01_IntroToComputation

7/30/2019 01_IntroToComputation

1/37

A Gentle Introduction to

Theory of Computation:

Ravindra V. Joshi


2/37

Goals of the session

What is meant by Computation?

A model for Computation ( Turing Machine )

Why Turing Machine is Universal?

Relevance of Church-Turing thesis

Halting theorem and its proof ( informal )

Complexity

P, NP, NP-Hard, NP-Complete

Most Important Problem in Computer Science!!


3/37

Theory of Computations

What are different types of Computers (Automata ) and what

kind of problems can be solved by them?

FSM RegEx Pattern Matching

PDA CFG -> Language Parsers

TM -> TM-Languages -> All computer languages and Programs


4/37

Turing Machine (Revised )


5/37

Universal Turing Machine


6/37

Universal Turing Machine

In other words, Any UTM with any number of Tapes and

Heads can be simulated by a 3-Tape Head in PolynomialTime


7/37

How Universalis UTM ?

Church-Turing thesis : Every form of computation can be expressedas a TM with Polynomial overhead

Parallel

Real ( Precision )

Randomness

A Tape with Random Bits

Quantum Computation

Other Exotic Theories ( String Theory!!! )

Natural-BIO_Inspired

ANN, ACO, Swarm_Intelligence, Cellular Automata and otheralgorithms

Apporximate Solutions

for subset of all Inputs

for some Sub Optimal Solution


8/37

UnComputability

0-Normal Idea = Input Program Output

PROGRAM TO COUNTNO OF SEMICOLONS

PROGRAM TO REVERSERS(REVERSE A STRING)

Get(s);

While( *s )

S.Push(s);

While ( !S.Empty() )

Print S.Pop();

Hello,

Hyderabad;

Hello, Bangalore;Hello, Chennai;

Hello, Mumbai;

Hello

3

olleH


9/37

UnComputability

1- Idea-Program as Input =

Another Program Program Output

PROGRAM TO COUNT NO

OF SEMICOLONS

Get (S); Count = 0;

Foreach ( char c in S )

If ( c==; )Count++;

Print Count;

3

RS(REVERSE A STRING)

Get(s);

While( *s )

Stack.Push(*s);

While ( !Stack.Empty() )

Print Stack.Pop();


10/37

UnComputability

Idea 2- Same-Program as Input To Itself =

Program Program Output

PROGRAM TO COUNT NO

OF SEMICOLONS

Get (S); Count = 0;


If ( c==; )Count++;

Print Count;

5

PROGRAM TO COUNT NO

OF SEMICOLONS

Get (S); Count = 0;


If ( c==; )Count++;

Print Count;


11/37

UnComputabilityHello, World

Consider a program to print Hello, World

Print Hello, World Hello, World

While(true) ;

Print Hello, World;


12/37

UnComputabilityDependency of

Ouptut on both Program and Input This program will print Hello, world for positive values and 0 but not

for negative values

The point is same program may print Hello, World for some data

and may not, for some other and vice versa

Get(x);

While(x != 0 )

x-- ;

Print Hello, World

Hello, World


13/37

The UnComputability Problem

Given a Program P and Input I can we say whether it will print

Hello, World or not?

P

I

PRINTS_HW

If P prints HW for input

I return TRUE else

FALSE

TRUE

FALSE


14/37

UnComputability Problem - Proof

Let us consider the pseudo-code for H The Undecidability

function

H( P, I )

{

If ( P halts over I )

Return Yes

Else

Return No

}


15/37

UnComputabilityProof ( contd )

Let us construct a function H1 on top of H as follows

H1( P, I)

{

IF (H( P, I) == TRUE )

Print YES;

ELSE

Print Hello, World;

}


16/37

UnComputability

Let us build a function H2 on top of H1.

H2( P )

{

H1(P,P)

}

Here a program is given as input to the same program.

This eliminates the need for Data.

H2(P) If a Program P ( /* when P itself is given as input to it*/) prints Hello World, it prints YES else it prints Hello, World

Now, the twist comes, what happens when we invoke

following function

H2(H2)?


17/37

UnComputability( contd )

// IF H2 can print HW, then Yes

// ELSE should Print Hello, World

// CASE-A: H2(H2) will print HW I.e., H(H2,H2)==> YES

H2(H2)

H1(H2,H2)

H(H2,H2) returns TRUEH1 prints YES

Output is YES

Where is "Hello, World?"


