1. Database Design Dr. M.E. Fayad, Professor
5/1/2019
Database Design
Dr. M.E. Fayad, Professor
Computer Engineering Department, Room #283I
College of Engineering
San José State University
One Washington Square
San José, CA
© M.E. Fayad
SJSU -- CmpE

2. Lesson 11: Constraint Databases
5/1/2019
Lesson 11:
Constraint Databases 2
© M.E. Fayad
SJSU – CmpE M.E. Fayad

3. 3 Lesson Objectives Understand Constraint Databases
5/1/2019
Lesson Objectives
Understand Constraint Databases
Learn How to Deal with Constraints
Arithmetic Atomic Constraints
Boolean Atomic Constraints
Constraint Formulas
Interpretations of Free Boolean Constraints
Understand the Constraint Data Model
Explore Data Abstraction 3
© M.E. Fayad
SJSU – CmpE M.E. Fayad

4. View and Logical Levels
5/1/2019
View and Logical Levels
Are The highest level of data abstraction
Are not suitable for computer to represent data. Why?
Because the allow an infinite number of tuples or points.
In this lecture we discuss constraint data model that is used to represent the logical level of data in a concise way. 4
© M.E. Fayad
SJSU – CmpE M.E. Fayad

5. 5 Constraint Databases Next level of data abstraction:
5/1/2019
Constraint Databases
Next level of data abstraction:
Constraint level – finitely represents
by constraints
the logical level 5
© M.E. Fayad
SJSU – CmpE M.E. Fayad

6. Two Most Common Types of Constraints
5/1/2019
Two Most Common Types of Constraints
Constraints apply to two or more variables.
Each variable takes its value from a domain
Common domains are:
+ The set of nonnegative numbers (or natural numbers) N
+ The set of integer numbers Z
+ The set of rational number Q
+ The set of real numbers R
Other domains – Set of English words in the dictionary W 6
© M.E. Fayad
SJSU – CmpE M.E. Fayad

7. 7 Constraints (1) Assume u and v are variables or constants
5/1/2019
Constraints (1)
Assume u and v are variables or constants
B and ci are constants
Let ± denote + or -
Let p be a polynomial function over variables X1, …., Xn
Any valid expression with constants, variables, addition, and multiplication symbols
We define the most basic type of constraints, called the atomic constraints over any of the domains N, Z, Q, R as following: 7
© M.E. Fayad
SJSU – CmpE M.E. Fayad

8. 8 Constraints (2) Arithmetic Atomic Constraints Inequality: u ≠ v
5/1/2019
Constraints (2)
Arithmetic Atomic Constraints
Equality: u = v
Inequality: u ≠ v
Lower bound: u ≥ b
Upper bound: – u ≥ b
Order: u ≥ v
Gap-order: u – v ≥ b
where b ≥ 0
Difference: u – v ≥ b 8
© M.E. Fayad
SJSU – CmpE M.E. Fayad

9. 9 Constraints (3) Half-Addition: ± u ± v ≥ b where b ≥ 0
5/1/2019
Constraints (3)
Half-Addition: ± u ± v ≥ b
where b ≥ 0
Addition: ± u ± v ≥ b
Linear: c1x1 + ……+ cnxn ≥ b
Polynomial: p( x1,……..,xn ) ≥ b
Note: Instead of ≥ we can also use >
b is a bound which is restricted to be nonnegative in the case of gap-order and half addition constraints.
ci is a coefficient in the linear constraint. 9
© M.E. Fayad
SJSU – CmpE M.E. Fayad

10. 5/1/2019
Constraints (4)
If a constraint C is any of these types of constraints and the domain of the variables is Z and Q we talk of integer C constraints or rational C constraints, respectively.
We consider each = constraint as the conjunction of two ≥ constraints.
For example: linear equation 5x+7y = 8 can be expressed as the conjunction of the linear inequality constraints:
5x +7y ≥ 8 and 5x +7y ≤ 8
This is why we do not consider linear equality as atomic constraints. Nevertheless, we will write constraints with = signs. 10
© M.E. Fayad
SJSU – CmpE M.E. Fayad

11. Boolean Atomic Constraints (1)
5/1/2019
Boolean Atomic Constraints (1)
Boolean Algebra < D, Λ, V, ‘, 0, 1 >
D – domain,
0 – zero element,
1 – one element.
such that the following axioms hold: 11
© M.E. Fayad
SJSU – CmpE M.E. Fayad

12. Boolean Atomic Constraints (2)
5/1/2019
Boolean Atomic Constraints (2)
Commutative:
x V y = y V x
x Λ y = y Λ x
Distributive:
x V ( y Λ z ) = ( x V y ) Λ ( x V z )
x Λ ( y V z ) = ( x Λ y ) V ( x Λ z )
Inverse:
x V x’ = 1
x Λ x’ = 0
Zero and one element properties:
x V 0 = x
x Λ 1 = x
0 ≠ 1 12
© M.E. Fayad
SJSU – CmpE M.E. Fayad

