Description

Assignment 4
Dirk Van Gucht
This assignment relies on the lectures
• Functions and expressions in SQL; • Aggregate functions and partitioning; and
• Triggers.
You will also see several problems that are listed as practice problems. You should not include your solutions for such problems in the materials you submit for this assignments. Only solutions for problems 1 through 10 need to be submitted.
Database schema and instances
For the problems in this assignment we will use the following database schema:
Person(pid, pname, city)

Company(cname,headquarter) Skill(skill) worksFor(pid, cname, salary) companyLocation(cname,city) personSkill(pid,skill) hasManager(eid,mid) Knows(pid1,pid2)
In this database we maintain a set of persons (Person), a set of companies (Company), and a set of (job) skills (Skill). The pname attribute in Person is the name of the person. The city attribute in Person specifies the city in which the person lives. The cname attribute in Company is the name of the company. The headquarter attribute in Company is the name of the city wherein the company has its headquarter. The skill attribute in Skill is the name of a (job) skill.
A person can work for at most one company. This information is maintained in the worksFor relation. (We permit that a person does not work for any company.) The salary attribute in worksFor specifies the salary made by the person.
The relation Knows maintains a set of pairs (p1,p2) where p1 and p2 are pids of persons. The pair (p1,p2) indicates that the person with pid p1 knows the person with pid p2. We do not assume that the relation Knows is symmetric: it is possible that (p1,p2) is in the relation but that (p2,p1) is not.
The domain for the attributes pid, pid1, pid2, salary, eid, and mid is integer. The domain for all other attributes is text. We assume the following foreign key constraints:
• pid is a foreign key in worksFor referencing the primary key pid in Person;
• cname is a foreign key in worksFor referencing the primary key cname in Company;
• cname is a foreign key in companyLocation referencing the primary key cname in Company;
• pid is a foreign key in personSkill referencing the primary key pid in Person;
• skill is a foreign key in personSkill referencing the primary key skill in Skill;
• eid is a foreign key in hasManager referencing the primary key pid in Person;
• mid is a foreign key in hasManager referencing the primary key pid in Person;
• pid1 is a foreign key in Knows referencing the primary key pid in Person; and
• pid2 is a foreign key in Knows referencing the primary key pid in
Person
Pure SQL and RA SQL
In this assignemt, we distinguish between Pure SQL and RA SQL. Below we list the only features that are allowed in Pure SQL and in RA SQL.
In particular notice that
• join, NATURAL join, and CROSS join are not allowed in Pure SQL.
• The predicates [not] IN, SOME, ALL, [not] exists are not allowed in RA SQL.
The only features allowed in Pure SQL

select … from … where WITH …
union, intersect, except operations exists and not exists predicates
IN and not IN predicates
ALL and SOME predicates
VIEWs that can only use the above RA SQL features
The only features allowed in RA SQL

select … from … where WITH …
union, intersect, except operations
join … ON …, natural join, and CROSS join operations VIEWs that can only use the above RA SQL features commas in the from clause are not allowed
Full SQL

