Relational Calculus
Chapter 4 – Part II
1
Formal Relational Query Languages
Two mathematical Query Languages form the
basis for “real” languages (e.g. SQL), and for
implementation:
 Relational Algebra:

• how to compute it , operational,
Representing execution plans (inside)

Relational Calculus:
• what you want, declarative.
Influent SQL design (outside)
2
Relational Calculus

Calculus has
•
•
•
•

variables, constants,
comparison ops (<,>,=,,.),
logical connectives ( ,)
Quantifiers (,)
Comes in two flavours:
•
•
Tuple relational calculus (TRC) : Variables range over tuples.
(SQL)
Domain relational calculus (DRC). Variables range over
domain elements. (QBE)
Both formula in TRC and DRC are simple subsets of firstorder logic.
3
Tuple Relational Calculus (TRC)
Tuple variable: variable that takes on tuples
of a relation schema as values.
 Query form:
T| pT 

T = tuple variable
p(T) = formula that describes T
Result = set of all tuples t for which p(t) with
T=t evaluates to be true
 Formula P(T) specified in First Order Logic

4
Find all sailors with a rating above 7
S|S  Sailors S.rating7











Variable S bound to a tuple in relation Sailors
5
TRC Formulas

Formula:
 an atomic formula
R  Rel,

R.a op S.b, R.a op constant, costant op R.a
Or recursively defined
p, pq, pq, pq , where p, q are formulas
R(p(R)), where R is a tuple variable
R(p(R)), where R is a tuple variable

Terms.
 bind --- the quantifier  and  are bind to variable R.
 free --- a variable is free if no quantifier binds to it in a
formula
6
Semantics of TRC Queries

F is an atomic formula R  Rel, and R is assigned a tuple in the instance of
relation Rel
 {S|SSailors}




F is a comparison R.a op S.b, R.a op constant, or constant op R.a,
and the tuples assigned to R and S have field names R.a and S.b that
make the comparison true.
F is of the form p, and p is not true;
the form p  q, and both p and q are true;
the form p  q, and one of them is true;
the form p  q, and q is true whenever p is true.
F is of the form R(p( R )), and there is some assignment of tuples to the
free variables in p ( R ), including the variable R that make the formula p(
R ) true;
F is of the form R (p ( R )), and there is some assignment of tuples to the
free variables in p( R ) that makes the formula p( R ) true no matter what
tuple is assigned to R.
7
TRC Example
Reserves
sid bid
day
22 101 10/10/96
58 103 11/12/96
Sailors
sid
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
8
Find the names and ages of sailors
with rating > 7
{ P | S  Sailors (S.rating > 7  P.name = S.sname  P.age = S.age)}
• P is a Tuple variable with two fields: name and age.
- the only fields mentioned in P
- P does not range over any relation in the query.
Sailors
sid
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
name
age
Lubber
55.0
rusty
35.0
9
Find the names of the sailors who
reserved boat 103
{ P | S  Sailors R  Reserves
(R.sid = S.sid  R.bid = 103  P.sname = S.sname)}
Retrieve all sailor tuples for which there exists a tuple in
Reserves having the same value in the sid field and with
bid =103
Question:
How is the answer tuple looks like? (columns? Rows?)
What is difference?
{ S | S  Sailors  R  Reserves (R.sid = S.sid  R.bid = 103)}
10
Find the names of the sailors who
have reserved a red boat
{ P | S  Sailors R  Reserves B  Boats
(R.sid = S.sid  B.bid = R.bid  B.color ='red'
 P.sname = S.sname)}
Retrieve all sailor tuples S for which there exists tuples R in
Reserves and B in Boats such that R.sid = S.sid  B.bid = R.bid 
B.color ='red'
{ P | S  Sailors R  Reserves
(R.sid = S.sid  P.sname = S.sname 
B  Boats(B.bid = R.bid  B.color ='red'))}
11
Find the names of the sailors who
have reserved all boat
{ P | S  Sailors B  Boats
(R  Reserves (S.sid = R.sid  R.bid = B.bid
 P.sname = S.sname))}
Find sailors S such that for all boats B there is a Reserves tuple
showing that sailor S has reserved boat B.
 (Tempsids, (
sid, bid
Re serves) / (
bid
Boats))
 sname (Tempsids  Sailors)
12
Find the sailors who have reserved all
red boat
{ P | S  Sailors  B  Boats
(B.color = 'red'
 (R  Reserves (S.sid = R.sid  R.bid = B.bid))}
Question: How to change the algebraic representation?
 (Tempsids, ( 
sid, bid
Re serves) / ( 
bid
Boats))
 sname (Tempsids  Sailors)
13
Find the sailors who have reserved all
red boat
 Logically p  q is equivalent to pq (Why?)
{ P | S  Sailors  B  Boats
(B.color = 'red'
 (R  Reserves (S.sid = R.sid  R.bid = B.bid))}
 is rewritten as:
{ P | S  Sailors  B  Boats
(B.color 'red'  (R  Reserves (S.sid = R.sid  R.bid = B.bid))}
 to restrict attention to those red boat.
14
Domain Relational Calculus

Query has the form:











x1, x2,..., xn | p x1, x2,..., xn






Answer includes all tuples x1, x2,..., xn that



make the formula p x1, x2,..., xn  be true.


