Big Data Integration Theory
Preface
Dependencies Between the Chapters
Detailed Plan
Acknowledgements
Notational Conventions
References
Contents
Chapter 1: Introduction and Technical Preliminaries
1.1 Historical Background
1.2 Introduction to Lattices, Algebras and Intuitionistic Logics
1.3 Introduction to First-Order Logic (FOL)
1.3.1 Extensions of the FOL for Database Theory
1.4 Basic Database Concepts
1.4.1 Basic Theory about Database Observations: Idempotent Power-View Operator
1.4.2 Introduction to Schema Mappings
1.5 Basic Category Theory
1.5.1 Categorial Symmetry
References
Chapter 2: Composition of Schema Mappings: Syntax and Semantics
2.1 Schema Mappings: Second-Order tgds (SOtgds)
2.2 Transformation of Schema Integrity Constraints into SOtgds
2.2.1 Transformation of Tuple-Generating Constraints into SOtgds
2.2.2 Transformation of Equality-Generating Constraints into SOtgds
2.3 New Algorithm for General Composition of SOtgds
2.3.1 Categorial Properties for the Schema Mappings
2.4 Logic versus Algebra: Categoriο¬cation by Operads
2.4.1 R-Algebras, Tarski's Interpretations and Instance-Database Mappings
2.4.2 Query-Answering Abstract Data-Object Types and Operads
2.4.3 Strict Semantics of Schema Mappings: Information Fluxes
2.5 Algorithm for Decomposition of SOtgds
2.6 Database Schema Mapping Graphs
2.7 Review Questions
References
Chapter 3: Deο¬nition of DB Category
3.1 Why Do We Need a New Base Database Category?
3.1.1 Introduction to Sketch Data Models
3.1.2 Atomic Sketch's Database Mappings
3.2 DB (Database) Category
3.2.1 Power-View Endofunctor and Monad T
3.2.2 Duality
3.2.3 Symmetry
3.2.4 (Co)products
3.2.5 Partial Ordering for Databases: Top and Bottom Objects
3.3 Basic Operations for Objects in DB
3.3.1 Data Federation Operator in DB
3.3.2 Data Separation Operator in DB
3.4 Equivalence Relations in DB Category
3.4.1 The (Strong) Behavioral Equivalence for Databases
3.4.2 Weak Observational Equivalence for Databases
3.5 Review Questions
References
Chapter 4: Functorial Semantics for Database Schema Mappings
4.1 Theory: Categorial Semantics of Database Schema Mappings
4.1.1 Categorial Semantics of Database Schemas
4.1.2 Categorial Semantics of a Database Mapping System
4.1.3 Models of a Database Mapping System
4.2 Application: Categorial Semantics for Data Integration/Exchange
4.2.1 Data Integration/Exchange Framework
4.2.2 GLAV Categorial Semantics
4.2.3 Query Rewriting in GAV with (Foreign) Key Constraints
4.2.3.1 Query Rewriting Coalgebra Semantics
4.2.4 Fixpoint Operator for Finite Canonical Solution
4.3 Review Questions
References
Chapter 5: Extensions of Relational Codd's Algebra and DB Category
5.1 Introduction to Codd's Relational Algebra and Its Extensions
5.1.1 Initial Algebras and Syntax Monads: Power-View Operator
5.2 Action-Relational-Algebra RA Category
5.2.1 Normalization of Terms: Completeness of RA
5.2.2 RA versus DB Category
5.3 Relational Algebra and Database Schema Mappings
5.4 DB Category and Relational Algebras
5.5 Review Questions
Reference
Chapter 6: Categorial RDB Machines
6.1 Relational Algebra Programs and Computation Systems
6.1.1 Major DBMS Components
6.2 The Categorial RBD Machine
6.2.1 The Categorial Approach to SQL Embedding
6.2.2 The Categorial Approach to the Transaction Recovery
6.3 The Concurrent-Categorial RBD Machine
6.3.1 Time-Shared DBMS Components
6.3.2 The Concurrent Categorial Transaction Recovery
6.4 Review Questions
Reference
Chapter 7: Operational Semantics for Database Mappings
7.1 Introduction to Semantics of Process-Programming Languages
7.2 Updates Through Views
7.2.1 Deletion by Minimal Side-Effects
7.2.2 Insertion by Minimal Side-Effects
7.3 Denotational Model (Database-Mapping Process) Algebra
7.3.1 Initial Algebra Semantics for Database-Mapping Programs
7.3.2 Database-Mapping Processes and DB-Denotational Semantics
7.4 Operational Semantics for Database-Mapping Programs
7.4.1 Observational Comonad
7.4.2 Duality and Database-Mapping Programs: Speciο¬cation Versus Solution
7.5 Semantic Adequateness for the Operational Behavior
7.5.1 DB-Mappings Denotational Semantics and Structural Operational Semantics
7.5.2 Generalized Coinduction
7.6 Review Questions
References
Chapter 8: The Properties of DB Category
8.1 Expressive Power of the DB Category
8.1.1 Matching Tensor Product
8.1.2 Merging Operator
8.1.3 (Co)Limits and Exponentiation
8.1.4 Universal Algebra Considerations
8.1.5 Algebraic Database Lattice
8.2 Enrichment
8.2.1 DB Is a V-Category Enriched over Itself
8.2.2 Internalized Yoneda Embedding
8.3 Database Mappings and (Co)monads: (Co)induction
8.3.1 DB Inductive Principle and DB Objects
8.3.2 DB Coinductive Principle and DB Morphisms
8.4 Kleisli Semantics for Database Mappings
8.5 Review Questions
References
Chapter 9: Weak Monoidal DB Topos
9.1 Topological Properties
9.1.1 Database Metric Space
9.1.2 Subobject Classiο¬er
9.1.3 Weak Monoidal Topos
9.2 Intuitionistic Logic and DB Weak Monoidal Topos
9.2.1 Birkhoff Polarity over Complete Lattices
9.2.2 DB-Truth-Value Algebra and Birkhoff Polarity
9.2.3 Embedding of WMTL (Weak Monoidal Topos Logic) into Intuitionistic Bimodal Logics
9.2.4 Weak Monoidal Topos and Intuitionism
9.3 Review Questions
References
Index