all the features of Pure SQL and RA SQL
user-defined functions aggregate functions group … by … having …
1 Solving queries using aggregate functions
Formulate the following queries in SQL. You should use aggregate functions to solve these queries. You can use views, temporary views, parameterized views, and user-defined functions.
1. Find each pair (c,n) where c is the cname of a company that pays an average salary between 50000 and 55000 and where n is the number of employees who work for company c.
2. Find the pid and name of each person who lacks at least 4 job skils and who knows at least 4 persons.
3. Find the pid and name of each person who has fewer than 2 of the combined set of job skills of persons who work for Google. By combined set of jobskills we mean the set
{s | s is a jobskill of an employee of Google}.
4. Find the cname of each company that employs at least 3 persons and that pays the lowest average salary among such companies.
5. Find each pair (c1,c2) of different company cnames such that, among all companies, company c2 pays the closest average salary compared to the average salary paid by company c1.
6. Without using set predicates, find each pid of a person who does not know each person who (1) works for Apple and (2) who makes less than 55000.
7. Without using set predicates, find each pairs (s1,s2) of skills such that the set of persons with skill s1 is the same as the set of persons with skill s2.
8. Find each pairs (s1,s2,n) of different skills s1 an s2 and such that (1) the number of persons with skill s1 is the same as the number of persons with skill s2 and (2) where n is the number of such persons associated with c1 and c2.
9. (a) Using the GROUP BY counting method, define a function
create or replace function numberOfSkills(c text)
returns table (pid integer, salary int, numberOfSkills bigint) as
$$ …
$$ language sql;
(b) Test your function for Problem 9a for the companies Apple, Amazon, and ACM.
(c) Write the same function numberOfSkills as in Problem 9c but this time without using the GROUP BY clause.
(d) Test your function for Problem 9c for the companies Apple, Amazon, and ACM.
(e) Using the function numberOfSkills but without using set predicates, write the following query: “Find each pair (c,p) where c is the name of a company and where p is the pid of a person who (1) works for company c, (2) makes more than 50000 and (3) has the most job skills among all the employees who work for company c.”
2 Introduction to Dynamic SQL
The examples in this section introduce you to Dynamic SQL. Dynamic SQL is a powerful generalization of SQL: it permits writing programs that generate queries, and furthermore, these queries can subsequently be evaluated in the programs that generated them.
Intuitively, in a dynamic program, a string is constructed that represents a SQL query. When an execute statement in the dynamic program is then applied to that string, the corresponding query is evaluated.
We will consider a simple example illustrating Dynamic SQL. (For more information about Dynamic SQL, we refer to the PostgreSQL manual under the topic of Dynamic SQL.)
Example 1 Assume that there are 3 unary relations
P(p : boolean)
Q(q : boolean)
R(r : boolean)
create table P (p boolean); create table Q (q boolean); create table R (r boolean);
insert into P values (true), (false); insert into Q values (true), (false); insert into R values (true), (false); So we have the following situation:
P Q R

t t t f f f
Next, we consider the set of boolean propositions (propositions, for short) over these variables P, Q, and R. An inductive definition of these propositions is as follows:
• P, Q, and R are propositions.
• If E and F are propositions then (E or F) is a proposition.
• If E and F are propositions then (E and F) is a proposition.
• If E is a propositon then (not E) is a proposition.
• If E is a propositon then (E) is a proposition.
Here are some examples of propositions over P, Q, and R.
P
Q
R
(not P)
(not (P))
(P or R)
(P and P)
(P and (not (R and Q)))
We will now consider the truth table of a proposition. (For the concept of truth table of a propositon, we refer to the document Propositional Logic in the course module Notes on Discrete Mathematics.) For example, the truth table for the proposition (PorR) is a follows:
p q r truthValue

t t t t t t f t t f t t t f f t f t t t f t f f f f t t f f f f
and that for (P and (not (R and Q))) is
p q r truthValue

