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1

An explanation of the Landauer bound and its ineffectiveness with regard to multivalued logic

Kycia Radosław A., Niemczynowcz Agnieszka

Technical Transactions

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2020

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Vol. 117, iss. 1

art. no. e2020042

EN

We discuss, using recent results on the thermodynamics of multivalued logic, the difficulties and pitfalls of how to apply the Landauer’s principle to thermodynamic computer memory models. The presentation is based on Szilard’s version of Maxwell’s demon experiment and use of equilibrium Thermodynamics. Different versions of thermodynamic/mechanical memory are presented – a one-hot encoding version and an implementation based on a reversed Szilard’s experiment. The relationship of the Landauer’s principle to the Galois connection is explained in detail.

PL

Opisujemy, używając niedawne badania związane z termodynamiką dla logiki wielowartościowej, problemy związane z zastosowaniem reguły Landauera dla termodynamicznego modelu pamięci komputera. Analiza jest oparta na wersji Szilarda demona Maxwella z termodynamiki równowagowej. Zostały zaprezentowane różne wersje termodynamicznej/mechanicznej pamięci – wersja gorąco jedynkowa i implementacja bazująca na odwróconym eksperymencie Szilarda. Zaprezentowano również związek pomiędzy regułą Landauera i koneksją Galois.

2

Rough sets based on Galois connections

Madrid Nicolás, Medina Jesús, Ramírez-Poussa Eloísa

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 2

299--313

EN

Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of approximation operators are based on an indiscernibility relation, which is an equivalence one. Lately, different papers have motivated the possibility of considering arbitrary relations. Nevertheless, when those are taken into account, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution to the arising problems by presenting an alternative definition of approximation operators based on the closure and interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks.

3

Rough Fuzzy Concept Analysis

Bartl E., Konecny J.

Fundamenta Informaticae

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2017

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Vol. 156, nr 2

141--168

EN

We provide a new approach to fusion of Fuzzy Formal Concept Analysis and Rough Set Theory. As a starting point we take into account a couple of fuzzy relations, one of them represents the lower approximation, while the other one the upper approximation of a given data table. By defining appropriate concept-forming operators we transfer the roughness of the input data table to the roughness of corresponding formal fuzzy concepts in the sense that a formal fuzzy concept is considered as a collection of objects accompanied with two fuzzy sets of attributes— those which are shared by all the objects and those which at least one object has. In the paper we study the properties of such formal concepts and show their relationship with concepts formed by well-known isotone and antitone operators.

4

Duality for Quasilattices and Galois Connections

Romanowska A. B., Smith J. D. H.

Fundamenta Informaticae

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2017

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Vol. 156, nr 3/4

331--359

EN

The primary goal of the paper is to establish a duality for quasilattices. The main ingredients are duality for semilattices and their representations, the structural analysis of quasilattices as Płonka sums of lattices, and the duality for lattices developed by Hartonas and Dunn. Lattice duality treats the identity function on a lattice as a Galois connection between its meet and join semilattice reducts, and then invokes a duality between Galois connections and polarities. A second goal of the paper is a further examination of this latter duality, using the concept of a pairing to provide an algebraic equivalent to the relational structure of a polarity.

5

A comprehensive survey on formal concept analysis, its research trends and applications

Singh P. K., Aswani Kumar C., Gani A.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 2

495--516

EN

In recent years, FCA has received significant attention from research communities of various fields. Further, the theory of FCA is being extended into different frontiers and augmented with other knowledge representation frameworks. In this backdrop, this paper aims to provide an understanding of the necessary mathematical background for each extension of FCA like FCA with granular computing, a fuzzy setting, interval-valued, possibility theory, triadic, factor concepts and handling incomplete data. Subsequently, the paper illustrates emerging trends for each extension with applications. To this end, we summarize more than 350 recent (published after 2011) research papers indexed in Google Scholar, IEEE Xplore, ScienceDirect, Scopus, SpringerLink, and a few authoritative fundamental papers.

