Contents Introduction and Technical Preliminaries Composition of Schema Mappings: Syntax and Semantics
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1 1 Introduction and Technical Preliminaries Historical Background Introduction to Lattices, Algebras and Intuitionistic Logics Introduction to First-Order Logic (FOL) Extensions of the FOL for Database Theory Basic Database Concepts Basic Theory about Database Observations: Idempotent Power-ViewOperator Introduction to Schema Mappings Basic Category Theory CategorialSymmetry References Composition of Schema Mappings: Syntax and Semantics Schema Mappings: Second-Order tgds (SOtgds) Transformation of Schema Integrity Constraints into SOtgds Transformation of Tuple-Generating Constraints into SOtgds Transformation of Equality-Generating Constraints into SOtgds New Algorithm for General Composition of SOtgds Categorial Properties for the Schema Mappings Logic versus Algebra: Categorification by Operads R-Algebras, Tarski s Interpretations and Instance-Database Mappings Query-Answering Abstract Data-Object Types and Operads Strict Semantics of Schema Mappings: Information Fluxes Algorithm for Decomposition of SOtgds Database Schema Mapping Graphs xvii
2 xviii 2.7 ReviewQuestions References Definition of DB Category Why Do We Need a New Base Database Category? Introduction to Sketch Data Models Atomic Sketch s Database Mappings DB (Database) Category Power-View Endofunctor and Monad T Duality Symmetry (Co)products Partial Ordering for Databases: Top and Bottom Objects Basic Operations for Objects in DB Data Federation Operator in DB Data Separation Operator in DB Equivalence Relations in DB Category The (Strong) Behavioral Equivalence for Databases Weak Observational Equivalence for Databases ReviewQuestions References Functorial Semantics for Database Schema Mappings Theory: Categorial Semantics of Database Schema Mappings Categorial Semantics of Database Schemas Categorial Semantics of a Database Mapping System Models of a Database Mapping System Application: Categorial Semantics for Data Integration/Exchange Data Integration/Exchange Framework GLAVCategorialSemantics Query Rewriting in GAV with (Foreign) Key Constraints Fixpoint Operator for Finite Canonical Solution ReviewQuestions References Extensions of Relational Codd s Algebra and DB Category Introduction to Codd s Relational Algebra and Its Extensions Initial Algebras and Syntax Monads: Power-View Operator Action-Relational-Algebra RA Category Normalization of Terms: Completeness of RA RA versus DB Category Relational Algebra and Database Schema Mappings DB Category and Relational Algebras
3 xix 5.5 ReviewQuestions Reference Categorial RDB Machines Relational Algebra Programs and Computation Systems Major DBMS Components The Categorial RBD Machine The Categorial Approach to SQL Embedding The Categorial Approach to the Transaction Recovery The Concurrent-Categorial RBD Machine Time-Shared DBMS Components The Concurrent Categorial Transaction Recovery ReviewQuestions Reference Operational Semantics for Database Mappings Introduction to Semantics of Process-Programming Languages Updates Through Views DeletionbyMinimalSide-Effects InsertionbyMinimalSide-Effects Denotational Model (Database-Mapping Process) Algebra Initial Algebra Semantics for Database-Mapping Programs Database-Mapping Processes and DB-Denotational Semantics Operational Semantics for Database-Mapping Programs Observational Comonad Duality and Database-Mapping Programs: Specification VersusSolution Semantic Adequateness for the Operational Behavior DB-Mappings Denotational Semantics and Structural Operational Semantics Generalized Coinduction ReviewQuestions References The Properties of DB Category Expressive Power of the DB Category Matching Tensor Product MergingOperator (Co)Limits and Exponentiation Universal Algebra Considerations Algebraic Database Lattice Enrichment
4 xx DB Is a V-Category Enriched over Itself Internalized Yoneda Embedding Database Mappings and (Co)monads: (Co)induction DB Inductive Principle and DB Objects DB Coinductive Principle and DB Morphisms Kleisli Semantics for Database Mappings ReviewQuestions References Weak Monoidal DB Topos Topological Properties Database Metric Space Subobject Classifier Weak Monoidal Topos Intuitionistic Logic and DB Weak Monoidal Topos Birkhoff Polarity over Complete Lattices DB-Truth-Value Algebra and Birkhoff Polarity Embedding of WMTL (Weak Monoidal Topos Logic) into Intuitionistic Bimodal Logics Weak Monoidal Topos and Intuitionism ReviewQuestions References Index
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