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Mereotopologies with Predicates of Actual Existence and Actual Contact

Vakarelov D.

Fundamenta Informaticae

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2017

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Vol. 156, nr 3/4

413--432

EN

We discuss in this work the importance of some predicates of ontological existence in mereology and in mereotopology especially for systems incorporating time. Tarski showed that mereology can be identified in some sense with complete Boolean algebras with zero 0 deleted. If one prefers to use only first-order language, the first-order theory for Boolean algebras can be used with zero included for simplicity. We extend the language of Boolean algebra with a one-place predicate AE(x), called ”actual existence” and satisfying some natural axioms. We present natural models for Boolean algebras with predicate AE(x) motivating the axioms and prove corresponding representation theorems. Mereotopology is considered as an extension of mereology with some relations of topological nature, like contact. One of the standard mereotopological systems is contact algebra, which is an extension of Boolean algebra with a contact relation C, satisfying some simple and obvious axioms. We consider in this paper a natural generalization of contact algebra as an extension of Boolean algebra with the predicate AE(x) and a contact relation Cα called ”actual contact”, assuming for them natural axioms combining Cα and AE. Relational and topological models are proposed for the resulting system and corresponding representation theorems are proved. I dedicate this paper to my teacher in logic Professor Helena Rasiowa for her 100-th birth anniversary. Professor Rasiowa showed me the importance of algebraic and topological methods in logic and this was her main influence on me.

2

On the Problem of Boundaries from Mereology and Rough Mereology Points of View

Polkowski L., Semeniuk-Polkowska M.

Fundamenta Informaticae

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2014

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Vol. 133, nr 2/3

241--255

EN

The notion of a boundary belongs in the canon of the most important notions of mereotopology, the topological theory induced by mereological structures; the importance of this notion rests not only in its applications to practical spatial reasoning, e.g., in geographical information systems, where it is usually couched under the term of a contour and applied in systems related to economy, welfare, climate, wildlife etc., but also in its impact on reasoning schemes elaborated for reasoning about spatial objects, represented as regions, about spatial locutions etc. The difficulty with this notion lies primarily in the fact that boundaries are things not belonging in mereological universa of things of which they are boundaries. Various authors, from philosophers through mathematicians to logicians and computer scientists proposed schemes for defining and treating boundaries. We propose two approaches to boundaries; the first aims at defining boundaries as things possibly in the universe in question, i.e., composed of existing things, whereas the second defines them as things in a meta–space built over the mereological universe in question, i.e., we assume a priori that boundaries are in a sense ‘things at infinity’, in an agreement with the topological nature of boundaries. Of the two equivalent topological definitions of a boundary, the first, global, defining the boundary as the difference between the closure and the interior of the set, and the second, local, defining it as the set of boundary points whose all neighborhoods transect the set, the first calls for the first type of the boundary and the second is best fitted for the meta–boundary. In the text that follows, we discuss mereology and rough mereology notions (sects. 2, 3), the topological approach to the notion of a boundary and the model ROM with which we illustrate our discussion (sect. 4), the mereology approach (sect. 5), and the approach based on rough mereology and granular computing in the framework of rough mereology (sect. 6).

3

Boundaries, Borders, Fences, Hedges

Polkowski L., Semeniuk-Polkowska M.

Fundamenta Informaticae

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2014

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Vol. 129, nr 1/2

149--159

EN

In this essay, we analyze various often semantically identified notions of separating things. In doing this, we contrast the set–theoretical approach based on the notion of an element/point with the mereological approach based on the notion of a part, hence, pointless. We address time aspect of the notion of a boundary and related notions as well as approximate notions defined in the realm of rough (approximate) mereology.

4

A New Approach to the Concepts of Boundary and Contact: Toward an Alternative to Mereotopology

Breysse O., De Glas M.

