1. Relational Calculus
Chapter 4 – Part II

2. 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)

3. Relational Calculus Calculus has Comes in two flavours:
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 first-order logic.

4. Tuple Relational Calculus (TRC)
Tuple variable: variable that takes on tuples of a relation schema as values.
Query form:
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

5. Find all sailors with a rating above 7
Variable S bound to a tuple in relation Sailors

6. TRC Formulas Formula: an atomic formula Or recursively defined Terms.
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 varialbe
Terms.
bind --- the quantifier  and  are bind to variable R.
free a variable is free if no quantifier binds to it in a formula

7. 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.

8. TRC Example
Reserves
Sailors

9. 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
name
age
Lubber
55.0
rusty
35.0

10. 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)}

11. 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'))}

12. 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.

13. 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?

14. 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.

15. Domain Relational Calculus
Query has the form:
Answer includes all tuples that
make the formula be true.

16. DRC Formulas Or recursively defined 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 varialbe

17. Free and Bound Variables
The use of quantifiers and in a formula is said to bind X.
A variable that is not bound is free.
Let us revisit the definition of a query:
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(...).

18. Find all sailors with a rating above 7
the domain variables I, N, T and A are bound to fields of the same Sailors tuple.
that every tuple 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’.

19. Find sailors rating > 7 who’ve reserved boat #103
Reserves
Boats
bid
bname
color
101
interlake
red
103
marine
green
We have used as a shorthand for
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 )}

20. Find sailors rated > 7 who’ve reserved a red boat
Reserves
Boats
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

21. Find sailors who’ve reserved all boats
Reserves
Boats
bid
bname
color
101
interlake
red
103
marine
green
Find all sailors I such that for each 3-tuple either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved it.

22. 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.

23. 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.)