John G. Stell

A formal approach to imperfection in geographic information

Traditional computational models of geographic phenomena offer no room for imperfection. Underlying this tradition is the simplifying assumption that reality is certain, crisp, unambiguous, independent of context, and capable of quantitative representation. This paper reports on initial work which explicitly recognises that most geographic information is intrinsically imperfect. Based on an ontology of imperfection the paper explores a formal model of imperfect geographic information using multi-valued logic. The development of Java software able to assist with a geodemographic retail site assessment application is used to illustrate the utility of a formal approach.

Boolean connection algebras: a new approach to the Region-Connection Calculus

Abstract The Region-Connection 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.

Spatial relations between indeterminate regions

Abstract Systems of relations between regions are an important aspect of formal theories of spatial data. Examples of such relations are part-of, partially overlapping, and disjoint. One particular family of systems is that based on the region-connection calculus (RCC). These systems of relations were originally formulated for ideal regions, not subject to imperfections such as vagueness or indeterminacy. This paper presents two new methods for extending the relations based on the RCC from crisp regions to indeterminate regions. As a formal context for these two methods we develop an algebraic approach to spatial indeterminacy using Łukasiewicz algebras. This algebraic approach provides a generalisation of the “egg-yolk” model of indeterminate regions. The two extension methods which we develop are proved to be equivalent. In particular, it is shown that it is possible to define part-of in terms of connection in the indeterminate case. This generalises a well-known result about crisp RCC regions. Our methods of extension take a relation on crisp regions taking values in the set of two Boolean truth values, and produce a relation on indeterminate regions taking one of three truth values. We discuss how our work might be developed to give more detailed relations taking values in a six-element lattice.

A boundary-sensitive approach to qualitative location

Reasoning about the location of regions in 2-dimensional space is necessarily based on finite approximations to such regions. These finite approximations are often derived by describing how a region (the figure) relates to a frame of reference (the ground). The frame of reference generally consists of regions, or cells, forming a partition of the space under consideration. This paper presents a new approach to describing figure-ground relationships which is able to take account of how the figure relates to boundaries between cells as well as to their interiors. We also provide a general theory of how approximations to regions lead to approximations to operations on regions. This theory is applied to the case of our boundary-sensitive model of location. The paper concludes by indicating how interpreting boundaries in a more general sense should lead to a theory dealing with generalized partitions in which the cells may overlap. The applications of the theory developed here will include qualitative spatial reasoning, and should have practical relevance to geographical information systems.

Vagueness and Rough Location

This paper deals with the representation and the processing of information about spatial objects with indeterminate location like valleys or dunes (objects subject to vagueness). The indeterminacy of the location of spatial objects is caused by the vagueness of the unity condition provided by the underlying human concepts valley and dune. We propose the notion of rough, i.e., approximate, location for representing and processing information about indeterminate location of objects subject to vagueness. We provide an analysis of the relationships between vagueness of concepts, indeterminacy of location of objects, and rough approximations using methods of formal ontology. In the second part of the paper we propose an algebraic formalization of rough location, and hence, a formal method for the representation of objects subject to vagueness on a computer. We further define operations on those representations, which can be interpreted as union and intersection operations between those objects. The discussion of vagueness of concepts, indeterminacy of location, rough location and the relationships between these notions contributes to the theory about the ontology of geographic space. The formalization presented can provide the foundation for the implementation of vague objects and their location indeterminacy in GIS.

The Algebraic Structure of Sets of Regions

The provision of ontologies for spatial entities is an important topic in spatial information theory. Heyting algebras, co-Heyting algebras, and bi-Heyting algebras are structures having considerable potential for the theoretical basis of these ontologies. This paper gives an introduction to these Heyting structures, and provides evidence of their importance as algebraic theories of sets of regions. The main evidence is a proof that elements of certain Heyting algebras provide models of the Region-Connection Calculus developed by Cohn et al. By using the mathematically well known techniques of “pointless topology”, it is straight-forward to conduct this proof without any need to assume that regions consist of sets of points. Further evidence is provided by a new qualitative theory of regions with indeterminate boundaries. This theory uses modal operators which are related to the algebraic operations present in a bi-Heyting algebra.

Stratified rough sets and vagueness

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 model-oriented 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.

Granulation for Graphs

In multi-resolution data handling, a less detailed structure is often derived from a more detailed one by amalgamating elements which are indistinguishable at the lower level of detail. This gathering together of indistinguishable elements is called a granulation of the more detailed structure. When handling spatial data at several levels of detail the granulation of graphs is an important topic. The importance of graphs arises from their widespread use in modelling networks, and also from the use of dual graphs of spatial partitions. This paper demonstrates that there are several quite different kinds of granulation for graphs. Four kinds are described in detail, and situations where some of these may arise in spatial information systems are indicated. One particular kind of granulation leads to a new formulation of the boundary-sensitive approach to qualitative location developed by Bittner are Stell. Vague graphs and their connection with granulation are also discussed, and two kinds of vague graphs are identified.

Generalizing Graphs Using Amalgamation and Selection

This work is a contribution to the developing literature on multi-resolution data models. It considers operations for model-oriented 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.

Foundations of Equational Deduction: A Categorical Treatment of Equational Proofs and Unification Algorithms

We provide a framework for equational deduction based on category theory. Firstly, drawing upon categorical logic, we show how the compositional structure of equational deduction is captured by a 2-category. Using this formulation, algorithms for solving equations are derived from general constructions in category theory. The basic unification algorithm arises from constructions of colimits. We also consider solving equations in the presence of term rewriting systems and the combination of unification algorithms.