The Relational Calculus

The Relational Calculus

• The Relational Calculus– High-level, non-procedural language– Relational calculus expression declares the requirements

tuples in the resulting relation must satisfy.– Provides foundation for standard query language (SQL)

• Expressive power– Any expression in relational algebra can be expressed in

relational calculus.• The tuple relational calculus• The domain relational calculus

The Tuple Relational Calculus

• A tuple variable– Ranges over a database relation, denoted R(t)– Reference to an attribute of a tuple is denoted using the

‘.’ notation• Tuple relational expression

– Form: {t | COND(t)}, where COND(t) is a conditional expression involving t

The Tuple Relational Calculus

Example: {t.NAME | EMPLOYEE(t) AND t.SALARY>50000}– The range relation of t is EMPLOYEE– The select condition that must evaluate to true:

SALARY>50000– The requested attribute: NAME

Expressions and Formulas

• A general expression{t1.Aj, t2.Ak,…, tn.Ap | COND(t1, t2,…, tn, tn+1,…, tn+m)}

– t1, t2,…, tn, tn+1,…, tn+m: tuple variables

– ti.Aj: attribute Aj of the relation on which ti ranges– COND is a condition or formula made of predicate

calculus atoms

Expressions and Formulae

• Atom– R(t): R is a relation name and t is a tuple variable– ti.A op tj.B: A and B are attributes of the relations on which

ti and tj range, respectively; op {=, <, , >, , }– ti.A op c: c is a constant value in the domain of A

• Formula– An atom is a formula.– If F is a formula, then so is NOT(F). – If F1 and F2 are formulas, then so are (F1 AND F2), (F1 OR

F2), and (IF F1 THEN F2).

Quantifiers

• Existential quantifier – If F is a formula, then so is (t)(F), where t is a tuple

variable.– (t)(F) is true if there exists some tuple that makes F

true.• Universal quantifier

– If F is a formula, then so is (t)(F), where t is a tuple variable.

– (t)(F) is true if F is true for every substitution for t.

Other Notations

R(t) tR NOT F FF1 AND F2 F1 F2

F1 OR F2 F1 F2

IF F1 THEN F2 F1 F2

Equivalences and Implications

(F1 F2) F1 F2

(F1 F2) F1 F2

F1 F2 F1 F2

(x)(P(x)) (x)(P(x))(x)(P(x)) (x)(P(x))(x)(P(x)) (x)(P(x))(x)(P(x)) (x)(P(x))

Free and Bound Variables

• Informally– A tuple variable is bound if it is quantified, otherwise it

is free.

Example:F1: d.DNAME=‘RESEARCH’

F2: (t)(d.NUMBER=t.DNO)

F3: (d)(d.MGRSSN=‘333445555’)

Examples

Q1: “Retrieve the name and address of all employees who work for the ‘Research’ department.”

{t.FNAME, t.LNAME, t.ADDRESS | EMPLOYEE(t) AND (d)(DEPARTMENT(d) AND d.NAME=‘Research’ AND d.NUMBER=t.DNO)}

t is the only free tuple variable, bound successively to each tuple, d.NAME=‘Research’ corresponds to a SELECT operation, and d.NUMBER=t.DNO serves as a join condition in relation algebra.

Examples

Q2: “For every project in ‘Stafford’, list the project number, the controlling department number, and the department manager’s last name, and address.”

{p.PNUMBER, p.DNUM, m.LNAME, m.ADDRESS | EMPLOYEE(m) AND PROJECT(p) AND p.LOCATION=‘Stafford’ AND ((d)(DEPARTMENT(d) AND p.DNUM=d.DNUMBER AND d.MGRSSN=m.SSN))}

Examples

Q3: “Find the name of each employee who works on some project controlled by department number 5.”

{e.LNAME, e.FNAME | EMPLOYEE(e) AND ((p)(w) (PROJECT(p) AND WORKS_ON(w) AND p.DNUM=5 AND w.ESSN=e.SSN AND p.PNUMBER=w.PNO))}

Examples

Q4: “Make a list of project numbers for projects that involve an employee whose last name is “Smith”, either as a worker or as manager of the controlling department for the project.”

