DOCSLIB.ORG

How Fuzzy Set Theory Can Help Make Database Systems More Cooperative Aurélien Moreau

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
How Fuzzy Set Theory Can Help Make Database Systems More Cooperative Aurélien Moreau How fuzzy set theory can help make database systems more cooperative Aurélien Moreau To cite this version: Aurélien Moreau. How fuzzy set theory can help make database systems more cooperative. Databases [cs.DB]. Université Rennes 1, 2018. English. NNT : 2018REN1S043. tel-01985938 HAL Id: tel-01985938 https://tel.archives-ouvertes.fr/tel-01985938 Submitted on 18 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ANNÉE 2018 THÈSE / UNIVERSITÉ DE RENNES 1 sous le sceau de l’Université Bretagne Loire pour le grade de DOCTEUR DE L’UNIVERSITÉ DE RENNES 1 Mention : Informatique École doctorale MathSTIC présentée par Aurélien MOREAU préparée à l’unité de recherche IRISA – UMR 6074 Institut de Recherche en Informatique et Systèmes Aléatoires École Nationale Supérieure des Sciences Appliquées et de Technologie Thèse soutenue à Lannion How Fuzzy le 26 juin 2018 devant le jury composé de : Set Theory Arnaud MARTIN Professeur, Université de Rennes 1 / Président Allel HADJALI Can Help Make Professeur, ISAE-ENSMA / Rapporteur Marie-Jeanne LESOT Maître de conférences HDR, Sorbonne Université / Database Systems Rapporteur Dominique LAURENT More Cooperative Professeur, Université de Cergy-Pontoise / Examinateur Olivier PIVERT Professeur, ENSSAT / Directeur de thèse Grégory SMITS Maître de conférences HDR, Université de Rennes 1 / Co-directeur de thèse À mon père. iv Résumé de la thèse en français Les bases de données sont désormais omniprésentes, et nécessaires dans de nombreux cas d’utilisation : la gestion des employés d’une entreprise, le placement des cours d’une université, la gestion des stocks d’un entrepôt, la recherche de liens entre différents achats effectués, etc. Nées dans les années 1970 suite à la genèse du modèle relationnel par Codd, les bases de données relationnelles se sont progressivement imposées comme le standard permettant de stocker et de manipuler des informations. Le modèle relation- nel est basé sur la logique booléenne, ce qui se traduit par plusieurs conséquences pour l’utilisateur : pas de classement des résultats (qui sont soit satisfaisants soit non satis- faisants), pas de prise en compte de la sensibilité aux bornes (une modification d’une condition de sélection pouvant transformer un ensemble de résultats vide en un ensem- ble de résultats pléthorique, et vice-versa). Plusieurs propositions ont été faites pour traiter ces problèmes, telles que les requêtes à préférences, des mécanismes coopérat- ifs permettant de gérer la cardinalité des ensembles de réponses, ou divers paradigmes d’interrogation tels que la recherche par facettes, par mots-clés ou à partir d’exemples. La théorie des ensembles flous permet de modéliser l’appartenance partielle d’un objet à une classe, par opposition à l’appartenance binaire modélisée par la logique booléenne. Elle permet ainsi d’apporter plus de flexibilité dans l’interaction entre les humains et les SGBD. L’utilisateur peut donc exprimer des préférences, et obtenir des réponses ordonnées en fonction de celles-ci. Des variables linguistiques peuvent être utilisées pour modéliser des termes de la langue naturelle tels que jeune ou bien payé. Cela se fait à l’aide de fonctions d’appartenance déterminant les seuils de valeur à partir desquels un élément aura totalement, partiellement, ou pas du tout la propriété définie. La formulation de requêtes à l’aide de variables linguistiques permet de renforcer l’expressivité des langages de requêtes, et donc de rendre les SGBD plus “intelligents”. Dans ces travaux de thèse nous proposons de tirer parti de la théorie des ensembles flous afin d’améliorer les interactions entre les systèmes de bases de données et les utilisateurs. Les mécanismes coopératifs visent à aider les utilisateurs à mieux interagir avec les SGBD, notamment par le biais de réponses utiles et ne prêtant pas à confusion. Ces mécanismes doivent faire preuve de robustesse : ils doivent toujours pouvoir proposer des réponses à l’utilisateur. Empty set (0,00 sec) est un exemple typique de réponse qu’il serait désirable d’éradiquer, en indiquant par exemple à l’utilisateur quelles clauses de sa requête ont le plus contribué à la production d’une réponse vide. Plusieurs approches coopératives basées sur la théorie des ensembles flous ont déjà été proposées, permettant v vi notamment d’éviter les réponses vides à l’aide de techniques de relaxation par exemple. Par ailleurs, les réponses à certaines requêtes peuvent parfois laisser les utilisateurs dubitatifs : qu’il s’agisse de réponses vides ou de réponses pléthoriques, il apparaît souhaitable de pouvoir expliquer ces réponses. L’origine des réponses (parfois appelée provenance) n’est que rarement précisée et beaucoup de systèmes sont vus comme des boîtes noires avec lesquelles les modalités d’interaction sont limitées. Le caractère informatif des explications de réponses est parfois plus important que les réponses elles-mêmes : ce peut être le cas avec les réponses vides et pléthoriques par exemple, d’où l’intérêt de mécanismes coopératifs robustes, capables à la fois de contribuer à l’explication ainsi qu’à