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Algebraic Formalisations of Fuzzy Relations and Their Representation Theorems
, 1998
"... The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representati ..."
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The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...
A Calculational Approach to Mathematical Induction
, 1994
"... Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variablefree relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when co ..."
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Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variablefree relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two nontrivial examples are presented. The first is a proof of a Newman's lemma. The second is a calculation of a condition under which the union of two wellfounded relations is wellfounded. In both cases the calculations lead to generalisations of the known results. In the case of the latter example, one lemma generalises three different conditions.
Algorithms from Relational Specification
, 1996
"... Introduction The purpose of a specification is to state a problem as clearly as possible. In many cases, the most direct and intuitive way to specify a problem is by writing down a logical predicate like in the phonebook example in Chapt. ??[chapt:background] that describes its possible solutions. ..."
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Introduction The purpose of a specification is to state a problem as clearly as possible. In many cases, the most direct and intuitive way to specify a problem is by writing down a logical predicate like in the phonebook example in Chapt. ??[chapt:background] that describes its possible solutions. We employ the calculus of binary relations for developing efficient algorithms from logical problem specifications, using the procedure sketched below. The first task is to obtain a relational specification from the original problem description. There are a number of correspondences between logical and relationalgebraic operations, which in many practical examples yield the desired relational specification easily. In more complex cases one can use relation algebra extended with direct products and their associated projections (see Sect. 2.5[sect:hetrel prod]). Relational specifications are very compact, and if the carrier sets are
STRUCTURING PROBABILISTIC DATA BY GALOIS LATTICES
 MATH. & SCI. HUM. / MATHEMATICS AND SOCIAL SCIENCES (43 E ANNÉE, N ° 169, 2005(1), P. 77104)
, 2005
"... In this paper we address the problem of organising probabilistic data by Galois concept lattices. Two lattices are proposed, the union lattice and the intersection lattice, corresponding to two distinct semantics, by choosing accordingly the join and meet operators. A new algorithm is proposed to c ..."
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In this paper we address the problem of organising probabilistic data by Galois concept lattices. Two lattices are proposed, the union lattice and the intersection lattice, corresponding to two distinct semantics, by choosing accordingly the join and meet operators. A new algorithm is proposed to construct the concept lattice. Two real data examples illustrate the presented approach.
Relational Semiotic Methods For Design Of Intelligent Systems
, 1998
"... This paper was focused towards presenting and further examining some notions and technical features of nonassociative compositions of mathematical relations that are useful in fuzzy logic and its applications. Manyvalued logic based (fuzzy) extensions of relations can contribute on the theoretical ..."
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This paper was focused towards presenting and further examining some notions and technical features of nonassociative compositions of mathematical relations that are useful in fuzzy logic and its applications. Manyvalued logic based (fuzzy) extensions of relations can contribute on the theoretical side, by utilizing the elegant algebraic structure of relational systems. By replacing the usual Boolean algebra of crisp relations by manyvalued logic algebras, one obtains extensions that contain the classical relational theory as a special case.
The Secrets of Causality
, 1993
"... In this paper a model for a relational calculus for distributed program design is introduced. Here, the distributed programs are compositions of socalled processes. The construction of the model is guided by desired properties for several forms of composition of processes, such as sequential compos ..."
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In this paper a model for a relational calculus for distributed program design is introduced. Here, the distributed programs are compositions of socalled processes. The construction of the model is guided by desired properties for several forms of composition of processes, such as sequential composition and feedback. Functionality (determinism) and totality of processes are defined. After the observation that functionality and totality are not preserved by the feedback operator the class of causal processes is introduced. It is shown that causality guarantees the preservation by the feedback of functionality and totality.
Resource bisimilarity and gbisimilarity coincide ∗
, 2010
"... Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labelled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisi ..."
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Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labelled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisimilarity for graded modal logic. This note shows that these two notions of bisimilarity coincide over imagefinite Kripke frames. 1
Resource bisimilarity and graded bisimilarity coincide ✩
"... Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labeled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisim ..."
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Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labeled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisimilarity for graded modal logic. This note shows that these two notions of bisimilarity coincide over imagefinite Kripke frames. Keywords: coalgebras Concurrency, resource bisimilarity, graded bisimilarity, graded modal logic, Kripke frames, 1.