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1

Where Rough Sets and Fuzzy Sets Meet

Polkowski L., Semeniuk-Polkowska M.

Fundamenta Informaticae

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2015

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Vol. 142, nr 1-4

269--284

EN

The title of this note announces an attempt at pointing to particular points where rough set and fuzzy set approaches to decision making are close to each other, the closeness meaning that formal approaches to behaviour of the two at those points can be given in an analogous form. This by no means implies that the two can be unified as theories dealing with uncertainty. As the notion of truth for rough set decision rules is well established, we propose a notion of truth for fuzzy decision rules and we seek an analogy between the two. In order to introduce an analogous form of graded notion of truth for decision rules in both theories, we introduce a new context in which to set this notion. This context is based on our earlier results concerning rough mereological granular logics and their relevance for rough decision rules. To make our exposition satisfactorily complete, we recall our approach to granularity based on rough mereology. This note is partitioned into two parts. In Prologue, four sections present basic ideas on vagueness and ambiguity, rough sets, fuzzy sets and mereology along with rough mereology. Here also notions of truth for rough and fuzzy decision rules are presented. In Episode, main protagonists enter an analysis aimed at pointing to further close analogies between them, notably concerning notions of partial truth and dependency among attributes. To this end in four sections on similarity, granulation of knowledge, granular logics and dependencies,we give basic information on similarity, granulation and dependency and we point to analogies between the two theories with respect to those notions. We sum up the content of the note in Conclusions.

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

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.

5

Wspomagana komputerowo z użyciem RSES 2 analiza pewnych aspektów pracy biblioteki

Semeniuk-Polkowska M.

PTINT Praktyka i Teoria Informacji Naukowej i Technicznej

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2005

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T. 13, nr 4

11-23

PL

Artykuł stanowi kontynuację tematyki podjętej już na łamach kwartalnika PTINT, mianowicie zastosowań teorii zbiorów przybliżonych (TZP) w różnych problemach przekazu informacji, szczególnie w obszarze pewnych zagadnień na styku socjologii oraz takich dziedzin jak np. bibliologia. Obecnie przedstawiono zastosowanie nowej wersji dawnych programów ekspertowych RSES oraz Rosetta, mianowicie programu RSES 2. Programy powyższe są zbudowane w oparciu o TZP. Zastosowanie RSES 2 zilustrowano na przykładzie przeprowadzonych badań ankietowych dotyczących m.in. opinii na temat rozszerzania oferty usług kulturalno-oświatowych w bibliotece publicznej.

EN

In this article, that undertakes the analysis a/applications of rough set theory (TZP) to various aspects of information engineering, already presented in the PTINT Quarterly, we present a decision analysis of the problem of extending the offer by a public library in the area of cultural services to the community. The analysis is based on data resulted from a questionnaire and it is carried out by means of the new system RSES 2, a rough set-based tool for data analysis.

6

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.

7

Some Remarks on Sets of Commun

Polkowski L., Semeniuk-Polkowska M.

Fundamenta Informaticae

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2004

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Vol. 60, nr 1-4

291=-305

EN

Communicating Sequential Processes (CSP), is a theoretical framework for discussing concurrent phenomena [7]. In this note, we begin an investigation into the nature of sets of communicating sequential processes. Sets of sequential processes arise naturally when one considers processes contained within specified bounds which provides their approximate description. Adopting the trace formalism, we may express those bounds in a natural way by means of containment of traces. We endow families of process traces and a fortiori, families of processes, with a rough set topology. We show in this note that basic operators on processes preserve exact sets of processes and they are non-expansive (i.e., non-destructive in terminology of [7]) with respect to the metric D on rough sets [6], [5], restricted to exact sets of processes.

8

O zastosowaniach teorii zbiorów przybliżonych (TZP) w pewnych problemach przekazu informacji

Semeniuk-Polkowska M.

PTINT Praktyka i Teoria Informacji Naukowej i Technicznej

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2002

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T. 10, nr 2

3-12

PL

Przedstawiono zastosowania teorii zbiorów przybliżonych jako narzędzie badań w humanistyce, a dokładniej systemy software'owe zbudowane w oparciu o tę teorię. Omówiono tzw. systemy sprzężone oparte na TZP. Podano również możliwości zastosowań w obszarze pewnych zagadnień na tle styku socjologii i takich dziedzin jak bibliologia, lingwistyka, pedagogika, sztuka. Zaprezentowano przykłady przedstawionych zastosowań.

EN

Applications of the rough set theory (TZP) as a tool research in the Arts, exactly in software systems, based on the theory, are presented in the paper. There are discussed coupling systems based on TZP. Possibilities of applications and examples in sociology, bibliology, linguistics, pedagogies and arts are also described.

9

Towards usage of natural language in approximate computation : a granular semantics employing formal languages over mereological granules of knowledge

Polkowski L., Semeniuk-Polkowska M.

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Prace Informatyczne

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2000

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z. 10

131-146

EN

Granularity of Knowledge [15] has been proposed as a metaphor with which to capture the basic phenomenon of Reasoning under Uncertainty viz. the presence in the reasoning prosess and in its symbolic versions of constructs representing vaguely defined collections of objects drawn together by some similarity relations, e.g. fuzzy ones (cf. [14]). Similarly, the idea of Computing with Words has been proposed [14] as a computational paradigm in which certain words/phrases of Natural Language are labels for granules of knowledge and semantics of a chosen subset of Natural Language is realized via a calculus of granules of knowledge, e.g. by means of Fuzzy Logic. In this work, we would like to make a further step towards this end (cf. [9]) and we modify here a scheme for reasoning under uncertainity based on approximate meteorological calculus in distributed systems proposed earlier (cf.[10]). Vague specifications coding synthesis problems are rendered as phrases of Natural Language and interpreted as approximate formulae in logics for approximate reasoning over distributed systems of intelligent agents endowed with knowledge, in particular with approximate mereological predicates for constructing granules of knowledge. Calculi of those granules and the resulting formal grammars and languages denoting semanically the chosen phrases of Natural Language are invoked here (cf. [12]). The results presented here constitute a step towards a certain fulfillment of Computing with Words program. It is expected that those languages will be applicable in problems of Intelligent Control, e.g. in Mobile Robotics and in problems of Approximate Reasoning in Natural Language (cf. our earlier work [8]).