l’amélioration des résultats. Exemple 1 Un utilisateur formule la requête “restaurants à burgers bons marché à Lannion” et obtient la réponse Empty set (0,00 sec). Comme l’ensemble des résultats est vide, l’utilisateur pourrait souhaiter comprendre quelle(s) condition(s) de sa requête a (ont) contribué à la vacuité de cet ensemble de résultats (telle que “bon marché” par exemple), et obtenir des relaxations possibles pouvant mener à des résultats acceptables quoique pas entièrement satisfaisants vis-à-vis de la requête initiale (par exemple en relaxant l’interprétation de “bon marché”). L’exemple ci-dessus illustre une situation où il n’existe pas de réponse pleinement satisfaisante, et où une réponse proche pourrait tout à fait satisfaire un utilisateur. Cependant une explication reste nécessaire pour indiquer à l’utilisateur quelle condi- tion de sa requête empêchait l’obtention de résultats. De plus, l’utilisation de termes de la langue naturelle pour décrire les données permet de garantir l’interprétabilité des explications fournies. La possibilité pour l’utilisateur de raffiner un vocabulaire contribue à la personnalisation des explications et améliore l’interprétabilité. Dans l’exemple ci-dessus la modalité “bon marché” est utilisée pour définir les valeurs de prix désirées par l’utilisateur : l’utilisation de modalités définies par les utilisateurs contribue à l’interprétabilité de la requête pour l’utilisateur. Nous proposons de nous intéresser aux explications dans le contexte des réponses coopératives sous trois angles : • dans le cas d’un ensemble pléthorique de résultats ; • dans le contexte des systèmes de recommandation ; • dans le cas d’une recherche à partir d’exemples. Ces axes définissent des approches coopératives où l’intérêt des explications est de per- mettre à l’utilisateur de comprendre comment sont calculés les différents résultats pro- posés dans un effort de transparence. Le caractère informatif des explications apporte une valeur ajoutée aux résultats bruts, et en ce sens constitue une réponse coopérative. Explication de clusters de réponses Des solutions pour la gestion des réponses pléthoriques ont déjà été proposées : à base de renforcement ou d’augmentation de la requête, ou de résumés des résultats vii ou encore de diversification des résultats. Nous proposons ici une approche mêlant les résumés de données avec les explications pour permettre aux utilisateurs de comprendre la structure des réponses à leurs requêtes. Nous utilisons un vocabulaire flou composé de partitions décrivant chacun des attributs de la base. Chaque partition est composée de modalités dont les labels permettent la formulation d’explications avec des termes de la langue naturelle. L’objectif général de l’approche est de permettre à un utilisateur de comprendre l’ensemble des résultats de sa requête, s’il est possible d’y trouver des sous-ensembles intéressants, et le cas échéant quelles sont les propriétés qui caractérisent chacun de ces sous-ensembles et pas les autres. Exemple 2 Considérons un utilisateur recherchant une voiture de seconde main de marque Peugeot, modèle 407 sur un site en ligne dédié. L’ensemble de réponses com- prend plus de 1000 résultats, ce qui n’est pas évident à parcourir manuellement. Les attributs qui intéressent tout particulièrement l’utilisateur sont le prix et le kilométrage. Un partitionnement de ces résultats conduit à la formation de deux sous-ensembles dis- tincts : • un sous-ensemble de voitures chères avec un faible kilométrage ; • un sous-ensemble de voitures bon marché avec un fort kilométrage. De plus, des informations complémentaires (qui n’ont pas été demandées explicitement par l’utilisateur) spécifiques à chaque sous-ensemble sont disponibles : les voitures du premier sous-ensemble sont récentes, quand celles du second sont plus anciennes. Avec l’approche proposée nous visons trois objectifs : • robustesse de l’approche (fournir des explications pour permettre aux utilisateurs de comprendre
Load more

Recommended publications
  • 1 Elementary Set Theory
    1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
    [Show full text]
  • Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T
    University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 8-1-2008 Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich University of Texas at El Paso, [email protected] Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: Technical Report: UTEP-CS-08-27a Published in Proceedings of the 9th International Conference on Intelligent Technologies InTech'08, Samui, Thailand, October 7-9, 2008, pp. 47-55. Recommended Citation Nguyen, Hung T. and Kreinovich, Vladik, "Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms" (2008). Departmental Technical Reports (CS). Paper 94. http://digitalcommons.utep.edu/cs_techrep/94 This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich Department of Mathematical Sciences Department of Computer Science New Mexico State University University of Texas at El Paso Las Cruces, NM 88003, USA El Paso, TX 79968, USA [email protected] [email protected] Abstract—Subsethood A ⊆ B and set equality A = B are Thus, for two fuzzy sets A and B, it is reasonable to define among the basic notions of set theory. For traditional (“crisp”) degree of subsethood and degree of similarity.
