Relational Query Languages

A query language in which a user requests information from the database.

Query languages:

• imperative
• functional
• declarative

Pure languages:

• relational algebra (functional)
• tuple relational calculus (declarative)
• domain relational calculus (declarative)

Tuple Relational Calculus

Each query is of the form ${t | P(t)}$. It is the set of all tuples $t$ such that predicate $P$ is true for $t$. $t$ is a tuple variable. $t[A]$ denotes the value of tuple $t$ on attribute $A$. $t \in r$ denotes that tuple $t$ is in relation $r$. $P$ is a formula similar to that of the predicate calculus.

Predicate Calculus formula

• Set of attributes and constants
• Set of comparison operators
• set of connectives
• Implication, if-then
• Set of quantifiers, (there exists, for all)

Safety of Expressions

It is possible to write tuple calculus expressions that generate infinite relations. For example, ${t | \neg t \in r}$ results in an infinite relation if the domain of any attribute of relation $r$ is infinite. To guard against the problem, we restrict the set of allowable expressions to safe expressions. An expression {t|P(t)} in the tuple relational calculus is safe if every component of $t$ appears in one of the relations, tuples or constants that appear in $P$.

The expression ${ < x_1, x_2, ..., x_n > | P (x_1, x_2, ..., x_n)}$ is safe if all of the following hold:

1. All values that appear in tuples of the expression are values from dom($P$)
2. For every "there exists" sub formula of the form $\exists x (P_1 (x))$, the subformula is true if and only if there is a value of $x$ in dom($P_1$) such that $P_1(x)$ is true.
3. For every "for all" subformula of the form $\forall_x (P_1 (x))$, the subformula is true if and only if $P_1(x)$ is true for all values $x$ from dom($P_1$)

Domain Relational Calculus

• a nonprocedural query language equivalent in power to the tuple relational calculus
• Each query in an expression of the form: ${ < x_1, x_2, ..., x_n > | P (x_1, x_2, ..., x_n)}$
  • $x_i$ represents domain variables
  • $P$ represents a formula similar to that of the predicate calculus