18/37

Uncomputability( Contd. )

//CASE-B: H2(H2) will not print HW i.e.,

//PRINTS(H2,H2)==>FALSE

H2(H2)

H1(H2,H2)

H(H2, H2) returns FALSE

In H1, ELSE part is executed

"Hello, World" is printed

But according to our own starting condition,

H2(H2) should not print "Hello, World"


19/37

UnComputability Problem Proof

( Final )H2 is a contradiction

H2 can not exist

H1 can not exist

H can not exist

Our assumption was wrong

Hence the theorem


20/37

Uncomputability Problem

Implemented in LISP


21/37

UnComputability Problem

Implemented in LISP


22/37

From Computability To Complexity

Given that a function is Computable by Turing Machine how

many instances of Turing Instructions will be executed?

In other words, how efficient is given function?

Given a definition of Problem, can we tell what is the

minimum number of instructions required to implement it?


23/37

Some Complexity Classes

Let n be the size of input( like Length of Array ) P-Polynomial - No of steps required can be expressed as a

polynomial of (length of Array ) O(n^2)

Complement of Polynomial

Cannot expressed as Polynomial of n

Exponential of n

NP Problems whose correctness can be verified within Polynomial

number of steps

NP is not same as Complement of P, in fact it is super set of P

NP-Hard, a class of problems which are currently solvable inexponential time only, and satisfy a curious property that if one of

the problems belonging to this class can be solved, all the problems

in that class can be solved

NP-Complete : A problem that is both NP and NP-Hard


24/37

Examples

P - Sorting Problem

NP Dinner Party Problem

NP Hard Finding the Hamiltonian Circuit in a Graph

NP Complete Mine Sweeper, also Dinner Party


25/37

Example of NP Problem - Dinner PartyMOTHER OF ALL PARTIES


26/37

But---Guests should be carefully

invited


27/37


invited


28/37


invited


29/37


invited


30/37

Complexity Classes, Visualized

P

NP

NP-HARDNP-COMPLETE


31/37

Most Important Question in

Computer Science Today Is this relation true?

http://www.claymath.org/millennium/P_vs_NP/
http://www.claymath.org/millennium/P_vs_NP/http://www.claymath.org/millennium/P_vs_NP/http://www.claymath.org/millennium/P_vs_NP/http://www.claymath.org/millennium/P_vs_NP/http://www.claymath.org/millennium/P_vs_NP/


32/37

Some Historical background

http://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-

1978.html

In a letter written by Godel to Neumann, Godel raises the question

of P=NP
http://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.htmlhttp://blog.computationalcomplexity.org/2006/04/kurt-gdel-1906-1978.html


33/37

QUESTIONS??

THANK YOU

Testing Ourselves Have we really


34/37

Testing Ourselves

Have we really

understood Uncomputability

theorem1. Do we need to do all this twisting and turning? Cant be there asimpler proof?2. We reverse the functionalities of h in h1. Is this fair? Can't we

negate existence of any logic by this kind of approach? Justconsider following series functions

MoreThan100 ( P )

{

If P contains more than 100 Lines

Return TRUE

Else

Return FALSE

}


35/37

Testing Ourselves

h1( P )

{

If MoreThan100( P )

Print NoElse

Print Yes

}

h2( P )

{

h1( P )

}


36/37

Testing Ourselves

3. Answer following questions on programs above.

a. Is h2 necessary?

b. What will be the output when A program contains 50 lines?

150 lines?

c. hat will be output of h2(h2)? Will it not give wrong results ?Does this mean there cannot be a program to count the no. of

lines( We all know that there indeed exists a program. Then,

where did we go wrong?

4. When we map h2(p) ==> h1( p, p). Is it merely, an approach

to reduce two parameters to one? Could'nt we make h1accept only program and no data h1(p,null)? Or Program P

and Data some f(P). F may be any string transforming

function?


37/37

Testing Ourselves

3. In Final step, we give H2 only as input to H2. Is this a must? If

so, why? Can't we find a simpler version?

4. Prove the Undecidability of Halting Theorem

5. This is only out-line of the Proof. Formal forms techniques

used here are known as Diagonalization and Reduction.Study them

01_IntroToComputation

Documents

Transcript of 01_IntroToComputation