    [Show full text]
  • Mathematics 144 Set Theory Fall 2012 Version
    MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction
    [Show full text]
  • A Taste of Set Theory for Philosophers
    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
    Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes.
    [Show full text]
  • Learning About Women and Urban Services N Latin America and the Caribbean
    P w- N :jk)- (,. -2 LEARNING ABOUT WOMEN AND URBAN SERVICES N LATIN AMERICA AND THE CARIBBEAN A Report on the Women, Low-h;come Households and Urban Services Project of The Population Council Wikh Selected Contributions from The International Center for Research on Women The Equity Policy Center The Development Planning Unit of University College LEARNING ABOUT WOMEN AND URBAN SERVICES IN LATIN AMERICA AND THE CARIBBEAN A Report on the Wbmen, Low-income Householdsand UrbanServices Project of The Population Council With Selected Contributions from The InternationalCenter for Research on Women The Equity Policy Cenier The Development Planning Unit of University College Marianne Schmink Judith Bruce and Marilyn Kohn Editors © 1986 The Population Council, Inc. Selections of this volume may be reproduced for teaching purposes in naga7ines and newspapers with acknowledgment to this report, to the authors, and the sponsoring institutions. The majority of the articles in this volume are an outgrowth of working group deliberations, the research and related activities of the Women, Low-Income Households, and Urban Services Project of the Population Council. The first phase of this project was supported by the United States Agency for International Development under Cooperative Agreement No. AID/OTR-0007-A-00-1154-00. The views expressed by the authors are their own. CONTENTS LEARNING ABOUT WOMEN AND URBAN SERVICES IN LATIN AMERICA AND THE CARIBBEAN A Report on the Women, Low-Income Households and Urban Services Project of tho Population Council With Selected Contributions from: The International Center for Research on Women The Equity Policy Center The Development Planning Unit of University College Preface Judith Bruce Part I.
    [Show full text]
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Applied Mathematics for Database Professionals
    7451FM.qxd 5/17/07 10:41 AM Page i Applied Mathematics for Database Professionals Lex de Haan and Toon Koppelaars 7451FM.qxd 5/17/07 10:41 AM Page ii Applied Mathematics for Database Professionals Copyright © 2007 by Lex de Haan and Toon Koppelaars All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. ISBN-13: 978-1-59059-745-3 ISBN-10: 1-59059-745-1 Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. Lead Editor: Jonathan Gennick Technical Reviewers: Chris Date, Cary Millsap Editorial Board: Steve Anglin, Ewan Buckingham, Gary Cornell, Jonathan Gennick, Jason Gilmore, Jonathan Hassell, Chris Mills, Matthew Moodie, Jeffrey Pepper, Ben Renow-Clarke, Dominic Shakeshaft, Matt Wade, Tom Welsh Project Manager: Tracy Brown Collins Copy Edit Manager: Nicole Flores Copy Editor: Susannah Davidson Pfalzer Assistant Production Director: Kari Brooks-Copony Production Editor: Kelly Winquist Compositor: Dina Quan Proofreader: April Eddy Indexer: Brenda Miller Artist: April Milne Cover Designer: Kurt Krames Manufacturing Director: Tom Debolski Distributed to the book trade worldwide by Springer-Verlag New York, Inc., 233 Spring Street, 6th Floor, New York, NY 10013.