15
DRC Formulas

Atomic formula:





<x1,x2,…,xn>  Rel , where Rel is a relation with n variables
X op Y
X op constant
op is one of comparison ops (<,>,=,,.),
Or recursively defined
p, pq, pq, pq , where p, q are formulas
X(p(X)), where X is a domain variable
X(p(X)), where X is a domain variable
16
Free and Bound Variables

The use of quantifiers  X and  X in a formula is
said to bind X.


A variable that is not bound is free.
Let us revisit the definition of a query:










x1, x2,..., xn | p x1, x2,..., xn


 
 
There is an important restriction: the variables
x1, ..., xn that appear to the left of `|’ must be
the only free variables in the formula p(...).
17
Find all sailors with a rating above 7





I, N,T , A | I, N, T, A  Sailors  T  7
Sailors





sid
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
I, N, T , A  Sailors

the domain variables I, N, T and A are bound
to fields of the same Sailors tuple.
 that every tuple I, N, T , A that satisfies T>7 is
in the answer.
 Question: Modify to answer:

Find sailors who are older than 18 or have a rating
under 9, and are called ‘Joe’.
18
Find sailors rating > 7 who’ve reserved boat #103





I, N, T, A | I, N, T , A  Sailors  T  7 
 Ir, Br, D Ir, Br, D  Re serves  Ir  I  Br  103




Sailors


sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0





Boats
Reserves
sid
22
31
58




sid
22
58
bid
101
103
day
10/10/96
11/12/96
bid
bname
color
101
interlake
red
103
marine
green
We have used  Ir , Br , D  . . . as a shorthand for  Ir   Br   D . . . 
Note the use of  to find a tuple in Reserves that `joins with’
the Sailors tuple under consideration.
{<I,N,T,A>|<I,N,T,A> Sailors T>7
  Ir,Br,D
(<Ir,103,D>  Reserves  Ir =I )}
19
Find sailors rated > 7 who’ve reserved a red boat





I, N, T , A | I, N, T , A  Sailors  T  7 
 Ir, Br, D Ir, Br, D  Re serves  Ir  I 




 B, BN, C B, BN, C  Boats  B  Br  C  ' red '




Sailors
Boats
Reserves
sid
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
 
 
  
  

sid
22
58
bid
101
103
day
10/10/96
11/12/96
bid
bname
color
101
interlake
red
103
marine
green

Observe how the parentheses control the scope of each
quantifier’s binding.

Rewrite using 'red' as constant
• BN = red
20
Find sailors who’ve reserved all boats





I, N, T , A | I, N, T , A  Sailors 
 B, BN, C  Boats




 Ir, Br, D  Re serves I  Ir  Br  B




Sailors

Boats
Reserves
sid
22
31
58
sname rating age
dustin
7
45.0
lubber
8
55.5
rusty
10 35.0
 
 
 
 

sid
22
58
bid
101
103
day
10/10/96
11/12/96
bid
bname
color
101
interlake
red
103
marine
green
Find all sailors I such that for each 3-tuple {B, BN, C} either it is
not a tuple in Boats or there is a tuple in Reserves showing that
sailor I has reserved it.





I, N, T , A | I, N, T , A  Sailors 
 B, BN, C  B, BN, C  Boats 









 Ir, Br, D














Ir, Br, D  Re serves  I  Ir  Br  B
 
 
   
   

21
Algebra vs. Calculus
Two formal query languages are equal in the
power of expressivness?
 Algebra  Calculus?
YES!
 Calculus  Algebra?

• Unsafe query
{S | (S  Sailors)}
Correct???
• Safe TRC --- Dom(Q , I)
- answers for Q contains only values that are in Dom(Q, I)
- R(p(R)), tuple r contains only constant in Dom(Q,I)
- R(p(R)), tuple r contains a constant that is not in
Dom(Q,I), then p(r) is true.
22
Summary

Algebra and safe calculus have same expressive power

Relationally complete --if a query language can express all the queries that
can be expressed in relational algebra.

Relational calculus is non-operational, and users define
queries in terms of what they want, not in terms of
how to compute it. (Declarativeness.)
23