A Phase-Field Perspective on Mereotopology

in H. Prade (ed.), Proceedings of the 13th European Conference on Artificial Intelligence (ECAI 98), Chichester, Wiley, 1998

We provide a model-theoretic framework for investigating and comparing a variety of mereotopological theories with respect to (i) the intended interpretation of their connection primitives, and (ii) the composition of their intended domains (e.g., whether or not they allow for boundary elements).

Journal of Philosophical Logic, 2003

The paper outlines a model-theoretic framework for investigating and comparing a variety of mereotopological theories. In the first part we consider different ways of characterizing a mereotopology with respect to (i) the intended interpretation of the connection primitive, and (ii) the composition of the admissible domains of quantification (e.g., whether or not they include boundary elements). The second part extends this study by considering two further dimensions along which different patterns of topological connection can be classified-the strength of the connection and its multiplicity.

Annals of Mathematics and Artificial Intelligence, 2016

The notion of contact algebra is one of the main tools in the region based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. The elements of the Boolean algebra are considered as formal representations of spatial regions as analogs of physical bodies and Boolean operations are considered as operations for constructing new regions from given ones and also to define some mereological relations between regions as part-of, overlap and underlap. The contact relation is one of the basic mereotopological relations between regions expressing some topological nature. It is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact and others. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of the operation of Boolean complementation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement and replacing the Boolean part of the definition by distributive lattice. First steps in this direction were made in [8, 9] presenting the notion of distributive contact lattice based on contact relation as the only mereotopological relation. In this paper we consider as nondefinable primitives the relations of contact, nontangential inclusion and dual contact, extending considerably the language of distributive contact lattices. Part I of the paper is devoted to a suitable axiomatization of the new language called extended distributive contact lattice (EDC-lattice) by means of universal first-order axioms true in all contact algebras. EDClattices may be considered also as an algebraic tool for certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDC-lattice can be isomorphically embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDC-lattices, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space.

Proceedings of the international …, 2001

Representation theorems for systems of regions have been of interest for some time, and various contexts have been used for this purpose: Mormann [17] has demonstrated the fruitfulness of the methods of continuous lattices to obtain a topological representation theorem for his formalisation of Whiteheadian ontological theory of space; similar results have been obtained by Roeper [20]. In this note, we prove a topological representation theorem for a connection based class of systems, using methods and tools from the theory of proximity spaces. The key novelty is a new proximity semantics for connection relations. notion of "point", the basic primitive notion of classical geometry, is now (secondorder) definable in various ways as a special collection of regions; this way it becomes one of the very complex notions of the theory. We will elaborate briefly on this in Section 2, and refer the reader to the paper by Gerla [10] for a survey on pointless geometry.

in N. Guarino (ed.), Formal Ontology in Information Systems, Amsterdam, IOS Press, 1998

Mereotopology is today regarded as a major tool for ontological analysis, and for many good reasons. There are, however, a number of open questions that call for an answer. Some of them are philosophical, others have direct import for applications, but all are crucial for a proper assessment of the strengths and limits of mereotopology. This paper is an attempt to put some order into this still untamed area of research. I will not attempt any answers. But I shall try to give an idea of the problems, and of their relevance for the systematic development of formal ontological theories. P-reflexivity: everything is part of itself. P-antisymmetry: two distinct things cannot be part of each other. P-transitivity: any part of a part of a thing is itself part of that thing. C-reflexivity: everything is connected to itself. C-symmetry: if a thing is connected to a second thing, the second is connected to the first. Monotonicity: everything is connected to anything to which its parts are connected.

Studia Logica, 2001

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T 0 topological space with an additional "contact relation" C defined by

1996

The paper is a contribution to formal ontology. It seeks to use topological means in order to derive ontological laws pertaining to the boundaries and interiors of wholes, to relations of contact and connectedness, to the concepts of surface, point, neighbourhood, and so on. The basis of the theory is mereology, the formal theory of part and whole, a theory which is shown to have a number of advantages, for ontological purposes, over standard treatments of topology in set-theoretic terms.

2000

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.

2004

Abstract We develop a formal ontology within the four dimensionalist (4D) paradigm by showing how the algebraic description of spatial mereotopology-a Boolean algebra equipped with a connection relation-can be enriched to provide a mereotopology in which the entities arc spatio-temporal, rather than spatial, regions. With the 4D approach it is natural to identify a period of time with all of space during that time, and thus to model temporal relations by relations between spatio-temporal regions.