    [Show full text]
  • A Class of Fuzzy Theories*
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 85, 409-451 (1982) A Class of Fuzzy Theories* ERNEST G. MANES Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003 Submitted by L. Zadeh Contenfs. 0. Introduction. 1. Fuzzy theories. 2. Equality and degree of membership. 3. Distributions as operations. 4. Homomorphisms. 5. Independent joint distributions. 6. The logic of propositions. 7. Superposition. 8. The distributional conditional. 9. Conclusions. References. 0. INTR~DUCTTON At the level of syntax, a flowchart scheme [25, Chap. 41 decomposes into atomic pieces put together by the operations of structured programming [ 11. Our definition of “fuzzy theory” is motivated solely by providing the minimal machinery to interpret loop-free schemes in a fuzzy way. Indeed, a fuzzy theory T = (T, e, (-)“) is defined in Section 1 by the data (A, B, C). For each set X there is given a new set TX of “distributions on X” or “vague specifications (A) of elements of X.” For each set X there is given a distinguished function e, : X-+ TX, “a crisp 03) specification is a special case of a vague one.” For each “fuzzy function” a: X-, TY there is given a distinguished “extension” (Cl a#: TX-+ TY. The data are all subject to three axioms. This definition is motivated by the flowchart scheme l.E. Some fundamental examples are crisp set theory: TX = X, CD) fuzzy set theory: TX = [0, llx, (E) * The research reported in this paper was supported in part by the National Science Foun- dation under Grant MCS76-84477. 409 0022-247X/82/020409-43$02.0010 Copyright 0 1982 by Academic Press, Inc.
    [Show full text]
  • Zhang Xianliang, Samuel Beckett and Albert Camus
    DEATH IN THREE NOVELS BY ZHANG XIANLIANG, SAMUEL BECKETT AND ALBERT CAMUS BY MAK MEI KWAN ALISA B.A., Chinese University of Hong Kong, 1997 THESIS Submitted to the Graduate School of The Chinese University of Hong Kong In partial fulfillment of the requirements for the degree MASTER OF PHILOSOPHY IN ENGLISH Hong Kong 2000 i(njy；! 11 )i UNIVERSITY ^^XLIBRARY Table of Contents Abstract i 摘要 iii Acknowledgements v Chapter One: 1 The Displaced Man Chapter Two: 19 The Fragmented Self in Xiguan siwang Getting Used to Dying Chapter Three: 46 Deatti of the Author: An Abandoned Being in Malofie\ Dies Chapter Four: 67 Death of Sharing: A Man of Authenticity in The Outsider Chapter Five: 94 Conclusion: The Helplessness of Life Works Cited 108 Wor卜 Consulted 115 丨 I 1 I i Abstract This thesis examines the fictional treatment of the topic death in three writers, Zhang Xianliang, Samuel Beckett and Albert Camus, of the century, presenting an age of life-in-death. Their treatments of death are believed to be the result of pressure from the socio-political background in different parts of the world. The first chapter is a brief sketch of the socio-political conditions affecting the three writers. It points out that an individual is by no means trapped in the relationship with the society, which makes them suffer from a deep sense of helplessness. The writers' personal experiences of being exposed to multi-national environment adds complexity to such relationship. r� I • ；Chapter Two examines how political power penetrates into the human mind and causes psychological distortion in the novel Getting Used to Dying 習 1 貫死亡� Itreveals the real power of the Chinese politics.
    [Show full text]
  • Appendix a Basic Concepts of Fuzzy Set Theory A.I Fuzzy Sets ILA{X): X
    Appendix A Basic Concepts of Fuzzy Set Theory This appendix gives the definitions of the concepts of fuzzy set theory, which are used in this book. For a comprehensive treatment of fuzzy set theory, see, for instance, (Zimmermann, 1996; Klir and Yuan, 1995). A.I Fuzzy Sets Definition A.I (fuzzy set) A fuzzy set A on universe (domain) X is defined by the membership function ILA{X) which is a mapping from the universe X into the unit interval: ILA{X): X -+ [0,1]. (A.l) F{X) denotes the set of all fuzzy sets on X. Fuzzy set theory allows for a partial membership of an element in a set. If the value of the membership function, called the membership degree (grade), equals one, x belongs completely to the fuzzy set. If it equals zero, x does not belong to the set. If the membership degree is between 0 and 1, x is a partial member of the fuzzy set. In the fuzzy set literature, the term crisp is often used to denote nonfuzzy quantities, e.g., a crisp number, a crisp set, etc. A.2 Membership Functions In a discrete set X = {Xi I i = 1,2, ... ,n}, a fuzzy set A may be defined by a list of ordered pairs: membership degree/set element: (A.2) or in the form of two related vectors: In continuous domains, fuzzy sets are defined analytically by their membership func­ tions. In this book, the following forms of membership functions are used: 228 FUZZY MODELING FOR CONTROL Trapezoidal membership function: x-a d-X) /-L(x;a,b,c,d)=max ( O,min(b_a,l'd_c) , (A4) where a, b, c and d are coordinates of the trapezoid apexes.
    [Show full text]