t t t f t t f t t f t t t f f t f t t f f t f f f f t f f f f f
It is possible to generate the truth table of a proposition in SQL. For example, for the proposition ’(P and (not (R and Q)))’ we can write the SQL query (let’s call it Q1)
select p, q, r, (P and (not (R and Q))) from P, Q, R;
The problem with this approach is that to determine the truth table for another proposition such as ’(not (Q and (not P)))’, we would need to write a different query (let’s call it Q2)
select p, q, r, (P and (not (R and Q))) from P, Q, R;
even though the blue parts in Q1 and Q2 are same.
A more general approach to generate the truth table of an arbitrary proposition is to use Dynamic SQL. Here is the user-defined function Dynamic SQL that can facilitate this:
create or replace function truthTable(proposition text)
returns table(p boolean, q boolean, r boolean, truthValue boolean) as
$$ begin
return query
execute ’select p, q, r, ’ || proposition ||
’ from P, Q, R;’;
end;
$$ language plpgsql;
To generate the truth table for the proposition
“P and (not (R and Q))”
we would call the truthTable function with the string representation ’(P and (not (R and Q)))’
of this proposition as follows:
select * from truthTable(’(P and (not (R and Q)))’);
The critical component of the DynamicSQL code for the function truthTable is the statement
return query
execute ’select p, q, r, ’ || proposition ||
’ from P, Q, R;’;
What happens in this statement is that we build a string that represents an SQL query by concatenating the string
’select p, q, r, ’
with the string that is passed into the truthTable function’s proposition parameter (in this case the string
’(P and (not (R and Q)))’)
and, finally, with the string
’ from P, Q, R;’.
This concatenation of strings is accomplished using the string concatenation function ||. As such we get the string
’select p, q, r, (P and (not (R and Q))) from P, Q, R;’
This is a string that represents a valid SQL query which we can then evaluate with the execute operation. The rest of the code, i.e., the return query statement, then returns the result of this evaluation, i.e., the truth table of the proposition (P and (not (R and Q))).
Now that we have the truthTable function, we will consider in the next example the problem of verifying if two Propositional Logic propositions are logically equivalent.
Example 2 Consider two Propositional Logic propositions E and F. We say that E and F are logically equivalent if their respective truth tables are the same.
We will write a boolean user-defined SQL function logicallyEquivalent(E text, F text) returns boolean
which takes as arguments two strings that represent the propositions E and F respectively and that returns ’true’ if E and F are logically equivalent, and ’false’ otherwise.
The code for this function relies on the following insight. Let TE and TF be the truth table of E and F, respectively. Then, by definition E and F are logically equivalent, i.e., TE = TF. But the later set condition is equivalent with the condition
(TE − TF) = ∅ and (TF − TE) = ∅.
Here is the function:
create or replace function
logicallyEquivalent(E text, F text) returns boolean as
$$ select not exists (select truthTable(E) except (select truthTable(F))) and not exists (select truthTable(F) except (select truthTable(E))); $$ language sql;
3 Operations on polynomials and matrices using SQL aggregate functions
In the problems in this section, you will practice working with aggregate functions.
A useful other aspect of solving these problems is that you will learn how relations can be used to represent polynomials and matrices and how SQL can be used to define operations on such objects.
10. Let P(x) and Q(x) be 2 polynomials with integer coefficients.
Let P(coefficient, degree) and Q(coefficient, degree) be two binary relations representing P(x) and Q(x), respectively. E.g., if P(x) = 2×2 − 5x + 5 and Q(x) = 4×4 + 3×3 + x2 − x then their representations in the relations P and Q are as follows:
Q
coefficient degree
4 4
3 3
1 2
−1 1
0 0
coefficient degree
2 2
−5 1
5 0
P
(a) Write a dynamic SQL function
create or replace function multiplyPolynomials(polynomal1 text, polynomial p2 text) returns table(coefficient integer, degree integer) as
$$ …
$$ language plpgsql; that computes a binary relation representing the multiplication of P(x) and Q(x), i.e., the polynomial P(x) ∗ Q(x).
For example, consider P(x) = 2×2 − 5x + 5 and Q(x) = 4×4+3×3+x2−x. Then P(x)∗Q(x) = (8)x6+(6−20)x5+(2− 15+20)x4+(−2−5+15)x3+(5+5)x2+(−5)x = 8×6−14×5+ 7×4+8×3+10×2−5x. So, for these polynomials, your function when called with the name of the relation representing the polynomials should return the relation
coefficient degree
8 6
−14 5
7 4
8 3
10 2
−5 1
0 0
(b) Your solution should work for arbitrary polynomials P(x) and Q(x). Show how your function can be used to compute
i. the polynomial P ∗ Q;
ii. the polynomial P ∗ P; and
iii. the polynomial P ∗ (Q ∗ P).
11. (Practice problem – not graded)
Let P(x) and Q(x) be 2 polynomials with integer coefficients. We can consider their composition P(Q(x)).
For example, consider P(x) = 2×3−5x+5 and Q(x) = 4×4+3×3, then
P(Q(x)) = 2(Q(x))3− 5Q(x) + 5
= 2(4×4 + 3×3)3− 5(4×4 + 3×3) + 5
= 2(4×4 + 3×3)(4×4 + 3×3)(4×4 + 3×3) − 5(4×4 + 3×3) + 5
and = 128×12 + 288×11 + 216×10 + 54×9− 20×4− 15×3 + 5
Q(P(x)) = 4(P(x))4 + 3(Q(x))3
= 4(2×3− 5x + 5)4 + 3(2×3− 5x + 5)
= 64×12− 640×10 + 664×8 + 2400×8− 4980×7− 1420×6+
12450×5− 10400×4− 5925×3 + 16125×2− 11125x + 2875
(a) Write a dynamic SQL function
create or replace function compositionPolynomials(polynomial1 text, polynomial2 text) returns table(coefficient integer, degree integer) as
$$ …
$$ language plpgsql;
that computes a binary relation representing the composition of P(Q(x)).
Your solution should work for arbitrary polynomials P(x) and Q(x). Show how your function can be used to compute
i. the polynomial P(Q(x));
ii. the polynomial Q(P(x)); and
iii. the polynomial P(P(P(x))).
12. (Practice problem – not graded)
Let M be an n × n matrix of boolean values. (We will assume that n ≥ 0.) We will use T to denote ‘true’ and F to denote
‘false’.
For i,j ∈ [1,n], we will denote by M[i,j] the boolean value in
matrix M at row i and column j. (Notice that when n = 0, M has no elements.)
A boolean matrix M can be represented using a relation M with schema
(rw: integer, colmn: integer, value: boolean) and such that for each i,j ∈ [1,n]
(i,j,M[i,j]) ∈ M.
Notice that we use the attribute names ‘rw’ and ‘colmn’ since the words ‘row’ and ‘column’ are reserved words in PostgreSQL. For example if M is the 3 × 3 boolean matrix
T F T M = F T T T F T
then M is the following relation of 9 tuples:
M
rw colmn value

