Composition of Schema Mappings: Syntax and Semantics

Abstract

In Chap. 2, the formal logical framework for the schema mappings is defined, based on the second-order tuple generating dependencies (SOtgds), with existentially quantified functional symbols. Each tgd is a material implication from the conjunctive formula (with relational symbols of a source schema, preceded with negation as well) into a particular relational symbol of the target schema. It provides a number of algorithms which transform these logical formulae into the algebraic structure based on the theory of R-operads. The schema database integrity constraints are transformed in a similar way so that both the schema mappings and schema integrity-constraints are formally represented by R-operads. Then the compositional properties are explored, in order to represent a database mapping system as a graph where the nodes are the database schemas and the arrows are the schema mappings or the integrity-constraints for schemas. This representation is used to define the database mapping sketches (small categories), based on the fact that each schema has an identity arrow (mapping) and that the mapping-arrows satisfy the associative low for the composition of them.