Fundamenta Informaticae

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2007

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Vol. 78, nr 2

217-238

EN

Mereotopology is a class of formal theories devoted to the analysis of spatiotemporal entities and their interactions. It has produced important advances in the analysis of natural language, naive geography and computer vision, illustrating a broad range of applications. However, it has been shown that the modelling of interactions between spatiotemporal entities with mereotopology can lead to unsolvable problems, including disconnectedness of the representation space as well as a mix-up of the relationships of contact and overlap. The origin of these problems, which fundamentally limit the usefulness of mereotopology, has not been fully identified. In this paper, we first formally demonstrate that these problems originate from the incompatibility of the concepts of boundary, continuity and contact within the framework of mereotopology, as suggested by previous studies. Secondly, we prove that this incompatibility stems from the formalization of these concepts through topology. We show that a solution can be found by substituting for topology an alternative theory, known as locology, which provides new mathematical tools for the modelling of spatiotemporal entities.

5

Points in point-free mereotopology

Schoop D.J.

Fundamenta Informaticae

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2001

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Vol. 46, nr 1,2

129-143

EN

It is considered the virtue of mereotopology that it takes regions instead of points to be the primitive entities of space. As mereotopology is assumed to avoid the difficulties incurred by considering points as primitive entities, mereotopology is thought to provide the means for common-sense spatial representation and reasoning. However, we show that considering regions as primitive entities in mereotopology does not prevent us referring to points in a commonly used mereotopological first-order language. Therefore, the difficulties attributed to taking points as primitive entities must be considered even for point-free mereotopologies. Consequently, the virtue of mereotopology cannot lie in taking regions instead of points as primitive entities. On the contrary, we argue that the virtue of mereotopology is its capability to treat regions and points as primitive entities.

6

Empiricism and rationalism in region-based theories of space

Pratt-Hartman I.

Fundamenta Informaticae

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2001

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Vol. 46, nr 1,2

159-186

EN

Suppose we want to develop a theory of space in which the primary entities are not points, but regions. How do we proceed? Historically, the most popular approach is to select a group of spatial relations corresponding to familiar spatial concepts, and then to construct an axiomatic system governing these relations. The appropriateness of this axiomatic system is to be judged on the basis of its ability to chime with pre-theoretic intuition and of its success in a larger theory of scientific or commonsense spatial reasoning. In this paper, we argue that this approach is flawed, and leads only to labyrinthine systems of doubtful theoretical or practical value. We present an alternative approach which substitutes mathematical rigour for pre-theoretic intuition, and which at the same time provides a real guarantee of practical applicability.

7

Rough Mereology in information systems with applications to qualitative spatial

Polkowski L., Skowron A.

Fundamenta Informaticae

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2000

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Vol. 43, Nr 1-4

291-320

EN

Rough Mereology has been proposed as a paradigm for approximate reasoning in complex information systems. Its primitive notion is that of a predicate of rough inclusion which gives for any two entities of discourse the degree in which one of them is a part of the other. Rough Mereology may be regarded as an extension of Rough Set Theory as it proposes to argue in terms of similarity relations induced from a rough inclusion instead of reasoning in terms of more strict indiscernibility relations. Rough Mereology is also a generalization of Mereology i.e. a theory of reasoning based on the notion of a part. Classical languages of mathematics are of two-fold kind: the language of set theory (naive or formal) expressing classes of objects as sets consisting of ëlements", "points" etc. suitable for objects perceived as built of ätoms" and applied to structures perceived as discrete and the language of part relations suitable for e.g. continuous objects like solids, regions, etc. where two objects are related to each other by saying that one of them is a part of the other. Mereological theories for reasoning about complex structures are at the heart of Qualitative Spatial Reasoning. In this paper, we study basic aspects of Rough Mereology in Information Systems. Mereology makes the distinction between entities perceived as individuals (singletons), to which the part predicate may be applied, and entities perceived as distributive classes (sets, lists, general names etc.) of entities. This distinction is made formal and precise within Ontology i.e. Theory of Being based on the primitive notion of the copula is which is also a basic ingredient of theories for Spatial Reasoning. The practical aim of Ontology is to elaborate a system of concepts (notions, names, sets of entities) about which the reasoning is carried out. Therefore, we begin our study with an analysis of a simple rough set-based Ontology (the template ontology) in Information Systems and in this setting we present our approach to Mereology in Information Systems. In this framework we introduce Rough Mereology and we present some ways for defining rough inclusions. We demonstrate applications of Rough Mereology to approximate reasoning taking as the case subject Qualitative Spatial Reasoning. We address here some of its mereo-topological as well as mereo-geometrical aspects.