The Relational Algebra and Relational Calculus
In this chapter we discuss the two formal languages for the relational model: the relational algebra and the relational calculus. In contrast, Chapters 4 and 5 described the practical language for the relational model, namely the SQL standard. Historically, the relational algebra and calculus were developed before the SQL language. In fact, in some ways, SQL is based on concepts from both the algebra and the calculus, as we shall see. Because most relational DBMSs use SQL as their language, we presented the SQL language first.
Recall from Chapter 2 that a data model must include a set of operations to manipulate the database, in addition to the data model’s concepts for defining the data-base’s structure and constraints. We presented the structures and constraints of the formal relational model in Chapter 3. The basic set of operations for the relational model is the relational algebra. These operations enable a user to specify basic retrieval requests as relational algebra expressions. The result of a retrieval is a new relation, which may have been formed from one or more relations. The algebra operations thus produce new relations, which can be further manipulated using operations of the same algebra. A sequence of relational algebra operations forms a relational algebra expression, whose result will also be a relation that represents the result of a database query (or retrieval request).
The relational algebra is very important for several reasons. First, it provides a formal foundation for relational model operations. Second, and perhaps more important, it is used as a basis for implementing and optimizing queries in the query processing and optimization modules that are integral parts of relational database management systems (RDBMSs), as we shall discuss in Chapter 19. Third, some of its concepts are incorporated into the SQL standard query language for RDBMSs.
Although most commercial RDBMSs in use today do not provide user interfaces for relational algebra queries, the core operations and functions in the internal modules of most relational systems are based on relational algebra operations. We will define these operations in detail in Sections 6.1 through 6.4 of this chapter.
Whereas the algebra defines a set of operations for the relational model, the relational calculus provides a higher-level declarative language for specifying relational queries. A relational calculus expression creates a new relation. In a relational calculus expression, there is no order of operations to specify how to retrieve the query result—only what information the result should contain. This is the main distinguishing feature between relational algebra and relational calculus. The relational calculus is important because it has a firm basis in mathematical logic and because the standard query language (SQL) for RDBMSs has some of its foundations in a variation of relational calculus known as the tuple relational calculus.
The relational algebra is often considered to be an integral part of the relational data model. Its operations can be divided into two groups. One group includes set operations from mathematical set theory; these are applicable because each relation is defined to be a set of tuples in the formal relational model (see Section 3.1). Set operations include UNION, INTERSECTION, SET DIFFERENCE, and CARTESIAN PRODUCT (also known as CROSS PRODUCT). The other group consists of operations developed specifically for relational databases—these include SELECT, PROJECT, and JOIN, among others. First, we describe the SELECT and PROJECT operations in Section 6.1 because they are unary operations that operate on single relations. Then we discuss set operations in Section 6.2. In Section 6.3, we discuss JOIN and other complex binary operations, which operate on two tables by combining related tuples (records) based on join conditions. The COMPANY relational database shown in Figure 3.6 is used for our examples.
Some common database requests cannot be performed with the original relational algebra operations, so additional operations were created to express these requests. These include aggregate functions, which are operations that can summarize data from the tables, as well as additional types of JOIN and UNION operations, known as OUTER JOINs and OUTER UNIONs. These operations, which were added to the orig-inal relational algebra because of their importance to many database applications, are described in Section 6.4. We give examples of specifying queries that use rela-tional operations in Section 6.5. Some of these same queries were used in Chapters 4 and 5. By using the same query numbers in this chapter, the reader can contrast how the same queries are written in the various query languages.
In Sections 6.6 and 6.7 we describe the other main formal language for relational databases, the relational calculus. There are two variations of relational calculus. The tuple relational calculus is described in Section 6.6 and the domain relational calculus is described in Section 6.7. Some of the SQL constructs discussed in Chapters 4 and 5 are based on the tuple relational calculus. The relational calculus is a formal language, based on the branch of mathematical logic called predicate cal culus. In tuple relational calculus, variables range over tuples, whereas in domain relational calculus, variables range over the domains (values) of attributes. In Appendix C we give an overview of the Query-By-Example (QBE) language, which is a graphical user-friendly relational language based on domain relational calculus. Section 6.8 summarizes the chapter.
For the reader who is interested in a less detailed introduction to formal relational languages, Sections 6.4, 6.6, and 6.7 may be skipped.