13. Boolean Atomic Constraints (3)
5/1/2019
Boolean Atomic Constraints (3)
Boolean equality constraints: T =B 0
Boolean inequality constraints: T ≠B 0
where T is a Boolean term built from domain elements and operations of the boolean algebra.
Example:
( x’ Λ z ) V ( y Λ z ) V ( x Λ y’ Λ z’ ) =B 0 13
© M.E. Fayad
SJSU – CmpE M.E. Fayad

14. Constraint Formulas (1)
5/1/2019
Constraint Formulas (1)
each atomic constraint
if F and G are formulas, then
(F and G) is a formula and
(F or G) is a formula.
if F is a formula, then
(not F) is a formula. 14
© M.E. Fayad
SJSU – CmpE M.E. Fayad

15. Constraint Formulas (2)
5/1/2019
Constraint Formulas (2)
Let F(x1,…,xn) be a formula with n rows,
t = (c1,…,cn) a tuple with n constants.
t satisfies F (t F) –
substituting each xi by ci
makes F true. 15
© M.E. Fayad
SJSU – CmpE M.E. Fayad

16. Interpretation of free Boolean algebra
5/1/2019
Interpretation of free Boolean algebra
Symbols are interpreted to be concrete operators or values.
Example:
D – all subsets of 1
Λ – intersection
V – union
‘ – complement
0 – emptyset
1 – set of names of all animals {cat, dog, parrot,…}
Question: What would be the meaning of the example
Boolean constraint under this interpretation ? 16
© M.E. Fayad
SJSU – CmpE M.E. Fayad

17. Constants in constraints are also interpreted
5/1/2019
Constants in constraints are also interpreted
Example: Suppose we have the constraint
b Λ m = 0
and we interpret
b as {canary, parrot, falcon}
m as {cat, dog}
Question: Is the Boolean constraint true under
this interpretation ? 17
© M.E. Fayad
SJSU – CmpE M.E. Fayad

18. The Constraint Data Model (1)
5/1/2019
The Constraint Data Model (1)
Constraint data model provides a finite representation of the infinite relational data model.
In constraint data model each attribute is associated with an attribute variable, and the value of the attributes in a relation is specified implicitly using constraints.
A constraint database is a finite set of constraint relations.
A constraint relation is a finite set of constraint tuples,
Where each constraint tuple is a conjunction of atomic constraints using the same set of attribute variables. 18
© M.E. Fayad
SJSU – CmpE M.E. Fayad

19. The Constraint Data Model (2)
5/1/2019
The Constraint Data Model (2)
Constraint relation – a finite set of constraint tuples.
Constraint tuples – tuples with variables and
constraints that restrict the variables.
Example:
( i, t : ≤ i , i ≤ 24000, t = 0.15 i )
is a constraint tuple of Taxtable(Income, Tax). It means that the tax is 15% on any income below $24,000. 19
© M.E. Fayad
SJSU – CmpE M.E. Fayad

20. The Constraint Data Model (3)
5/1/2019
The Constraint Data Model (3)
Income
Tax i t
0 ≤ i , i ≤ 24000, t = 0.15 i
24000 < i , i ≤ 58150, t = (i – 24000)
58150 < i , i ≤ t = (i – 58150)
< i , i ≤ , t = (i – )
< i , t = (i – ) 20
Taxtable
© M.E. Fayad
SJSU – CmpE M.E. Fayad

21. The Constraint Data Model (4)
5/1/2019
The Constraint Data Model (4)
The taxtable constraint relation consists of only five contraint tuples.
It represents the infinite relation that we described early.
The constraints in the table are order constraints and linear equations over the rational numbers. 21
© M.E. Fayad
SJSU – CmpE M.E. Fayad

22. The Constraint Data Model (5)
5/1/2019
The Constraint Data Model (5)
The Go relation can be presented by the following constraint relation.
Assumption: If the shortest path from city1 to city2 is some m minutes, then if we leave city1 at time t1 we may arrive at city2 at some time t2, such that
t2 – t1 ≥ m 22
© M.E. Fayad
SJSU – CmpE M.E. Fayad

23. The Constraint Data Model (6)
5/1/2019
The Constraint Data Model (6)
City 1
time1
City 2
Time2
Omaha t1
Lincoln t2
t2 – t1 ≥ 60
Mpls
t2 – t1 ≥ 650
De_Moines
t2 – t1 ≥ 400
Chicago
t2 – t1 ≥ 370
t2 – t1 ≥ 600
Kansas_City
t2 – t1 ≥ 180 23 Go
© M.E. Fayad
SJSU – CmpE M.E. Fayad

24. The Constraint Data Model (7)
5/1/2019
The Constraint Data Model (7)
In Go, we represent each connection between two cities by two constraint tuples. 24
© M.E. Fayad
SJSU – CmpE M.E. Fayad