6

The Construction of Fuzzy Concept Lattices Based on (θ,σ)-Fuzzy Rough Approximation Operators

Yao Y., Li Z., Mi J., Xie B.

Fundamenta Informaticae

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2011

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Vol. 111, nr 1

33-45

EN

Formal concept analysis and rough set analysis are two complementary approaches for analyzing data. This paper studies approaches to constructing fuzzy concept lattices based on generalized fuzzy rough approximation operators. For a residual implicator θ satisfying θa, b) = *theta;(1 -b, 1 -a) and its dual σ, a pair of (θ,σ)-fuzzy rough approximation operators is defined. We then propose three kinds of fuzzy operators, and examine some of their basic properties. Thus, three complete fuzzy concept lattices can be produced, for which the properties are analogous to those of the classical concept lattices.

7

Maximal and minimal c-monoids

Vargas E. M.

Demonstratio Mathematica

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2011

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Vol. 44, nr 3

615-627

EN

In this paper a restricted version of the Galois connection between poly-morphisms and invariants, called Pol . C Inv, is studied, where the invariant relations are restricted to so-called clausal relations. The lattice of all clones arising from this Ga-lois connection, so-called C-clones, is investigated up to equality of their unary parts, denominated C-monoids. All atoms and co-atoms in the lattice of all C-monoids are characterized.

8

Subtraction-like operations in nearsemilattices

Cirulis J.

Demonstratio Mathematica

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2010

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Vol. 43, nr 4

725-738

EN

A nearsemilattice is a poset having the upper-bound property. A binary operation — on a poset with the least element 0 is said to be subtraction-like if x ≤ y if and only if x — y = 0 for all x, y. Associated with such an operation is a family of partial operations lp defined by lp(x) := p— x on every initial segment [0, p]; these operations are thought of as local (sectional) complementations of some kind. We study several types of subtraction-like operations, show that each of these operations can be restored in a uniform way from the corresponding local complementations, and state some connections between properties of a (sufficiently strong) subtraction on a nearsemilattice and distributivity of the latter.

9

On single-valued and multi-valued convergences

Bacher Ł., Kamiński A., Nalepa R.

Journal of Mathematics and Applications

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2009

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Vol. 31

19-27

EN

We present a review of ideas of a general theory of convergence, developed independently of topology, with the stress on the duality of convergence and topology. Results and problems concerning sufficient and necessary conditions for a convergence to be topological, both in case of the single- and multi-valued cases, are recalled. We reconstruct, filling certain gaps, an example given in [7] to show that one of sufficient conditions in the theorems proved in [1] and [9] for multi-valued convergences to be topological is not necessary.

10

Pawlak's Information Systems in Terms of Galois Connections and Functional Dependencies

Järvinen J.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

315-330

EN

In this paper we show that each Galois connection between two complete lattices determines an Armstrong system, that is, a closed set of dependencies. Especially, we study Galois connections and Armstrong systems determined by Pawlak's information systems.

11

Modal-Like Operators in Boolean Lattices, Galois Connections and Fixed Points

Järvinen J., Kondo M., Kortelainen J.

Fundamenta Informaticae

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2007

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Vol. 76, nr 1-2

129-145

EN

In this work, four modal-like operators on Boolean lattices are introduced and their theory is presented from lattice-theoretical, topological and algebraic point of view. It is also shown how rough set approximation operators, modal operators in temporal logic, and linguistic modifiers determined by L-sets can be interpreted as modal-like operators.

12

Galois Connections and Data Analysis

Wolski M.

Fundamenta Informaticae

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2004

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Vol. 60, nr 1-4

401--415

EN

We investigate Galois connections and their relations to the major theories of (qualitative) data analysis: Rough Set Theory (RST), Formal Concept Analysis (FCA) and John Stuart Mill Reasoning (JSM-Reasoning). Polarities, a type of contravariant Galois connections, and their relationships with data analysis has been already well-known. This paper shows how axialities, a type of covariant Galois connections, are related to problems adressed by data analysis and prove that they give rise to a number of interesting lattices.