Introduction to Relational Calculus

by K. Yue

1. Introduction

• Non-procedural, declarative, and high level.
• Two kinds:
  1. Domain Relational Calculus (DRC)
  2. Tuple Relational Calculus (TRC)
• Queries specified by the set builder form: {s | cond(s) }
• cond(s) is known as a formula.
• Constructs:
  • Variables:
    1. TRC: tuples (bound to tuples): e.g. s, t, student, class, etc.
    2. DRC: attributes (bound to domain value): e.g. a, b, c, stuId, firstName, etc.
    3. The variables are sometime known as 'dummy variables.' They can assume any names.
  • Constants: string, int, etc. E.g. 12, 'csci', 3.7.
  • Comparison operators: <, <, =, etc.
  • Boolean operators: and (conjunction, ∧ or just ,), or (disjunction ∨), not (¬), implies (⇒), etc.
  • Membership functions: belongs to, ∈, not belongs to, ∉, etc.
  • Quantifiers: there exists (existential, ∃), not exists (∄), for all (universal ∀).
• An atom can be thought of as a simple Boolean expression: x op y, where x and y are attributes or constants, and op is a comparison operation.
  • Example: R.a>=2, sname='BBC', t ∉ T, etc.
• A formula is either an atom or formulas connected by Boolean operators or qualifiers.
  • Example: ∃e(e ∈ R), a=1 ∨ b=2, ∃a,b,c((a,b,c) ∈ R) or simply (a,b,c) ∈ R
• A formula that is not an atom can be thought of a compound Boolean expression.
• A variable is bound if it has a finite number of values., such as appearing in qualifier expressions (without negation). A free variable is not bound.
• Relational Calculus expressions need to be safe: results should be a finite set of tuples.
• Care should be taken especially for the negation operation. E.g. {s |¬ (s ∈ Student) } is unsafe.
• For a given implementation of relational calculus:
  • There may be restrictions of supported constructs.
  • There may be certain canonical requirements: e.g. conjunction of disjunction.
• Relational Calculus and Relational Algebra:
  1. All RA expressions can be expressed in RC.
  2. RA and RC have the same expressive power.
  3. Any query language that can express all RA queries is known to be relational complete.

Example:

{i | i ∈ I ∧ i % 2 =0}
{i | i ∈ I, i % 2 =0} -- set builder form.

{t | ∃r ∈R, r.firstname = t.firstname, r.lastname = t.lastname}

• t is a free variable.
• It will have two attributes: t.firstname and t.lastname.

Alternatively, we can use the set builder form in the LHS before |:

{(r.firstname, r.lastname) | r ∈ R}

R(A,B,C,D) / S(C,D)

{(a,b) | ∀(c,d) ∈ S((a,b,c,d) ∈ R))}

• In this class, with the query is complicated, you may use intermediate relations to construct a sequence of RC expressions to provide the result.

Exercises:

How do you use RC to implement RA operations?

2. TRC

• The variables in TORC are tuples.
• SQL is based on TRC.

Exercise:

Work on some of the query questions listed in the Supplies database example in TRC.

3. DRC

• The variables in DRC are attributes (domain values).
• May use anonymous variable: _ as a placeholder.
• Query By Example (QBE) is based on DRC.

Exercise:

Work on some of the query questions listed in the Supplies database example in DRC.