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1
Content available remote Rough Granular Computing in Modal Settings : Generalised Approximation Spaces
Wolski M., Gomolińska A.
Fundamenta Informaticae
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2016
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Vol. 148, nr 1/2
157--172
EN
The paper studies the rough granular computing paradigm within the conceptual settings of multi-modal logic. The main idea is to express a generalised approximation space (U; I; κ), where U is the universe of objects, I is an uncertainty function, and κ is a rough inclusion function, in terms of binary relations, and then to consider the corresponding modal operators. The new modal structure obtained in this way is rich enough to define closure and interior operators corresponding to the classical rough approximation operators and their well-known uni-modal generalisations. In contrast to the standard modal interpretation of rough set approximations, in the new settings one can directly deal with information granules and their properties, which is crucial for granular computing paradigm. More precisely, we are provided with means of describing features of objects and information granules, as well as inclusion degrees between granules.
2
Content available remote Rough Inclusion Functions and Similarity Indices
Gomolińska A., Wolski M.
Fundamenta Informaticae
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2014
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Vol. 133, nr 2/3
149--163
EN
Rough inclusion functions are mappings considered in rough set theory with which one can measure the degree of inclusion of a set (information granule) in a set (information granule) in line with rough mereology. On the other hand, similarity indices are mappings used in cluster analysis with which one can compare clusterings, and clustering methods with respect to similarity. In this article we show that a large number of similarity indices, known from the literature, can be generated by three simple rough inclusion functions, the standard rough inclusion function included.
3
Content available remote Concept Formation : Rough Sets and Scott Systems
Wolski M., Gomolińska A.
Fundamenta Informaticae
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2013
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Vol. 127, nr 1-4
17--33
EN
The paper addresses the problem of concept formation (knowledge granulation) in the settings of rough set theory. The original version of rough set theory implicitly accommodates a lot of well-established philosophical assumptions about concept formation as presented by A. Rand. However, as suggested by S. Hawking and L. Mlodinow, one has also to consider the dynamics of the universe of objects and different scales at which concepts may be formed. These both aspects have already been discussed separately in rough set theory. Different forms of dynamics have been addressed explicitly – especially the case of extending the universe by new objects; in contrast, different scales of description have been addressed implicitly, mainly within the Granular Computing (GrC) paradigm. Following the example of Life, the famous game invented by J. Conway, we describe the corresponding dynamics in Pawlak information systems using a GrC driven methodology. Having dynamics discussed, we address the problem of concept formation at zoom-out scales of description. To this end, we build Scott systems as information systems describing the universe at a coarser scale than the original scale of Pawlak systems. We regard these systems as a special type of classifications, which have already been studied in the context of rough sets by A. Skowron et al.
4
Content available remote An Incidence Algebra Approach to Knowledge Granulation in Pawlak Information Systems
Wolski M., Gomolińska A.
Fundamenta Informaticae
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2013
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Vol. 128, nr 1-2
223--238
EN
Representation theory is a branch of mathematics whose original purpose was to represent information about abstract algebraic structures by means of methods of linear algebra (usually, by linear transformations and matrices). G.-C. Rota in his famous Foundations defined a representation of a locally finite partially ordered set (locally finite poset) P in terms of a module over a ring \mathbbA, which can further be extended by the addition of a convolution operation to an associative \mathbbA-algebra called an incidence algebra of P. He applied this construction to solve a number of important problems in combinatorics. Our goal in this paper is to discuss the concept of an incidence algebra as a representation of a Pawlak information system. We shall analyse both incidence algebras and information systems in the context of granular computing, a paradigm which has recently received a lot of attention in computer science. We discuss therefore the concept of an incidence algebra on two levels: the level of objects which form a preordered set and the level of information granules which form a poset. Since incidence algebras induced on these two levels are Morita equivalent, we may focus our attention on the incidence algebra of information granules. We take the lattice of closed ideals of this algebra, where the maximal elements serve as a representation of information granules. The poset of maximal closed ideals obtained in this way is isomorphic to the set of information granules of the Pawlak information system equipped with a natural information order.
