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1
Content available remote A Discrete Representation for Dicomplemented Lattices
Düntsch I., Kwuida L., Orłowska E.
Fundamenta Informaticae
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2017
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Vol. 156, nr 3/4
281--295
EN
Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of general lattices.
2
Content available remote A Relational Logic for Spatial Contact Based on Rough Set Approximation
Düntsch I., Orłowska E., Wang H.
Fundamenta Informaticae
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2016
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Vol. 148, nr 1/2
191--206
EN
In previous work we have presented a class of algebras enhanced with two contact relations representing rough set style approximations of a spatial contact relation. In this paper, we develop a class of relational systems which is mutually interpretable with that class of algebras, and we consider a relational logic whose semantics is determined by those relational systems. For this relational logic we construct a proof system in the spirit of Rasiowa-Sikorski, and we outline the proofs of its soundness and completeness.
3
Content available remote Lattice Machine Classification based on Contextual Probability
Wang Hongbo, Düntsch I., Trindade L.
Fundamenta Informaticae
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2013
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Vol. 127, nr 1-4
241--256
EN
In this paper we review Lattice Machine, a learning paradigm that “learns” by generalising data in a consistent, conservative and parsimonious way, and has the advantage of being able to provide additional reliability information for any classification. More specifically, we review the related concepts such as hyper tuple and hyper relation, the three generalising criteria (equilabelledness, maximality, and supportedness) as well as the modelling and classifying algorithms. In an attempt to find a better method for classification in Lattice Machine, we consider the contextual probability which was originally proposed as a measure for approximate reasoning when there is insufficient data. It was later found to be a probability function that has the same classification ability as the data generating probability called primary probability. It was also found to be an alternative way of estimating the primary probability without much model assumption. Consequently, a contextual probability based Bayes classifier can be designed. In this paper we present a new classifier that utilises the Lattice Machine model and generalises the contextual probability based Bayes classifier. We interpret the model as a dense set of data points in the data space and then apply the contextual probability based Bayes classifier. A theorem is presented that allows efficient estimation of the contextual probability based on this interpretation. The proposed classifier is illustrated by examples.
4
Content available remote Discrete Duality for Rough Relation Algebras
Düntsch I., Orłowska E.
Fundamenta Informaticae
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2013
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Vol. 127, nr 1-4
35--47
EN
Rough relation algebras are a generalization of relation algebras such that the underlying lattice structure is a regular double Stone algebra. Standard models are algebras of rough relations. A discrete duality is a relationship between classes of algebras and classes of relational systems (frames). In this paper we prove a discrete duality for a class of rough relation algebras and a class of frames.
5
Content available remote Structures with Multirelations, their Discrete Dualities and Applications
Düntsch I., Orłowska E., Rewitzky I.
Fundamenta Informaticae
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2010
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Vol. 100, nr 1/4
77-98
EN
In this paper we show that the problem of discrete duality can be extended beyond the clasical setting of duality between a class of algebras and a class of relational structures. Namely, for some classes of algebras, the relevant dual structures are the structures with multirelations. Several applications of multirelations will be described.
6
Content available remote Complex Algebras of Arithmetic
Düntsch I., Pratt–Hartmann I.
Fundamenta Informaticae
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2009
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Vol. 97, nr 4
347-367
EN
An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic, and logical operations to be performed on sets of natural numbers. Arithmetic circuits can, also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of, natural numbers. In the present paper we investigate the algebraic structure of complex algebras of, natural numbers and make some observations regarding the complexity of various theories of such algebras.
7
Content available remote A Multi-modal Logic for Disagreement and Exhaustiveness
Düntsch I., Konikowska B.
Fundamenta Informaticae
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2007
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Vol. 75, nr 1-4
215-238
EN
The paper explores two basic types of relations betwen objects of a Pawlak-style information system generated by the values of some attribute of those objects: disagreement (disjoint sets of values) and exhaustiveness (sets of values adding up to the whole universe of the attribute). Out of these two fundamental types of relations, most other types of relations on objects of an information system considered in the literature can be derived - as, for example, indiscernibility, similarity and complementarity. The algebraic properties of disagreement and indiscernibility relations are explored, and a representation theorem for each of these two types of relations is proved. The notions of disagreement and exhaustiveness relations for a single attribute are extended to relations generated by arbitrary sets of attributes, yielding two families of relations parametrized by sets of attributes. They are used as accessibility relations to define a multi-modal logic with modalities corresponding to the lower and upper approximation of a set in Pawlak's rough set theory. Finally, a complete Rasiowa-Sikorski deduction system for that logic is developed.
8
Content available remote Rough Relation Algebras Revisited
Düntsch I., Winter M.
Fundamenta Informaticae
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2006
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Vol. 74, nr 2,3
283-300
EN
Rough relation algebras arise from Pawlak's information systems by considering as object ordered pairs on a fixed set X. Thus, the subsets to be approximated are binary relations over X, and hence, we have at our disposal not only the set theoretic operations, but also the relational operators ;, ˇ , and the identity relation 1˘. In the present paper, which is a continuation of [6], we further investigate the structure of abstract rough relation algebras.
9
Content available remote Algebras of approximating regions
Düntsch I., Orłowska E., Wang H.
Fundamenta Informaticae
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2001
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Vol. 46, nr 1,2
71-82
10
Content available remote Relations algebras in qualitative spatial reasoning
Düntsch I., Wang H., McCloskey S.
Fundamenta Informaticae
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1999
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Vol. 39, Nr 3
229-248
EN
The formalization of the ``part - of'' relationship goes back to the mereology of S. Leśniewski, subsequently taken up by [34], and [11]. In this paper we investigate relation algebras obtained from dixfferent notions of ``part-of'', respectively, ``connectedness'' in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra.
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