Dr. Naveed Riaz Design and Analysis of Algorithms 1 1 Formal Methods in Software Engineering Lecture...

Dr. Naveed Riaz

Design and Analysis of Algorithms

11

Formal Methods in Software Engineering

Lecture # 24

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Use of Formal Methods in SE Branch of computer science in which we argue and

analyze the computer programs using formal techniques.

Programs specification and verification. Constructing programs.

Each step written is valid and lead to a correct solution

Number of tools and techniques are available.

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Can we properly test the software? Famous saying by Dijesktra -> By testing we can

only prove the present of bug not the absence of bug In order to prove mathematical theorem, counter

examples are provided 1. All numbers are < 1000

You can either provide formal prove or counter example

Overhead with test cases: Not possible to write huge number of test cases

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Test Cases VS F-M Test two equal Strings .

isEqual (“Cat”, Dog”) – Expected output: False

isEqual (“Testing”, Testing”) – Expected output: True

isEqual (“House”, Home”) – Expected output: False

Many number of possibilities

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Test Cases VS F-M Programming Exp:.

Equal = strlen(String1)== strlen(String2);

If (equal)

for ( i =0; i<strlen(String1); i++)

equal = String1[i] = = String2 [i];

return equal;

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Structure Programming Same phenomena can be seen with structure

programming. Usage of flow graph to understand program structure

Components of flow graph (Sequence, selection and loop)

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Flow Graphs Identify the flow diagram

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Flow Graphs Bubble sort Flow graph

Sorted = false ; //1

While (!sorted){ //2

Sorted = true;

For(int i=0;i<size-1;i++){//3

If(a[i]>a[i+1]){ //4

Swap(a[i]>a[i+1]); //5

Sorted = false;

} //6

}//7

} //8

1

2

3

4

65

7

8

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Flow Graphs Base upon flow graphs we try to write test cases to test the

different components of flow graph

Limitations of these kind of testing

3 types of test coverage's that we have when using flow graphs

Statement coverage: We make sure that every statement must be executed at least once.

Branch Coverage: Identify branches and try to test all the branches

Path Coverage: Execute all possible paths and then write test cases

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Path Coverage overhead For(int i=0; i<N; i++) { //1

If(condition1)

// some statement //2

Else

Some statement //3

//some statement

} //5

How many paths are there

For N iteration?

1

2 3

4

5

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Model-based solution We try to develop model of the system

We analyze the correctness of these models using mathematical techniques

What is a Model? Why Model required and How it help us?

Diskstra’s Game

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Model-based solution Color of the last ball?

1 Ball Game : Suppose there is only one ball in jar

2 Ball Game: Suppose there is two balls in jar -> 3 possibilities

Both White

Both Black

1 W, 1 B

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Model-based solution 3 Balls Game

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Model-based solution If even number of white balls -> Last ball color Black

If odd number of white balls -> Last ball color White

Can you verify this for 100 balls in the jar? i.e. Our hypothesis is true or not?

To verify this we need formal proof -> built mathematical model

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Mathematical Model If even number of white balls -> Last ball color Black

If odd number of white balls -> Last ball color White

(2 B out, 1 B in ) B-2+1, W=b-1, W

F(B,W) = (2 W out, 1 B in) B+1, W-2

(1 of each out, 1 W in) B-1, W-1+1 = B-1, W

In tow cases we reduces the black ball by 1 and maintain the number of white balls

In third case, we reduce the white ball by two and increases the black ball by 1.

Total number of balls is reduced by exactly one in each move.

Parity of the white ball does not change ( either 0 or 2)

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Propositional Calculus Two tools will be used to built mathematical models ?

Logic and Set theory

What is a proposition? Statement which is either true or false e.g. My is name is Ali

(true or false)

A = B (true or false)

Which statement is not a proposition? You should read this book carefully.

Logic is built on proposition

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Propositional Calculus Two tools will be used to built mathematical models ?

Logic and Set theory

What is a proposition? Statement which is either true or false e.g. My is name is Ali

(true or false)

A = B (true or false)

Which statement is not a proposition? You should read this book carefully.

Logic is built on proposition

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Truth Table Implication

Only false when P (T) and F (F) otherwise implication is

always True.

If P is true only then we can make a statement about Q.

If P is false then we do not need to say any thing about Q.

Originally conditional statement (if or If-then-Else)

P Q P Q

T T T

T F F

F T T

F F T

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Truth Table Bi-Conditional – (If and only If)

P > means p-> Q and Q ->P

Outcome is true if P and Q have the same value

If P and Q do not have same values then Bi-conditional is not true

Compare it with Exclusive OR

Can be used for equal statements

P Q P < Q

T T T

T F F

F F T

F T F

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Tautology A Compound proposition which is always true,

irrespective of the truth values of the comprising propositions. p V ┐p

Contradiction -> Opposite to tautology. Proposition which is always false

p ˄ ┐p (outcome always false)

It is an important tool to prove the theorem

For example; We can take a hypothesis and then we can show that if we can assume something that end up in a logical contradiction and hence the hypothesis is incorrect else we can prove it.

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Logical Equivalence

The proposition P and Q are called logically equivalent if p>q is tautology

It is written as;

P ≡Q

For example ; ┐(p v q) ≡ ┐p ˄ ┐q ( Demorgans Law)

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Logic Puzzle The Island of Knights Knaves

Two types of resident -> Knights and Knaves

They have some special characteristics

Knights -> Always speak truth

Knaves -> Always speak lie

Based upon these information we need to solve some problems

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Logic problem for the day Some one asks person A,

“Are you a knight?”

He replies, “If I am a knight then I will eat my hat”.

Prove that A has to eat his hat. Use the logic to solve this problem; First you have identify the proposition – then

hypothesis is required Think that what does this mean that a person A make a

statement “S” on island If person A is knight then statement “S” is true else

False

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Logic problem for the day Two solution possible: Truth table and Logical

deduction rule First conversion to proposition is required

A is a knight: A

A eats his hat: H

If I am a knight then I will eat my hat.

A=> H

We have seen that ( X S)

Both have the same truth values there for;

(A A => H)

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Using Truth Table Truth table column

A H A=> H A (A => H)

A H A=> H

T T T

F T T

T F F

F F T

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A H A=> H A (A => H)

A A=> H A (A => H)

T T T

F T F

T F F

F T F

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A H A=> H A (A => H)

A H A=> H A (A => H)

T T T T

F T T F

T F F F

F F T F

If another H having true result but with False value then?

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Problem with Truth Table Any problem with the truth table? Perform good when the number of proposition is

small but if the number of proposition is large then? In truth table number of rows depends on the number

of variables: If we have two variables then … Another method is to built logical argument using

deduction rules.

Dr. Naveed Riaz Design and Analysis of Algorithms 1 1 Formal Methods in Software Engineering Lecture...

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