1 1 T
1 2 F
1 3 T
2 1 F
2 2 T
2 3 T
3 1 T
3 2 F
3 3 T
Let M and N be two n × n boolen matrices represented by the two relations M and N.
(a) Write a dynamic SQL function
create or replace function booleanMatrixMultiplication (M text, N text) returns table (rw integer, colmn integer, value boolean) as
$$ begin return query execute …;
end;
$$ language plpgsql;
that computes a relation over schema (rw, column, value) that represents the matrix M ∗ N when given the names ‘M’ and ‘N’ of the relations that represent M and N, respectively. Your solution should work for any n ≥ 1.
For example if M and N are by the following 3 × 3 boolean matrices stored as the relations
M
rw colmn value
N
rw colmn value

1
1
1
2
2
2
3
3
3 1 T 2 F 3 T 1 F
2 T
3 T
1 T 2 F
3 T 1
1
1
2
2
2
3
3
3 1
2
3
1
2
3
1
2
3 T
T
T F
F
F
T
F
T
then your function should produce the relational representation of M ∗ N, i.e., the relation
M∗N
rw colmn value

1 1 T
1 2 T
1 3 T
2 1 T
2 2 F
2 3 T
3 1 T
3 2 T
3 3 T
Consider the Person relation. Then the relation Knows, which is a binary relation over Person, can be modeled by a boolean matrix M Knows by using the Person pids as row and column indices for this matrix, and by using the value T at position (i,j) if (i,j) ∈ Knows, and the value F at position (i,j) if (i,j) 6∈ Knows. We can then represent this boolean matrix with a relation M Knows(rw integer, colmn integer, value) by using the following SQL statements: create table M_Knows(rw integer, colmn integer, value boolean);
insert into M_Knows select rw.pid, colmn.pid, exists (select 1 from Knows k where rw.pid = k.pid1 and colmn.pid = k.pid2)
from person rw, person colmn;
(b) Compute the matrix MKnows ∗ MKnows. What does the matrix represent?
(c) MKnows ∗ (MKnows ∗ M Knows). What does the matrix represent?
4 Triggers (Practice problems – not graded)
To begin the problems in this section, you should first remove the entire database, including the relation schemas. You should then create the relations but without specifying the primary and foreign key constraints. You should also not yet populate the relations with data. Solve the following problems:
13. Develop appropriate insert and delete triggers that implement the primary key and foreign key constraints that are specified for the Person, Company, and Worksfor relations.
Your triggers should report appropriate error conditions.
For this problem, implement the triggers such that foreign key constraints are maintained using the cascading delete semantics.
For a reference on cascading deletes associated with foreign keys maintenance consult the PostgreSQL manual page https:www.postgresql.orgdocs9.2ddl-constraints.html
Test your triggers using appropriate inserts and deletes.
15. Consider the following view definition
create or replace view PersonIsKnownByNumberofPersons as (select p1.pid, p1.name,
(select count(1) from Person p2 where (p2.pid, p1.pid) in (select k.pid1, k.pid2 from Knows k)) as NumberofKnownByPersons
from Person p1);
This view defines the set of tuples (p,n,c) where p and n are the pid and name of a person and c is the number of persons who know the person with pid p.
You should not create this view! Rather your task is to create a relation PersonIsKnownByNumberofPersons that maintains a materialized view in accordance with the above view definition under insert and delete operations on the Person and Knows relation.
Your triggers should be designed to be incremental. In particular, when an insert or delete occurs, you should not always have to reevaluate all the number of persons who know persons.
Rather the maintenance of PersonIsKnownByNumberofPersons should only apply to the person information that is affected by the insert or delete.
Provide tests that show that your solution works.