{p.PNUMBER | PROJECT(p) AND (((e)(w)(EMPLOYEE(e) AND WORKS_ON(w) AND w.PNO=p.PNUMBER AND e.LNAME=‘Smith’ AND e.SSN=w.ESSN)) OR ((m)(d)(EMPLOYEE(m) AND DEPARTMENT(d) AND p.DNUM=d.DNUMBER AND d.MGRSSN=m.SSN AND m.LNAME=‘Smith’)))}

Examples

Q5: “Find the names of employees who work on all projects controlled by department number 5.”

{e.LNAME, e.FNAME | EMPLOYEE(e) AND ((x) ((PROJECT(x) AND (x.DNUM=5))

((w)(WORKS_ON(w) AND w.ESSN=e.SSN AND x.PNUMBER=w.PNO)) )}

Examples

Q6: “Find the names of employees who have no dependents.”

{e.LNAME, e.FNAME | EMPLOYEE(e) AND (NOT(d) (DEPENDENT(d) AND e.SSN=d.ESSN))}

Examples

Q7: “List the names of managers who have at least one dependent.”

{e.LNAME, e.FNAME | EMPLOYEE(e) AND ((d)(p) (DEPARTMENT(d) AND DEPENDENT(p) AND e.SSN=d.MGRSSN AND e.SSN=p.ESSN))}

Safe Expressions

• Safe expression– Yields a result with a finite number of tuples– All values in its result are from the domain of the

expression, which is the set of all values referenced by the expression or exist in any tuple in the relations referenced in the expression.

Examples: – dom(EMPLOYEE(t) AND t.SALARY>25000): 25000

and all values appear in EMPLOYEE}– {t | NOT(EMPLOYEE(t))} is not safe.

The Domain Relational Calculus

• Variables– Range over single values from domains of attributes

• Domain relational expression– {x1, x2,…,xn | COND(x1, x2,…,xn, xn+1, xn+2,…,xn+m)}

– x1, x2,…,xn, xn+1, xn+2,…,xn+m are domain variables– COND is a condition or formula of the domain

relational calculus,

The Domain Relational Calculus

• Atom– R(x1, x2,…,xj): R is a relation of degree j and xi is a

domain variable.– xi op xj: xi and xj are domain variables and op {=, <,

, >, , }– xi op c: c is a constant value in the domain of xi

• Formula– Made of atoms, variables, and quantifiers

Examples

Q1: “Retrieve the birth date and address of the employee whose name is ‘John B. Smith’.”

{uv | (q)(r)(s)(t)(w)(x)(y)(z) (EMPLOYEE(qrstuvwxyz) AND q=‘John’ AND r=‘B’ AND s=‘Smith’}

Examples

Q2: “Retrieve the name and address of the employees who work for the ‘Research’ department.”

{qsv | (z)(l)(m)(EMPLOYEE(qrstuvwxyz) AND DEPARTMENT(lmno) AND l=‘Research’ AND m=z}

Examples

Q3: “For every project located in ‘Stafford’, list the project number, the controlling department number, and the department manager’s last name and address.”

{iksv | (j)(m)(n)(t)(PROJECT(hijk) AND EMPLOYEE(qrstuvwxyz) AND DEPARTMENT(lmno) AND k=m AND n=t AND j=‘STAFFFORD’}

Examples

Q4: “Retrieve the names of employees who have no dependents.”

{qs | (t)(EMPLOYEE(qrstuvwxyz) AND (NOT(l) (DEPENDENT(lmnop) AND t=l)))}

Examples

Q4: “List the names of managers who have at least one dependents.”

{sq | (t)(j)(l)(EMPLOYEE(qrstuvwxyz) AND DEPARTMENT(hijk) AND DEPENDENT(lmnop) AND t=j AND l=t}