1998, Stell Worboys 1998, Stell Worboys 1999) 1.3 STRUCTURE OF THE PAPER The present paper proposes a formal framework for the representation of discrete spatial regions at multiple levels of detail. In presenting this ....

Bittner, T. & Stell, J. G. (1998), `A boundarysensitive approach to qualitative location', Annals of Mathematics and Articial Intelligence 24, 93{ 114.

.... will focus on qualitative aspects [FR93, Fre91, Coh97] i.e. different kinds of things [SM98] qualitative relations between lines [All83] and qualitative relations between regions [CBGG97] At the formal level a language based on the qualitative notion of boundary sensitive rough approximations [BS98] is T. Bittner The qualitative structure of built environments 3 used to describe built environments. This notion provides the basis for the formal description of built environments and for the evaluation of the complexity of navigation. This paper is structured as follows. It starts with an ....

....hold between objects x and y approximated by X and Y . This will be used in the remainder of the paper in order to give a formal account of the ontological constraints O1 O3 informally discussed in the previous section. The notions we are discussing in this section were originally defined in [BS98] and [BS00] 3.1. Approximations 3.1.1. Boundary insensitive approximation Suppose a space R of spatial regions. By imposing a partition, G, on R we can approximate elements of R by elements of G 3 . That is, we approximate regions in R by functions from G to the set 3 = ffo; po; nog. The ....

[Article contains additional citation context not shown here]

T. Bittner and J. G. Stell. A boundary-sensitive approach to qualitative location. Annals of Mathematics and Artificial Intelligence, 24:93--114, 1998.

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T Bittner and J G Stell, `A boundary-sensitive approach to qualitative location', Annals of Mathematics and Artificial Intelligence, 24(1/4), 93--114, (1998).

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