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1

On the Compactness Property of Mereological Spaces

Polkowski Lech

Fundamenta Informaticae

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2020

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Vol. 172, nr 1

73--95

EN

Continuing our work on mass-based rough mereologies, we make use of the Stone representation theorem for complete Boolean algebras and we exhibit the existence of a finite base in each mereological space. Those bases in turn allow for the introduction of distributed mereologies; regarding each element of the base as a mereological space, we propose a mechanism for fusing those mereological spaces into a global distributed mereological space. We define distributed mass-assignments and rough inclusions pointing to possible applications.

2

S-approximation Spaces : A Three-way Decision Approach

Shakiba A., Hooshmandasl M. R.

Fundamenta Informaticae

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2015

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Vol. 139, nr 3

307--328

EN

In this paper, we investigate properties of the lower and upper approximations of an Sapproximation space under different assumptions for its S operator. These assumptions are partial monotonicity, complement compatibility and functional partial monotonicity. We also extend the theory of three way decisions to non-inclusion relations. Also in this work, a new representation for partial monotone S-approximation spaces, called inflections, is introduced. We will also discuss the computational complexity of representing an S-approximation space in terms of inflection sets. Finally, the usefulness of the introduced concepts is illustrated by an example.

3

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).

4

NAO Soccer Robots Path Planning Based on Rough Mereology

Przytuła Ł.

Fundamenta Informaticae

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2014

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Vol. 131, nr 2

241--251

EN

Soccer game is a good playground for testing artificial intelligence of robots and methods for spatial reasoning in real conditions. Decision making and path planning are only two of many tasks performed while playing soccer. This paper describes an application of rough mereology introduced by Polkowski and Skowron (1994) for path planning in robotic soccer game. Our path planning method was based on mereological potential fields introduced by Polkowski and Ośmiałowski (2008) and Ośmiałowski (2009) but was redesigned due to conditions of dynamic soccer environment, so an entirely new method was developed.

5

On a Notion of Extensionality for Artifacts

Semeniuk-Polkowska M., Polkowski L.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

65--80

EN

The notion of extensionality means in plain sense that properties of complex things can be expressed by means of their simple components, in particular, that two things are identical if and only if certain of their components or features are identical; e.g., the Leibniz Identitas Indiscernibilium Principle: two things are identical if each applicable to them operator yields the same result on either; or, extensionality for sets, viz., two sets are identical if and only if they consist of identical elements. In mereology, this property is expressed by the statement that two things are identical if their parts are the same. However, building a thing from parts may proceed in various ways and this, unexpectedly, yields various extensionality principles. Also, building a thing may lead to things identical with respect to parts but distinct with respect, e.g., to usage. We address the question of extensionality for artifacts, i.e., things produced in some assembling or creative process in order to satisfy a chosen purpose of usage, and, we formulate the extensionality principle for artifacts which takes into account the assembling process and requires for identity of two artifacts that assembling graphs for the two be isomorphic in a specified sense. In parallel, we consider the design process and design things showing the canonical correspondence between abstracta as design products and concreta as artifacts. In the end, we discuss approximate artifacts as a result of assembling with spare parts which analysis does involve rough mereology.

6

Applications of comparators in data processing systems

Sosnowski Ł.

Czasopismo Techniczne. Automatyka

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2013

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R. 110, z. 2-AC

81--98

EN

This paper shows practical examples of compound object comparators and the application of the theory in various fields related to data processing systems. One can also find the necessary theoretical background needed to understand the examples.

PL

Niniejszy artykuł przedstawia praktyczne zastosowania teorii komparatorów obiektów złożonych w różnych aspektach dotyczących systemów przetwarzania danych. Dodatkowo został umieszczony skrót materiału teoretycznego pozwalającego na zrozumienie przykładów oraz ogólnej tematyki komparatorów.

7

On path planning for mobile robots: Introducing the mereological potential field method in the framework of mereological spatial reasoning

Ośmiałowski B.

Journal of Automation Mobile Robotics and Intelligent Systems

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2009

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Vol. 3, No. 2

24-33

EN

Path planning is one of the most vital problems in mobile robotics; it falls into general province of planning, however, due to specificity of the subject of mobile robotics, it has emerged as a discipline per se with its own solutions. Among many methods of probabilistic, geometrical and topological nature, the methodology of potential fields introduced by Krogh (1984) and Khatib (1985), based on physical analogies with gravitational or electromagnetic fields, has emerged. We adhere to this methodology, however, contrary to the practice of building the potential on the basis of Coulomb, or gravitational force fields, we apply the novel idea of building the potential function by means of mereological distance over a juxtaposition of grids of fixed diameter, i.e., over a discrete structure. We describe our implementation of the relevant mereological functors in the Player/Stage system as SQL predicates accessible in Player/Stage cooperating with PostgreSQL database. We present also the results of simulations with mobile robots.

