Relational Algebra.

Relational algebras received little attention until the publication of E.F. Codd's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages. The first query language to be based on Codd's algebra was ISBL, and this pioneering work has been acclaimed by many authorities as having shown the way to make Codd's idea into a useful language. Business System 12 was a short-lived industry-strength relational DBMS that followed the ISBL example. In 1998 Chris Date and Hugh Darwen proposed a language called Tutorial D intended for use in teaching relational database theory, and its query language also draws on ISBL's ideas. Rel is an implementation of Tutorial D. Even the query language of SQL is loosely based on a relational algebra, though the operands in SQL (tables) are notexactly relations and several useful theorems about the relational algebra do not hold in the SQL counterpart (arguably to the detriment of optimisers and/or users).

Because a relation is interpreted as the extension of some predicate, each operator of a relational algebra has a counterpart in predicate calculus. For example, the natural join is a counterpart of logical AND (). If relations R and S represent the extensions of predicates p1 and p2, respectively, then the natural join of R and S (R S) is a relation representing the extension of the predicate p1 p2.

It is important to realise that Codd's algebra is not in fact complete with respect to first-order logic. Had it been so, certain insurmountable computational difficulties would have arisen for any implementation of it. To overcome these difficulties, he restricted the operands to finite relations only and also proposed restricted support for negation (NOT) and disjunction (OR). Analogous restrictions are found in many other logic-based computer languages. Codd defined the term relational completeness to refer to a language that is complete with respect to first-order predicate calculus apart from the restrictions he proposed. In practice the restrictions have no adverse effect on the applicability of his relational algebra for database purposes.
As in any algebra, some operators are primitive and the others, being definable in terms of the primitive ones, are derived. It is useful if the choice of primitive operators parallels the usual choice of primitive logical operators. Although it is well known that the usual choice in logic of AND, OR and NOT is somewhat arbitrary, Codd made a similar arbitrary choice for his algebra.

The six primitive operators of Codd's algebra are the selection, the projection, the Cartesian product (also called the cross product or cross join), the set union, the set difference, and the rename. (Actually, Codd omitted the rename, but the compelling case for its inclusion was shown by the inventors of ISBL.) These six operators are fundamental in the sense that none of them can be omitted without losing expressive power. Many other operators have been defined in terms of these six. Among the most important are set intersection, division, and the natural join. In fact ISBL made a compelling case for replacing theCartesian product by the natural join, of which the Cartesian product is a degenerate case.

Altogether, the operators of relational algebra have identical expressive power to that of domain relational calculus or tuple relational calculus. However, for the reasons given in the Introduction above, relational algebra has strictly less expressive power than that of first-order predicate calculus without function symbols. Relational algebra actually corresponds to a subset of first-order logic that is Horn clauses without recursion and negation. The Relational Algebra is a procedural query language. Six fundamental operations :
Unary
 Select
 Project
 Rename
Binary
 Cartesian product
 Union
 Set-difference Several other operations, defined in terms of the fundamental operations :
 Set-intersection
 Natural join
 Division
 Assignment Operations produce a new relation as a result.



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