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Seminar Details

Date 5-12-2003
Time 9:30
Room/Location 218
Title Denotational semantics for View-based Database Mappings
Speaker Zoran Majkic
Affiliation Univ. di Roma
Abstract This document presents a categorical denotational semantics for database mapping, based on views, in the most general framework, database-integration/exchange and Peer-to-peer. The base database category DB (instead of traditional Set category) , with objects database-instances and with morphisms (mappings which are not functions) between them, is used (at instance level) as a proper semantic domain for a database mappings based on a set of complex query computations. The higher (logical) level schema mappings between databases, usually written in some high expressive logical language (for example, LAV, GAV, GLAV, tuple generating dependency) may be functorially translated into this base "computation" category. The new approach based on the behavioral point of view for databases is assumed, and are established behavioural equivalences for databases and their mappings. The introduction of observations, which are computations without side-effects, also defines the fundamental (from Universal algebra) monad endofunctor T, which is also the closure operator for objects and for morphisms such that the database lattice is an algebraic (complete and compact) lattice, where Ob{DB} is a set of all objects (database instances) of DB category and "\preceq" is a preorder relation between them. The join and meet operators of this database lattice are Merging and Matching database operators respectively. The resulting 2-category DB is symmetric (also a mapping is represented as an object, that is, a database instance), so, a mappings between mappings are a 1-cell morphisms for all higher meta-theories: each mapping is a homomorphism from a Kleisli monadic T-coalgebra into the cofree monadic T-coalgebra. The database category DB has nice properties: it is equal to its dual, complete and cocomplete, locally small and locally finitely presentable, and monoidal biclosed V-category enriched over itself. The monad derived from the endofunctor T is an enriched monad. Generally, database mappings are not simply programs from values (relations) into computations (views) but an equivalence of computations: because of that each mapping, from any two databases A and B, is equivalently reversible and gives a duality property to the category DB. The denotational semantics of database mappings is given by morphisms of the Kleisli category DB_T which may be "internalized" in DB category as "computations". Special attention is devoted to some practical examples: query definition, query rewriting in the Database-integration environment, P2P mappings and their equivalent GAV translation.
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