the respective websites or databases) into an easily processed form for the data integration solution (see figure 2). When an application-user queries the mediated schema, the data-integration solution transforms this query into appropriate queries over the respective data sources. Finally, the virtual database combines the results of these queries into the answer to the user's query. This solution offers the convenience of adding new sources by simply constructing an adapter or an application software blade for them. It contrasts with ETL systems or with a single database solution, which require manual integration of entire new data set into the system. The virtual ETL solutions leverage virtual mediated schema to implement data harmonization; whereby the data are copied from the designated "master" source to the defined targets, field by field. Advanced data virtualization is also built on the concept of object-oriented modeling in order to construct virtual mediated schema or virtual metadata repository, using hub and spoke architecture. Each data source is disparate and as such is not designed to support reliable joins between data sources. Therefore, data virtualization as well as data federation depends upon accidental data commonality to support combining data and information from disparate data sets. Because of the lack of data value commonality across data sources, the return set may be inaccurate, incomplete, and impossible to validate. One solution is to recast disparate databases to integrate these databases without the need for ETL. The recast databases support commonality constraints where referential integrity may be enforced between databases. The recast databases provide designed data access paths with data value commonality across databases. Theory The theory of data integration forms a subset of database theory and formalizes the underlying concepts of the problem in first-order logic. Applying the theories gives indications as to the feasibility and difficulty of data integration. While its definitions may appear abstract, they have sufficient generality to accommodate all manner of integration systems, including those that include nested relational / XML databases and those that treat databases as programs. Connections to particular databases systems such as Oracle or DB2 are provided by implementation-level technologies such as JDBC and are not studied at the theoretical level. Definitions Data integration systems are formally defined as a tuple where is the global (or mediated) schema, is the heterogeneous set of source schemas, and is the mapping that maps queries between the source and the global schemas. Both and