DBMS : Domain Relational Calculus (DRC)

 

Domain Relational Calculus (DRC):

• The domain relational calculus is a calculus that was introduced as a declarative database query language for the relational data model.
• It uses domain variables that take on value from an attribute domain rather than value for an entire tuple.
• To form a relation of degree for a query result, we must have ‘n’ of these domain variable one for each attribute.
• An expression is in the form:

{< x₁,x₂,...,x_n> | P(x₁,x₂,...,x_n)}

 

where x_i, 1<=i<=n, represent domain variables and P is a formula.

• An atom in the domain relational calculus is of the following form:

1. {< x₁,x₂,...,x_n > ∈ r, where ‘r’ is a relation on ‘r’ attributes and x_i, 1<=i<=n are domain variables or constants.
2. x Ө y, where x and y are domain variables, and  Ө  is a comparison operator.
3. x Ө c, where c is a constant.

• Formulae are built up from atoms using the following rules:

1.  An atom is a formula.
2.  If P is a formula, then so are ㄱP and (P).
3. If P1 and P2 are formulae, then so are P1 ⋃ P2, P1 ⋂ P2 and P1=>P2.
4. If P(x) is a formula, where x is a domain variable, then so are ∃ x (P(x)) and ∀ x (P(x)).

 

Examples:

 

a)    Find branch name, loan number, customer name and amount for loans of over $1200.

 

{<b,l,c,a>|<b,l,c,a> ∈ borrow  ⋂ a>1200 }

 

b)    Find all customers who have a loan for an amount greater than $1200.

 

{<c>| ∃ b,l,a <b,l,c,a> ∈ borrow  ⋂ a>1200 }

 

c)    Find all customers having a loan from the Miami branch and the city in which they live.

 

{<c,x>| ∃ b,l,a <b,l,c,a> ∈ borrow  ⋂ b= “Miami” ⋂ ∃ Y(<c,y,x> ∈ customer))) }

 

 

Safety of Expressions in Domain Relational Calculus:

• In tuple relational calculus, it is possible to write expressions that may generate infinite relations. The same situation arises for a domain relational calculus.
• In the domain relational calculus, we must concern about the formula within “there exists” and “for all”.
• Consider the expression:

 

{x|∃ (<x,y> r) Λ ∃ z (ㄱ(<x,z>r) Λ P (x,z))}

• In the second part of this expression; ∃z (ㄱ(<x,z>r)ΛP(x,z))} we need to consider values for z that do not appear in r .
• This set of values actually is an infinite set.
• This case does not appear in the tuple relational calculus because, in tuple relational calculus, the universal and existential quantifiers are applied on variables that already range over a specific relation (in the form ∃ t r (P (t)) and ∀ t r (P (t))).
• Therefore, in DRC, an expression {<x₁, x₂, …,x_n> | P(x₁, x₂, …,x_n)} is safe if all the following condition holds:

 

1. All values in the result are values from DOM (P).
2. For every “there exists” sub formula of the form ∃ x (P1(x)), the sub formula is true if there is a value x in DOM (P1) such that P1(x) is true.
3. For every “for all” sub formula of the form ∀ x (P2(x)), the sub formula is true if for all values x from DOM (P2), P2(x) is true.