25. The Constraint Data Model (8)
5/1/2019
The Constraint Data Model (8)
Radio X Y 1 x y
(x – 8)2 + (y – 9)2 ≤ 25 2
(x – 8)2 / 52 + (y – 3)2/32 ≤ 1 3
(y – 1) ≥ (x – 3)2 , y ≤ 10
The first radio station broadcasts within a circle with center at location (8, 9)
The second in an elliptic area
The third in a parabolic area is cut off by a line.
The broadcast relation can be expressed as a constraint relation with
Quadratic constraints over the real number. 25
© M.E. Fayad
SJSU – CmpE M.E. Fayad

26. The Constraint Data Model (9)
5/1/2019
The Constraint Data Model (9)
Corn
Rye
Sunflower
Wheat x1 x2 x3 x4
15x1 + 8x2 + 10x3 + 15x4 ≤ 3000
30x1 + 25x2 + 5x3 + 10x4 ≤ 8000
The data can be represented in the Crops relation using linear inequality constraints over rational numbers. 26
© M.E. Fayad
SJSU – CmpE M.E. Fayad

27. 5/1/2019
Data Abstraction (1)
Each data model provides some abstraction about stored data.
There are four levels of data abstraction, namely
View level
Logical level
Constraint Level
Physical level
The various levels of data abstraction are linked by translation methods. 27
© M.E. Fayad
SJSU – CmpE M.E. Fayad

28. Data Abstraction: View Level
5/1/2019
Data Abstraction: View Level
How users can view their data
For different sets of users different views can be provided by a database
We will see more examples of visualization used in the MLPQ and PReSTO constarint databases systems in chapter 18 and 20. 28
© M.E. Fayad
SJSU – CmpE M.E. Fayad

29. Data Abstraction: Logical Level
5/1/2019
Data Abstraction: Logical Level
Logical level – infinite relation r, tuple t
Describes what data are stored in terms of sets of infinite relations.
Discussed in Chapter 1 29
© M.E. Fayad
SJSU – CmpE M.E. Fayad

30. Data Abstraction: Constraint Level
5/1/2019
Data Abstraction: Constraint Level
Describes the data stored using one or more constraint relations.
There is a natural link between the logical and constraint levels. Let r be the infinite relational model and R be the constraint model of a relation. Then a tuple t is in r iff there is a constraint tuple C in R such that after substitution of the constants in t into the variables of C, we get a true formula, that is t satisfies C.
Constraint level – finite relation R, constraint tuple C,
t є r iff t ╞ C for some C є R 30
© M.E. Fayad
SJSU – CmpE M.E. Fayad

31. Data Abstraction: Physical Level
5/1/2019
Data Abstraction: Physical Level
Physical level – the way the data is actually stored in the computer system
We will see example of this in Chapter 17 31
© M.E. Fayad
SJSU – CmpE M.E. Fayad

32. 32 Discussion Questions T/F
5/1/2019
Discussion Questions
T/F
Constraint data model is used to represent the logical level of data in a concise way.
Logical level is not suitable for computer to represent data.
View level is suitable for computer to represent data.
Constraint level describes the data stored using one or more constraint relations.
Physical level corresponds to the way the data is actually stored in computer system 32
© M.E. Fayad
SJSU – CmpE M.E. Fayad

33. 5/1/2019
Discussion Questions
Exercise 2.1 Represent the Constraint relation Free in a relational form assuming that d is an integer.
Person
Day
Andrew d
3 ≤ d, d ≤ 6
16 ≤ d, d ≤ 26
Barbara
6 ≤ d, d ≤ 18
25 ≤ d, d ≤ 30 33
© M.E. Fayad
SJSU – CmpE M.E. Fayad

34. 34 Solution Person Free Andrew 3, 4, 5, 6
5/1/2019
Solution
Person Free
Andrew 3, 4, 5, 6
Andrew 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26
Barbara 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Barbara 25, 26, 27, 28, 29, 30 34
© M.E. Fayad
SJSU – CmpE M.E. Fayad

35. 5/1/2019
Discussion Questions
Exercise 2.3 Using a constraint relation represent the street map shown in Figure 1.3 As following 35
© M.E. Fayad
SJSU – CmpE M.E. Fayad

36. 36 Solution Name X Y Willow x y y=19, x<=20, x>=4 Hare x y
5/1/2019
Name X Y
Willow x y
y=19, x<=20, x>=4
Hare x y
y=-2x+29, y>=11, y<=20
Bear x y
y=1/3 *x /3, 2<=x, x<=17
Vine x y
x=5, y>=2, y<=14
Oak x y
x=11, y>=2, y<=12 36
Maple x y
y=5, x<=20, x>=7
© M.E. Fayad
SJSU – CmpE M.E. Fayad

37. Questions for the Next Lecture
5/1/2019
Questions for the Next Lecture
Relational Algebra
SQL 37
© M.E. Fayad
SJSU – CmpE M.E. Fayad