Rough implication operator based on strong topological rough algebras
Information SciencesPublished 12 June 2010
Xiaohong Zhang, Yiyu Yao, Hong Yu
Citations31
SJR quartileQ1
SJR score1.80
SNIP1.98
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
The rough implication operator is introduced in strong topological rough algebra that is a generalization of classical rough algebra and a topological De Morgan algebra, and a new semantic interpretation of Lukasiewicz continuous-valued logic system Luk is constructed based on rough implication.
Abstract
The role of topological De Morgan algebra in the theory of rough sets is investigated. The rough implication operator is introduced in strong topological rough algebra that is a generalization of classical rough algebra and a topological De Morgan algebra. Several related issues are discussed. First, the two application directions of topological De Morgan algebras in rough set theory are described, a uniform algebraic depiction of various rough set models are given. Secondly, based on interior and closure operators of a strong topological rough algebra, an implication operator (called rough implication) is introduced, and its important properties are proved. Thirdly, a rough set interpretation of classical logic is analyzed, and a new semantic interpretation of Lukasiewicz continuous-valued logic system Luk is constructed based on rough implication. Finally, strong topological rough implication algebra (STRI-algebra for short) is introduced. The connections among STRI-algebras, regular double Stone algebras and RSL-algebras are established, and the completeness theorem of rough logic system RSL is discussed based on STRI-algebras.
Keywords
Computer Science
Rough Sets: Theoretical Aspects of Reasoning about Data
8,416 Citations1991Zdzisław Pawlak
Trends in logicMetamathematics of Fuzzy Logic
3,973 Citations1998Petr Hájek
This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.
International Journal of General SystemsROUGH FUZZY SETS AND FUZZY ROUGH SETS*
2,948 Citations1990Didier Dubois, Henri Prade
It is argued that both notions of a rough set and a fuzzy set aim to different purposes, and it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems.
Information SciencesRudiments of rough sets
2,036 Citations2006Zdzisław Pawlak, Andrzej Skowron
The basic concepts of rough set theory are presented and some rough set-based research directions and applications are pointed out, indicating that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences.
Trends in logicAlgebraic Foundations of Many-Valued Reasoning
1,335 Citations2000Roberto Cignoli, Ítala M. Loffredo D’Ottaviano +1 more
Information SciencesRough sets and Boolean reasoning
955 Citations2006Zdzisław Pawlak, Andrzej Skowron
Application of rough set methods for feature selection, feature extraction, discovery of patterns and their applications for decomposition of large data tables are presented as well as the relationship of rough sets with association rules.
Fuzzy Sets and SystemsMonoidal t-norm based logictowards a logic for left-continuous t-norms
916 Citations2001Francesc Esteva, Lluı́s Godo
This paper investigates a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua, and completeness of MTL with respect to linearly ordered MTL-algebras is proved.
Information SciencesConstructive and algebraic methods of the theory of rough sets
807 Citations1998Yiyu Yao
This paper reviews and compares constructive and algebraic approaches in the study of rough set algebras and states axioms that must be satisfied by the operators.
Information SciencesReduction and axiomization of covering generalized rough sets
698 Citations2003William Zhu, Fei‐Yue Wang
It has been proved that the reduct of a covering is the minimal covering that generates theSame covering lower approximation or the same covering upper approximation, so this concept is also a technique to get rid of redundancy in data mining.
International Journal of Approximate ReasoningTwo views of the theory of rough sets in finite universes
589 Citations1996Yiyu Yao
This paper presents and compares two views of the theory of rough sets: the operator-oriented and set-oriented views, which interprets rough set theory as an extension of set theory with two additional unary operators.
Intelligent Automation & Soft ComputingGeneralization of Rough Sets using Modal Logics
462 Citations1996Yiyu Yao, T. Lin
A number of extended rough set models are proposed and examined based on the properties satisfied by a binary relation, such as serial, reflexive, symmetric, transitive, and Euclidean, which correspond to different modal logic systems.
IEEE Transactions on Fuzzy SystemsA Survey on Fuzzy Implication Functions
378 Citations2007M. Mas, M. Monserrat +2 more
This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators.
Studies in fuzziness and soft computingIncomplete Information: Rough Set Analysis
313 Citations1998Ewa Orłowska
This book discusses Rough Sets, Rough Set-Based Deduction Methods, and Similarity-Based Reasoning, a Logic for Reasoning about Similarity Information Systems, Similarity Relations and Modal Logics.
CERN Document Server (European Organization for Nuclear Research)Rough Sets and Data Mining: Analysis of Imprecise Data
313 Citations1996Tzu‐Yu Lin, N. Cercone
Rough Sets and Data Mining: Analysis of Imprecise Data is an edited collection of research chapters on the most recent developments in rough set theory and data mining that focus on discovering dependencies among data, and reasoning about vague, uncertain and imprecise information.
Combination of Rough and Fuzzy Sets Based on α-Level Sets
150 Citations1997Yiyu Yao
This paper examines some fundamental issues involved in the combination of rough-set and fuzzy-set models, with emphasis on their structures in terms of crisp sets.
Lecture notes in computer scienceAlgebraic Structures for Rough Sets
113 Citations2004Gianpiero Cattaneo, Davide Ciucci
This work discusses a possible algebraization of the concrete algebra of the power set of X through quasi BZ lattices that enables us to define two rough approximations based on a similarity and on a preclusive relation.
Studia LogicaOn the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic
112 Citations2002Francesc Esteva, Joan Gispert +2 more
The issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL are addressed.