8

Possible Rough Ingredients of Concepts in Approximation Spaces

Gomolińska A.

Fundamenta Informaticae

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2006

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Vol. 72, nr 1-3

139-154

EN

We discuss the problem of rough ingredients and parts of concepts of an indiscernibility-based approximation space. The notion of a (rough) ingredient is extended to the notion of a possible (rough) ingredient, and analogously in the case of parts. The term "possible" means that a concept is perceived as a candidate for a future substitute of some ingredient. Our approach is in line with rough mereology except for allowing the empty concept for the sake of simplicity.

9

On Rough Set Logics Based on Similarity Relations

Polkowski L., Semeniuk-Polkowska M.

Fundamenta Informaticae

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2005

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Vol. 64, nr 1-4

379-390

EN

In this paper, dedicated to Professor Solomon Marcus on the occasion of His 80th birthday, we discuss the idea of intensional many-valued logic reflecting the logical content of rough set approach to analysis and treatment of uncertainty. In constructing the variety of logics presented in the paper, we make use of a certain kind of tolerance (similarity) relations called rough mereological tolerances. A study of tolerance relations that arise in rough set environments was initiated in 1994, with the paper [23], in which basic ideas pertaining to tolerance relations in the rough set framework were pointed to. The analysis of the role tolerance relations may play in machine learning based on rough set-theoretic ideas was carried out by Professor Solomon Marcus in His seminal paper, written during His stay in Warsaw in December of the year 1994. At the same time the first author had first ideas related to the applicability of ideas of mereology in the rough set analysis of uncertainty. In a later analysis it has turned out that mereological approach has led to a development of a new paradigm in reasoning under uncertainty, called rough mereology, proposed by Lech Polkowski and Andrzej Skowron. Within this paradigm, one is able to construct a variety of tolerance relations. Those tolerance relations, induced by rough mereological constructs called rough inclusions, serve as a basis for constructing a variety of logics, called rough mereological logics, that are related to the inherent structure of any rough set universe. In this paper, we introduce gradually all essential and necessary notions from the area of rough set theory, mereology and rough mereology, and then we discuss tolerance relations induced by rough inclusions along with some methods for inducing rough inclusions with desired properties. The paper culminates with a discussion of intensional logics based on rough mereological tolerance relations. In this way, we explore one of so many paths in scientific research, that have been either pointed to or threaded by Professor Solomon Marcus.

10

Rough Set Approach to Domain Knowledge Approximation

Nguyen T.T.

Fundamenta Informaticae

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2004

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Vol. 59, nr 2,3

261--270

EN

Classification systems working on large feature spaces, despite extensive learning, often perform poorly on a group of atypical samples. The problem can be dealt with by incorporating domain knowledge about samples being recognized into the learning process. We present a method that allows to perform this task using a rough approximation framework. We show how human expert's domain knowledge expressed in natural language can be approximately translated by a machine learning recognition system. We present in details how the method performs on a system recognizing handwritten digits from a large digit database. Our approach is an extension of ideas developed in the rough mereology theory.

11

Rough Mereology: A Rough Set Paradigm for Unifying Rough Set Theory and Fuzzy Set Theory

Polkowski L.

Fundamenta Informaticae

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2003

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Vol. 54, nr 1

67-88

EN

In this work, we would like to discuss rough inclusions defined in Rough Mereology - a paradigm for approximate reasoning introduced by Polkowski and Skowron [20] - as a basis for common models for rough as well as fuzzy set theories. We would like to adhere to the point of view that tolerance (or, similarity) is the leading motif common to both theories and in this area paths between the two lie. To this end, we demonstrate that rough inclusions (which represent a hierarchy of tolerance relations) induce rough set theoretic approximations as well as partitions and equivalence relations in the sense of fuzzy set theory. For completeness sake, we also discuss granulation mechanisms based on rough inclusions with applications to Rough-Neuro Computing and Computing with Words. These considerations are also carried out in specialized cases of Menger's as well as Łukasiewicz's rough inclusions introduced in the paper.

12

On connection synthesis via rough mereology

Polkowski L.

Fundamenta Informaticae

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2001

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Vol. 46, nr 1,2

83-96

EN

Rough mereology is a paradigm for reasoning under uncertainty whose primitive notion is that of being a part to a degree; hence, rough mereology falls in the province of mereology-based theories for reasoning about complex objects. Among mereological theories of objects, theories based on the primitive notion of a connection distinguish themselves by a variety of applications of which we would like to mention the area of Qualitative Spatial Reasoning. In this paper, we define rough mereologies within the realm of mereologies based on the primitive notion of a part and we show that in this framework one may induce notions of connection closely related to initial rough mereologies in the sense that they induce the same notion of a part. We also address the distributed environment proving some results about connection preservation throughout the reasoning system.

13

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.