Results 1 
8 of
8
A Theory of Granular Partitions
, 2001
"... This paper presents an application of the theory of granular partitions proposed in (Smith and Brogaard, to appear), (Smith and Bittner 2001) to the phenomenon of vagueness. We understand vagueness as a semantic property of names and predicates. This is in contrast to those views which hold that the ..."
Abstract

Cited by 72 (34 self)
 Add to MetaCart
This paper presents an application of the theory of granular partitions proposed in (Smith and Brogaard, to appear), (Smith and Bittner 2001) to the phenomenon of vagueness. We understand vagueness as a semantic property of names and predicates. This is in contrast to those views which hold that there are intrinsically vague objects or attributes in reality and thus conceive vagueness in a de re fashion. All entities are crisp, on de dicto view here defended, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being its extension. We show that the theory of granular partitions provides a general framework within which we can understand the relation between terms and concepts on the one hand and their multiple referents or extensions on the other, and we show how it might be possible to formulate within this framework a solution to the Sorites paradox. 1.
Boolean Connection Algebras: A New Approach to the RegionConnection Calculus
 Artificial Intelligence
, 1999
"... The RegionConnection Calculus (RCC) is a well established formal system for qualitative spatial reasoning. It provides an axiomatization of space which takes regions as primitive, rather than as constructions from sets of points. The paper introduces boolean connection algebras (BCAs), and prove ..."
Abstract

Cited by 43 (7 self)
 Add to MetaCart
The RegionConnection Calculus (RCC) is a well established formal system for qualitative spatial reasoning. It provides an axiomatization of space which takes regions as primitive, rather than as constructions from sets of points. The paper introduces boolean connection algebras (BCAs), and proves that these structures are equivalent to models of the RCC axioms. BCAs permit a wealth of results from the theory of lattices and boolean algebras to be applied to RCC. This is demonstrated by two theorems which provide constructions for BCAs from suitable distributive lattices. It is already well known that regular connected topological spaces yield models of RCC, but the theorems in this paper substantially generalize this result. Additionally, the lattice theoretic techniques used provide the first proof of this result which does not depend on the existence of points in regions. Keywords: RegionConnection Calculus, Qualitative Spatial Reasoning, Boolean Connection Algebra, Mer...
Generalizing Graphs using Amalgamation and Selection
 Advances in Spatial Databases. 6th International Symposium, SSD'99, volume 1651 of Lecture Notes in Computer Science
, 1999
"... . This work is a contribution to the developing literature on multiresolution data models. It considers operations for modeloriented generalization in the case where the underlying data is structured as a graph. The paper presents a new approach in that a distinction is made between generalization ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
. This work is a contribution to the developing literature on multiresolution data models. It considers operations for modeloriented generalization in the case where the underlying data is structured as a graph. The paper presents a new approach in that a distinction is made between generalizations that amalgamate data objects and those that select data objects. We show that these two types of generalization are conceptually distinct, and provide a formal framework in which both can be understood. Generalizations that are combinations of amalgamation and selection are termed simplifications, and the paper provides a formal framework in which simplifications can be computed (for example, as compositions of other simplifications). A detailed case study is presented to illustrate the techniques developed, and directions for further work are discussed. 1 Introduction Specialist spatial information systems (SIS) play an increasingly important role within the Information Tech...
