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1
Content available remote 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
Content available remote Introducing Mass-based Rough Mereology in a Mereological Universe with Relations to Fuzzy Logics and a Generalization of the Łukasiewicz Logical Foundations of Probability
Polkowski Lech
Fundamenta Informaticae
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2019
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Vol. 166, nr 3
227--249
EN
We investigate a model for rough mereology based reasoning in which things in the universe of mereology are endowed with positive masses. We define the mass based rough inclusion and establish its properties. This model does encompass inter alia set theoretical universes of finite sets with masses as cardinalities, probability universes with masses as probabilities of possible events, sets of satisfiable formulas with values of satisfiability, measurable bounded sets in Euclidean n -spaces with n -dimensional volume as mass, in particular complete Boolean algebras of regular open or closed sets – the playground for spatial reasoning and geographic information systems. We define a mass-based rough mereological theory (in short mRM-theory). We demonstrate affinities of the mass-based rough mereological mRM-theory with classical many-valued (‘fuzzy’) logics of Łukasiewicz, Gödel and Goguen and we generalize the theses of logical foundations of probability as given by Łukasiewicz. We give an abstract version of the Bayes theorem which does extend the classical Bayes theorem as well as the proposed by Łukasiewicz logical version of the Bayes formula. We also establish an abstract form of the betweenness relation which has proved itself important in problems of data analysis and behavioral robotics. We address as well the problem of granulation of knowledge in decision systems by pointing to the most general set of conditions a thing has to satisfy in order to be included into a formally defined granule of knowledge, the notion instrumental in our approach to data analysis. We address the problem of applications by pointing to our work on intelligent robotics in which the mass interpreted as the relative area of a planar region is basic for definition of a rough inclusion on regular open/closed regions as well as in definition of the notion of betweenness crucial for a strategy for navigating teams of robots.
3
Content available remote From Leśniewski, Łukasiewicz, Tarski to Pawlak : Enriching Rough Set Based Data Analysis. A Retrospective Survey
Polkowski L. T.
Fundamenta Informaticae
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2017
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Vol. 154, nr 1/4
343--358
EN
This work is dedicated to Profesor Andrzej Ehrenfeucht, the eleve of the Warsaw School of Logic and Mathematics on the occasion of His 85th Birthday. We propose to exploit certain of the milestone ideas created by this School and to apply them to data analysis in the framework of the rough set theory proposed by Professor Zdzisław Pawlak. To wit, we apply the idea of fractional truth states due to Jan Łukasiewicz, mereology created by Stanisław Leśniewski and the betweenness relation used by Alfred Tarski as one of primitive predicates in His axiomatization of Euclidean geometry. These ideas applied in problems of approximate reasoning permit us to formalize calculus of granules of knowledge and use it in preprocessing of data before applying a classification algorithm. Introduction of a mereological version of betweenness relation to data allows for partitioning of data into the kernel and the residuum, both sub-data sets providing a faithful representation of the whole data set and reducing the size of data without any essential loss of accuracy of classification. In the process of algorithmic construction of the partition of data into the kernel and the residuum, we exploit the Dual Indiscernibility Matrix which further allows us to introduce notions of a pair classifier and, more generally, k-classifier yet to be studied.
4
Content available remote 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).
5
Content available remote 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.
6
Content available remote Ontological Models Based on Mereology, Topology and Theoretical Data
Vasyliuk J., Teslyuk V., Mykhailiuk A.
Machine Dynamics Research
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2013
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Vol. 37, No. 2
91-96
EN
In this paper the ontological models based on mereology, topology and theoretical data about microelectromechanical system as ontology is described throughout the spectrum of mereology, topological principles. As result, it was illustrated the elementary structure, the interaction between microsystems components and internal processes between them.
7
Content available remote O kształtowaniu się intuicyjnej koncepcji zbioru Stanisława Leśniewskiego
Miszczyński R.
Kwartalnik Historii Nauki i Techniki
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2013
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R. 58, nr 3
99--119
EN
The first vague definitions of a set (Cantor, Dedekind, Hausdorfi) described it as bringing certain objects together. However, in the set theory there appeared some objects which did not originate in this way, for example the empty set. The disadvantages of this theory were clearly revealed in Russell’s antinomy. According to Leśniewski, the con tradictions will disappear when the concept of a set is based on intuition. Therefore, the distributive concept was replaced with the collective one: the relation of the element to the set is based on the relation between the part and the whole. In the new theory, the empty set no longer exists - the non-intuitive distinction between an object and the set composed of this object, disappears. These features allow to prevent Russell’s antinomy.
8
Content available remote 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.
9
Content available remote Towards a Pragmatic Mereology
Janicki R., Le D.T.M.
Fundamenta Informaticae
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2007
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Vol. 75, nr 1-4
295-314
EN
A version of mereology (i.e. theory of parts and fusions) is presented. Some applications to model software structures are discussed.
10
Content available remote 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.
11
Content available remote A categorical axiomatisation of Region-Based Geometry
Bennett B.
Fundamenta Informaticae
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2001
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Vol. 46, nr 1,2
145-158
EN
Region Based Geometry (RBG) is an axiomatic theory of qualitative configurations of spatial regions. It is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. Whereas in Tarski's theory the combination of mereological and geometrical axioms involves set theory, in RBG the interface is achieved by purely 1st-order axioms. This means that the elementary sublanguage of RBG is extremely expressive, supporting inferences involving both mereological and geometrical concepts. Categoricity of the RBG axioms is proved: all models are isomorphic to a standard interpretation in terms of Cartesian spaces over \mathbbR.
12
Content available remote 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.
13
Content available remote The Notion of Cube in theTheory of Squares
Sochacki R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2000/2001
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Vol. 8
67--79
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