L6-S1 Constraint DB 2003 SJSU -- CmpE Database Design Dr. M.E. Fayad, Professor Computer Engineering Department, Room #283I College of Engineering San

2003 SJSU -- CmpE L6-S1 Constraint DB

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 95192-0180

http://www.engr.sjsu.edu/~fayad

2003 SJSU – CmpE --- M.E. Fayad L6-S2 Constraint DB

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Lesson 06:Constraint Databases


Lesson Objectives

Objectives

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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


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

View and Logical Levels


Next level of data abstraction:

Constraint level – finitely represents

by constraints

the logical level

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Constraint Databases


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 W6

Two Most Common Types of Constraints


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

Constraints (1)


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

Constraints (2)


Half-Addition: ± u ± v ≥ b where b ≥ 0Addition: ± 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.

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Constraints (3)


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

Constraints (4)


Boolean Algebra < D, Λ, V, ‘, 0, 1 >

D – domain,

0 – zero element,

1 – one element.

such that the following axioms hold:

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Boolean Atomic Constraints (1)


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



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



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

Constraint Formulas (1)


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.

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Constraint Formulas (2)


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

Interpretation of free Boolean algebra


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

Constants in constraints are also interpreted


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

The Constraint Data Model (1)


Constraint relation – a finite set of constraint tuples.

Constraint tuples – tuples with variables and

constraints that restrict the variables.

Example:

( i, t : 0 ≤ 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



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Income Tax

i t 0 ≤ i , i ≤ 24000, t = 0.15 i

i t 24000 < i , i ≤ 58150, t = 3600 + 0.28 (i – 24000)

i t 58150 < i , i ≤ 121300 t = 13162 + 0.31(i – 58150)

i t 121300 < i , i ≤ 263750, t = 32738.5 + 0.36 (i – 121300)

i t 263750 < i , t = 84020.5 + 0.396 (i – 263750)

Taxtable


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.

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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

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t2 – t1 ≥ m


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Go

City 1 time1 City 2 Time2

Omaha t1 Lincoln t2 t2 – t1 ≥ 60

Omaha t1 Mpls t2 t2 – t1 ≥ 650

Omaha t1 De_Moines t2 t2 – t1 ≥ 400

Mpls t1 Chicago t2 t2 – t1 ≥ 370

De_Moines t1 Chicago t2 t2 – t1 ≥ 600

De_Moines t1 Kansas_City t2 t2 – t1 ≥ 180


In Go, we represent each connection between two cities by two constraint tuples.

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25


Radio X Y

1 x y (x – 8)2 + (y – 9)2 ≤ 25

2 x y (x – 8)2 / 52 + (y – 3)2/32 ≤ 1

3 x y (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 areaThe 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.


The data can be represented in the Crops relation using linear inequality constraints over rational numbers. 26


Corn Rye Sunflower Wheat

x1 x2 x3 x4

15x1 + 8x2 + 10x3 + 15x4 ≤ 3000

30x1 + 25x2 + 5x3 + 10x4 ≤ 8000


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.

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Data Abstraction (1)


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.

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Data Abstraction: View Level


Logical level – infinite relation r, tuple t

Describes what data are stored in terms of sets of infinite relations.

Discussed in Chapter 1

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Data Abstraction: Logical 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

Data Abstraction: Constraint Level


Physical level – the way the data is actually stored in the computer system

We will see example of this in Chapter 17

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Data Abstraction: Physical Level


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 view 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

Discussion Questions


Exercise 2.1 Represent the Constraint relation Free in a relational form assuming that d is an integer.

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Person Day

Andrew d 3 ≤ d, d ≤ 6

Andrew d 16 ≤ d, d ≤ 26

Barbara d 6 ≤ d, d ≤ 18

Barbara d 25 ≤ d, d ≤ 30


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Solution

Person FreeAndrew 3, 4, 5, 6Andrew 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26Barbara 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18Barbara 25, 26, 27, 28, 29, 30


Exercise 2.3 Using a constraint relation represent the street map shown in Figure 1.3 As following

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Solution

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 + 10 1/3, 2<=x, x<=17

Vine x y x=5, y>=2, y<=14

Oak x y x=11, y>=2, y<=12

Maple x y y=5, x<=20, x>=7


Relational Algebra

SQL

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Questions for the Next Lecture

Documents

L6-S1 Constraint DB 2003 SJSU -- CmpE Database Design Dr. M.E. Fayad, Professor Computer Engineering Department, Room #283I College of Engineering San