Theoretical Computer ScienceA logic for rough sets
107 Citations1997Ivo Düntsch
A logic for rough sets is presented and its algebraic semantics is presented in the spirit of Andreka and Nemeti (1994).
Studies in fuzziness and soft computingRough Set Theory and Logic-Algebraic Structures
83 Citations1998Piero Pagliani
This paper investigates Rough Set Systems within the framework of Nelson algebras and the structure of the resulting subclass is inherently described using the properties of Approximation Spaces, achieving in a quite “natural” way an interpretation based on the notion of Chain Based Lattice.
Information SciencesRough operations on Boolean algebras
57 Citations2004Guilin Qi, Weiru Liu
Two pairs of rough operations on Boolean algebras are introduced and a pair of generalized rough approximations based on a partition of the unity of a Boolean algebra are proposed.
Fuzzy Sets and SystemsTriangle algebras: A formal logic approach to interval-valued residuated lattices
56 Citations2007Bart Van Gasse, Chris Cornelis +2 more
This paper introduces triangle algebras, a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1, that serve as an equational representation of interval-valued residuates lattices (IVRLs), a system of many-valued logic capturing the tautologies of IVRLs.
Notre Dame Journal of Formal LogicA certain conception of the calculus of rough sets.
52 Citations1992Zbigniew Bonikowski
It is proved that the family of rough sets, determined by the unions of equivalence classes of the relation R with the operations of rough addition, rough multiplication, and complement, is a complete atomic Boolean algebra.
Workshops in computingOn Rough Sets in Topological Boolean Algebras
50 Citations1994Marek Chuchro
This work has focused on rough sets in topological Boolean algebras and characterization of the rough sets determined by regular open and regular closed elements.
Transactions on rough setsPropositional Logics from Rough Set Theory
46 Citations2007Mohua Banerjee, Md. Aquil Khan
An attempt is made to place the gamut of work on propositional logics with semantics based on rough sets, spread over the last 20 years, in one platform, and present the various relationships that emerge and indicate questions that surface.
Topological Rough Algebras
34 Citations1997Anita Wasilewska
Science in China Series F Information SciencesA fuzzy logic system based on Schweizer-Sklar t-norm
31 Citations2006Xiaohong Zhang, Huacan He +1 more
Based on the Schweizer-Sklar t-norm, a fuzzy logic system UL* is established, and its soundness theorem and completeness theorem are proved and the essential meaning of a parameter p inUL* is explained and the use of fuzzy logicSystem UL* in approximate reasoning is presented.
Lecture notes in computer scienceHeyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets
26 Citations2002Gianpiero Cattaneo, Davide Ciucci
It is shown that standard Pawlak approach to rough sets is a model of this structure and it is possible to obtain a de Morgan Brouwer-Zadeh (BZ) distributive lattice with respect to the partial order induced from the ?
Information SciencesThe pseudo-linear semantics of interval-valued fuzzy logics
21 Citations2008B VANGASSE, Chris Cornelis +2 more
It is proved that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper.
Fundamenta InformaticaeAutomated Deduction Techniques for Studying Rough Algebras
14 Citations1998Laurent Vigneron
A non-exhaustive state of the art in the domain of deduction in first-order logic with equality, describing different strategies introduced over the years and an application of those techniques to the study of non-classical logics.
2007 IEEE International Conference on Granular Computing (GRC 2007)Generalized Rough Set Model on De Morgan Algebras
10 Citations2007Xiaohong Zhang, Gang Yao
By introducing the notion of similar topological open subsystem in De Morgan algebra, a pair of general rough approximations based on De Morgangebra are defined and another rough set model is constructed, call it second type generalized rough set models on DeMorgan algebra.
2007 IEEE International Conference on Granular Computing (GRC 2007)Implication Operators Based on Rough Set Model over Boolean Algebras
10 Citations2007Xiaohong Zhang, Yingcang Ma +2 more
Some algebraic properties of rough approximation operators over Boolean algebras are investigated and the concept of rough Stone algebra and rough Nelson algebra are generalized and a new way for studying rough logic is pointed out.
Fundamenta InformaticaeRough Relation Algebras Revisited
9 Citations2006Ivo Düntsch, Michael Winter
The structure of abstract rough relation algebras is investigated and not only the set theoretic operations, but also the relational operators; ˘, and the identity relation 1'.
2007 IEEE International Conference on Granular Computing (GRC 2007)Comparison of Machine Learning and Pattern Discovery Algorithms for the Prediction of Human Single Nucleotide Polymorphisms
8 Citations2007Rui Yan, Paul C. Boutros +2 more
Fuzzy Sets and SystemsTopological soft algebra and its application
6 Citations1992Akira Nakamura
It is shown that a kind of decision problem of a modal fuzzy logic is solvable in this topological soft algebra and a solution of this problem is given not only theoretically but as a procedure.
2007 IEEE International Conference on Granular Computing (GRC 2007)On Systems Software Engineering with Application to Bioinformatics
5 Citations2007James Kuodo Huang
An integrated "systems software engineering" theory based on Boolean logic, Hilbert logic and set theory is introduced andBioinformatics, bioengineering and medical information systems are adopted as a special case to describe this integrated "software systems engineering".
Rough Set Algebra and Lattice Implication Algebra
3 Citations2004Keyun Qin
It is proved that rough set algebra becomes lattice implication algebra if proper implication and complement operators are selected.
HAL (Le Centre pour la Communication Scientifique Directe)Rough Algebras & Automated Deduction
1 Citations1998Anita Wasilewska, Laurent Vigneron
The rough R5 and R4 algebras investigated here are particular cases of topological rough algeBRas introduced by Anita Wasilewska, and are algebraic models for modal logics S4 and S5, respectively.