5
Content available remote On Graded Nearness of Sets
Gomolińska A., Wolski M.
Fundamenta Informaticae
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2012
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Vol. 119, nr 3/4
301-317
EN
In this article we present three inclusion functions which characterise the nearness relation between finite sets of objects defined in line with J. F. Peters, A. Skowron, and J. Stepaniuk [26]. By means of these functions we extend the notion of nearness to the graded case where one can measure the degree to which one set is near to another one.
6
Content available remote A Logic-Algebraic Approach to Graded Inclusion
Gomolińska A.
Fundamenta Informaticae
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2011
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Vol. 109, nr 3
265-279
EN
In this article we continue searching for functions which might be used as measures of inclusion of information granules in information granules. Starting with a 3-valued logic having an adequate logical matrix, we show how to derive a corresponding graded inclusion function. We report on the results of examination of several best known 3-valued logics in this respect. We also give some basic properties of the inclusion functions obtained.
7
Content available remote Satisfiability of Formulas from the Standpoint of Object Classification: The RST Approach
Gomolińska A.
Fundamenta Informaticae
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2008
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Vol. 85, nr 1-4
139-153
EN
In this article we discuss judgment of satisfiability of formulas of a knowledge representation language as an object classification task. Our viewpoint is that of the rough set theory (RST), and the descriptor language for Pawlak's information systems of a basic kind is taken as the study case. We show how certain analogy-based methods can be employed to judge satisfiability of formulas of that language.
8
Content available remote Approximation Spaces Based on Relations of Similarity and Dissimilarity of Objects
Gomolińska A.
Fundamenta Informaticae
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2007
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Vol. 79, nr 3-4
319-333
EN
In this article, we aim at extension of similarity-based approximation spaces to the case, where both similarity and dissimilarity of objects are taken into account. Apart from the well-known notions of lower rough approximation, upper rough approximation, and variable-precision positive regions of concepts, adapted to our case, the notions of exterior, possibly negative region, and ignorance region of concepts are introduced and investigated.
9
Content available remote 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.
10
Content available remote Satisfiability and Meaning of Formulas and Sets of Formulas in Approximation Spaces
Gomolińska A.
Fundamenta Informaticae
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2005
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Vol. 67, nr 1-3
77--92
EN
In this paper, we study general notions of satisfiability and meaning of formulas and sets of formulas in approximation spaces. Rather than proposing one particular form of rough satisfiability and meaning, we present a number of alternative approaches. Approximate satisfiability and meaning are important, among others, for modelling of complex systems like systems of adaptive social agents. Finally, we also touch upon derivative concepts of meaning and applicability of rules.
11
Content available remote A Graded Meaning of Formulas in Approximation Spaces
Gomolińska A.
Fundamenta Informaticae
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2004
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Vol. 60, nr 1-4
159--172
EN
The aim of the paper is to introduce degrees of satisfiability as well as a graded form of the meaning of formulas and their sets in the approximation space framework.
12
Content available remote A Comparative Study of Some Generalized Rough Approximations
Gomolińska A.
Fundamenta Informaticae
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2002
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Vol. 51, nr 1,2
103-119
EN
In this paper we focus upon a comparison of some generalized rough approximations of sets, where the classical indiscernibility relation is generalized to any binary reflexive relation. We aim at finding the best of several candidates for generalized rough approximation mappings, where both definability of sets by elementary granules of information as well as the issue of distinction among positive, negative, and border regions of a set are taken into account.
13
Content available remote Credibility of information for modelling belief state and its change
Gomolińska A.
Fundamenta Informaticae
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1998
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Vol. 34, nr 1,2
33-51
EN
Credibility of information is investigated for the purpose of modelling belief state and its change.
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