Fuzzy Preference Queries to Relational Databases

The following sections are included:

• Foreword

• Acknowledgments

• Contents

The following sections are included:

• Databases and their Evolution

• Preferences and Fuzzy Sets

• Overview of the Book

The following sections are included:

• Basic Notions and Vocabulary

• Algebraic Operations
  • Set operations
  • Relational operations
  • Properties

• An Overview of SQL
  • The base block
  • Combining base blocks
  • Partitioning
  • Expressing division and antidivision

The following sections are included:

• Introduction

• Definitions and Notations

• Composition of Fuzzy Sets
  • Intersection and union of fuzzy sets
  • Difference between fuzzy sets
  • Cartesian product of fuzzy sets
  • Trade-off operators
  • Nonsymmetric operators
    • “And if possible”
    • “Or else”
    • “All the more as”

• Inclusions and Implications
  • Fuzzy implications
    • Introduction
    • S-implications
    • R-implications
    • Contraposition of R-implications
    • Some characteristics of fuzzy implications
    • Semantic aspects of fuzzy implications
  • Inclusions
    • Introduction
    • Regular inclusion between fuzzy sets
    • Graded inclusion
    • Graded equality of fuzzy sets
    • Tolerant inclusion

• Fuzzy Measures and Integrals
  • Introduction
  • Fuzzy measures
  • Fuzzy integrals

• The Extension Principle

• Fuzzy Quantified Propositions
  • Fuzzy linguistic quantifiers
  • Quantified propositions

The following sections are included:

• Introduction

• Quantitative Approaches
  • Distances and similarity
  • Linguistic preferences
  • Explicit scores attached to entities
  • Top-k queries
  • Outranking

• Qualitative Approaches
  • Secondary preference criterion
  • Pareto-order-based approaches
  • CP-nets
  • Domain linearization
  • Possibilistic-logic-based approach

• Conclusion

The following sections are included:

• Introduction

• An Extended Relational Algebra

• An Overview of a Basic Version of SQLf
  • Introduction
  • The multiple relation base block
  • Subqueries
    • Nesting with “in”
    • Nesting with “exists”
    • Nesting with “all” and “any”
    • Nesting and scalar comparison
  • Set-oriented operators
  • Relation partitioning
    • Qualification of partitions by means of aggregate functions

• Interface for User-Defined Terms and Operators

• Contextual Queries
  • Queries with one level of context
  • Queries with several levels of context

• Evaluation of Simple Fuzzy Queries
  • Derivation principle
    • Transformation of base predicates
    • Single-block projection-selection-join queries
    • Nested queries
  • Derivation-based processing of SQLf queries
    • General strategy
    • Single-block selection-projection-join queries
    • Nested queries
    • Efficiency of the derivation-based processing strategy

• Conclusion

The following sections are included:

• Introduction

• Quantified Statements
  • Introduction
  • Quantified statements and fuzzy integral theory
    • Introduction
    • Fuzzy integrals and increasing quantifiers
  • Interpretation of statements of the type “Q X are A”
    • Interpretation according to Zadeh
    • Interpretation according to Prade
    • Yager’s competitive type aggregation
    • Interpretation based on the OWA operator
    • Interpretation of quantifiers of the type “about S”
  • Integration into SQLf
    • Horizontal quantification
    • Nested queries (extension of any and all)
    • Quantified statement in the having clause
  • Evaluation of SQLf queries involving quantified statements
    • Zadeh’s interpretation
    • Interpretation based on Sugeno’s integral
    • Interpretation based on Choquet’s integral

• Aggregates
  • Introduction
  • The case of monotonic predicates and aggregates
  • Dealing with the general case
    • A straightforward interpretation
    • The proposed interpretation
    • Algebraic properties related to negation
  • SQLf queries involving aggregates
    • Aggregates in the having clause
    • Nested queries
  • Evaluation of SQLf queries involving aggregates
    • Queries involving aggregates over fuzzy sets

• Conclusion

The following sections are included:

• Introduction

• Division of Fuzzy Relations
  • Principles
  • On the choice of implication
  • Primitivity of the extended division operator
  • Expressing extended division in SQLf

• Tolerant Division
  • Exception-based tolerant division
    • Quantitative tolerant division
    • Qualitative tolerant division
  • Resemblance-based tolerant division

• Stratified Division
  • Introduction
  • The queries
    • Conjunctive queries (CJ)
    • Disjunctive queries (DJ)
    • Full discrimination-based queries (FD)
  • Quotient property of the result delivered

• Queries Mixing Division and Antidivision
  • Motivation
  • Mixed stratified queries

• Evaluation of Division Queries
  • Processing the division of fuzzy relations
  • Processing the tolerant divisions of fuzzy relations
    • Quantitative exception-tolerant division
    • Qualitative exception-tolerant division
    • Resemblance-based tolerant division
  • Processing the conjunctive stratified division
    • Principle of the algorithms
    • Experiments and results

• Conclusion

The following sections are included:

• Introduction

• Preliminaries
  • About bipolarity

• Extended Algebraic Operators
  • Intersection
  • Union
  • Cartesian product
  • Negation
    • Approach inspired by twofold fuzzy sets
    • Counter-example
    • From “and if possible” to “or else”
    • An approach based on product and division
    • Negation based on antonymy
  • Difference
    • A logical view of difference
    • A non-logical view of the difference
    • Counter-example (first property).
    • Conclusion
  • Selection
    • Counter-example
  • Projection
  • Join
  • Division
    • Reminder about “classical” division
    • Reminder about the division of fuzzy relations
    • Bipolar division of bipolar fuzzy relations
    • About the quotient property
    • About the non-primitivity of the operator

• Implementation Aspects

• Conclusion

The following sections are included:

• Introduction

• An Extended Group By Clause
  • Use of a crisp partition
  • Use of a fuzzy partition

• Having Clause
  • Inclusion constraint
  • Aggregate₁ θ aggregate₂
  • Aggregate is ψ

• Application to Association Rule Mining
  • Rules of the type A is L_i → B is L′
    • Computation of the support
    • Computation of the confidence
  • Rules of the type A is L → B is L′_i

• Evaluation of a Fuzzy Group By

• Related Work
  • Extended group by
  • Fuzzy OLAP
  • Fuzzy database summarization techniques
  • Mining association rules with SQL

• Conclusion

The following sections are included:

• Introduction

• Empty Answer Problem
  • Query relaxation
  • Relaxation by predicate weakening
    • Principle
    • Dilation mechanism
    • Relaxing an atomic criterion
    • Relaxing a conjunctive query
  • Case-based reasoning approach
    • Introduction and principle
    • Predicate substitution
    • Conjunctive query replacement strategy

• Plethoric Answer Problem
  • Introduction
  • Approach based on predicate strengthening
    • Principle
    • Erosion operation
    • Erosion of an atomic query
    • Intensifying a conjunctive query
  • Approach based on query expansion
    • Fuzzy cardinalities and association rules
    • Shared vocabulary
    • Strengthening steps
    • Strengthening process
    • Experimentation

• Conclusion

To the best of our knowledge, this book is the first extensive monograph devoted to fuzzy preference queries. The main objectives we pursued were:

• to show that fuzzy set theory constitutes a suitable and highly expressive framework for modeling database preference queries,

• to study the algorithmic aspects of the evaluation of fuzzy queries, in order to demonstrate that this framework offers a good trade-off between expressivity and efficiency,

• to tackle some specific issues somewhat related to fuzzy querying, such as bipolarity (Chapter 8) and cooperative answering (Chapter 10).

The following sections are included:

• Bibliography

• Index