Stratified Rough Sets And Vagueness
 Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science. International Conference COSIT’03
, 2003
"... The relationship between less detailed and more detailed versions of data is one of the major issues in processing geographic information. Fundamental to much work in modeloriented generalization, also called semantic generalization, is the notion of an equivalence relation. Given an equivalence re ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
The relationship between less detailed and more detailed versions of data is one of the major issues in processing geographic information. Fundamental to much work in modeloriented generalization, also called semantic generalization, is the notion of an equivalence relation. Given an equivalence relation on a set, the techniques of rough set theory can be applied to give generalized descriptions of subsets of the original set. The notion of equivalence relation, or partition, has recently been significantly extended by the introduction of the notion of a granular partition. A granular partition provides what may be thought of as a hierarchical family of partial equivalence relations. In this paper we show how the mechanisms for making rough descriptions with respect to an equivalence relation can be extended to give rough descriptions with respect to a granular partition. In order to do this, we also show how some of the theory of granular partitions can be reformulated; this clarifies the connections between equivalence relations and granular partitions. With the help of this correspondence we then can show how the notion of hierarchical systems of partial equivalence classes relates to partitions of partial sets, i.e., partitions of sets in which not all members are known. This gives us new insight into the relationships between roughness and vagueness. 1
Part and Complement: Fundamental Concepts in Spatial Relations
 ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE
, 2004
"... The spatial world consists of regions and relationships between regions. Examples of such relationships are that two regions are disjoint or that one is a proper part of the other. The formal specification of spatial relations is an important part of any formal ontology used in qualitative spatial ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
The spatial world consists of regions and relationships between regions. Examples of such relationships are that two regions are disjoint or that one is a proper part of the other. The formal specification of spatial relations is an important part of any formal ontology used in qualitative spatial reasoning or geographical information systems. Various schemes of relationships have been proposed and basic schemes have been extended to deal with vague regions, coarse regions, regions varying over time, and so on. The principal aim of this paper is not to propose further schemes, but to provide a uniform framework within which several existing schemes can be understood, and upon which further schemes can be constructed in a principled manner. This framework is based on the fundamental concepts of part and of complement. By varying these concepts, for example allowing a partof relation taking values in a lattice of truth values beyond the twovalued Boolean case, we obtain a family of schemes of spatial relations. The viability of this approach to spatial relations as parameterized by the concepts of part and complement is demonstrated by showing how it encompasses the RCC5 and RCC8 schemes as well as the case of `eggyolk regions'. The use of the approach for discrete regions is discussed briefly.
TIL: A typedirected optimizing compiler for ML
 In ACM SIGPLAN '96 Conference on Programming Language Design and Implementation
, 2006
"... Summary. MultiVMap is a compact framework from which plane graphs representing geographic maps at different levels of detail can be extracted. Its main feature is that the scale of the extracted map can be variable through its domain, while each entity maintains consistent combinatorial relations w ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Summary. MultiVMap is a compact framework from which plane graphs representing geographic maps at different levels of detail can be extracted. Its main feature is that the scale of the extracted map can be variable through its domain, while each entity maintains consistent combinatorial relations with the rest of entities represented in the map. The model is based on a set of operators, called updates, which modify the level of detail in a small portion of a map. The set of updates is partially ordered, and can therefore be represented as a Directed Acyclic Graph, which defines our multiscale structure. An algorithm to extract a map at the required resolution is proposed, and a lower bound for the number of different maps which can be extracted from the model is given. The model supports map data processing operations (e.g., querying), as well as progressive and selective transmission of maps over a network. 1
Relations in Mathematical Morphology with applications to Graphs and Rough Sets
"... Abstract. Rough sets have been applied in spatial information theory to construct theories of granularity – presenting information at different levels of detail. Mathematical morphology can also be seen as a framework for granularity, and the question of how rough sets relate to mathematical morphol ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. Rough sets have been applied in spatial information theory to construct theories of granularity – presenting information at different levels of detail. Mathematical morphology can also be seen as a framework for granularity, and the question of how rough sets relate to mathematical morphology has been raised by Bloch. This paper shows how by developing mathematical morphology in terms of relations we obtain a framework which includes the basic constructions of rough set theory as a special case. The extension of the relational framework to mathematical morphology on graphs rather than sets is explored and new operations of dilations and erosions on graphs are obtained. 1
ABSTRACT Applying Hierarchical Graphs to Pedestrian Indoor Navigation
"... In this paper we propose to apply hierarchical graphs to indoor navigation. The intended purpose is to guide humans in large public buildings and assist them in wayfinding. We start by formally defining hierarchical graphs and explaining the particular benefits of this approach. In the main part, we ..."
Abstract
 Add to MetaCart
In this paper we propose to apply hierarchical graphs to indoor navigation. The intended purpose is to guide humans in large public buildings and assist them in wayfinding. We start by formally defining hierarchical graphs and explaining the particular benefits of this approach. In the main part, we suggest an algorithm to automatically construct such a multilevel hierarchy from floor plans. The algorithm is guided by the idea to exploit domainspecific characteristics of indoor environments. Besides this, two particular problems are addressed: first, how to incorporate threedimensional elements in the hierarchy, and second, the need for extending the hierarchy at complex geometrical regions with implicit decision points. An extended version of this paper is also available [1].