Theory of computation Lec1

Theory of Computation

Welcome to a journey to

Lecture1: Introduction

Cairo UniversityFCI

Dr. Hussien SharafComputer Science [email protected]

Part oneIntroduction

Dr. Hussien M. Sharaf

What is TC and how old?TC is an accumulation of mathematicians

work to make a model for a machine that can do thinking and calculations.

The concept of a machine at early 1900 was a device that does physical work.

Scientists effort started with a machine that can do specific calculations like encrypting text using specific set of steps.

Alan Turing believed he could invent a generic machine that can solve more than one type of problems.


Let s hear a StoryIt started before World War II, Germans

army used Enigma encryption.Alan Turing and many mathematicians

tried to break the Enigma encryption.Their efforts resulted in emergence of a

mechanical device that was dedicated for deciphering Enigma encrypted messages.

As a result many German submarines were attacked and destroyed.


Enigma machine


Enigma wiring


Back to TCVon Newman, Alan Turing and many

others continued working on creating a model for a generic machine that can solve different types of problems.

The accumulation of their work resulted in emergence of a collection of theorems called theory of computation.


What is TC?TC emerged to give answers for

“What are the fundamental capabilities and limitations of computing machines?”

Most powerful and modern super computers can NOT solve some problems!!

No matter how much processors get fast , no matter how much memory can be installed; the unsolved problems remain unsolved!

We might need life time of the universe to find prime factors of a 500-digits number!


Why we still need TC?Technologies become obsolete but basic

theories remain forever.TC provides tools for solving computational

problems like regular expressions for string parsing and pattern matching.

Studying different types of grammars like CFG would help in many other areas like compilers design and natural language processing.


Branches of TC


Branches of TCTC consists of:1. Automata theory: mathematical

models for computational problems such as pattern recognition and other problems.

2. Computability theory: computational models and algorithms for general purpose.

3. Complexity theory: classifying problems according to their difficulty.


Part twoMathematical notations and

Terminology


Sets-[1]A set is a group of elements

represented as a unit.For example

S ={a, b, c} a set of 3 elements

Elements included in curly brackets { }a S a belongs to S

f S f does NOT belong to S


Sets-[2]If S ={a, b, c} and

T = {a, b}Then

T ST is a subset of S

T S ={a, b}T intersects S ={a,

b}

T S ={a, b, c}T Union S ={a, b, c}

Venn diagram for S and T


Sequences and TuplesA sequence is a list of elements in some order.

(2, 4, 6, 8, ….) parentheses A finite sequence of K-elements is called k-

tuple.(2, 4) 2-tuple or pair(2, 4, 6) 3-tuple

A Cartesian product of S and P (S x P)is a set of 2-tuples/pairs (i, j)where i S and j P


Example for Cartesian ProductExample (1)If N ={1,2,…} set of integers; O ={+, -}

N x O ={(1,+), (1,-), (2,+), (2,-), …..}Meaningless?Example (2)

N x O x N={(1,+,1), (1,-,1), (2,+,1), (2,-,2), …..}

Does this make sense?


Continue Cartesian ProductExample (3)If A ={a, b,…, z} set of English alphabets;

A x A ={(a, a), (a, b), ..,(d, g), (d, h), …(z, z)}

These are all pairs of set A.Example (4)If U={0,1,2,3…,9} set of digits then U x U

x U ={(1,1,1), (1,1,2),...,(7,4,1), ….., (9,9,9)}


Example (5)If U={0, 1, 2, 3…, 9} set of digits then U x… x U ={(1,..,1), (1,..,2),...,(7,..,1), ...,

(9,..,9)}

Continue Cartesian Product

nCan be written as Un


Relations and functionsA relation is more general than a function.Both maps a set of elements called

domain to another set called co-domain.In case of functions the co-domain can be

called range.R : D CA relation has no restrictions.f : D RA function can not map one element to

two differnet elements in the range.


Surjective function

t1

T

t3

t2

P1

P2

Pn

P4

P3

s1

s3

s2t1

t3

t2

S T

Bijective function

Many planes fly at the same time

Only one plane lands on one runway at a time


What is the use of functions in TC?Helps to describe the

transition function that transfer the computing device from one state to another.

Any computing device must have clear states.

s1

s3

s2

Ds2

shalt

s3

R


GraphsIs a visual

representation of a set and a relation of this set over itself.

G = (V, E)V ={1, 2, 3, 4, 5}E = {(i, j) and (j, i)| i, j

belongs to V}E is a set of pairs ={(1,

3), (3, 1) …(5, 4), (4, 5)}

1

35

2 4


Graphs ConstructIs there a formal language

to describe a graph?G =(V, E)Where :V is a set of n vertices

={i| i < n-1}E is a set of edges. Each

edge is a pair of elements in V={(i, j), (j, i)|i, j V}

or={(i, j) |i, j V }

1

35

2 4


Definitions (S Alphabet) : a finite set of letters,

denoted SLetter: an element of an alphabet SWord: a finite sequence of letters from

the alphabet SL (empty string): a word without

letters.Language S * (Kleene ‘s Star): the set

of all words on SDr. Hussien M. Sharaf

Strings and languagesA string w1 over an alphabet Σ is a finite

sequence of symbols from that alphabet.1. Σ: is a set of symbols i.e. {a, b, c, …, z} English letters;{0,1, 2,…,9,.} digits of Arabic numbers{AM, PM}different clocking system{1, 2, …, 12}hours of a clock;


Strings -22.1 String: is a sequence of Σ (sigma) symbols

Σ Σ to the power?

Example Description

{a, b, c, …, z}

Σ5 apple English

string

{0,1, 2,…,9,.}

Σ2 35 the oldest

age for girls

{AM, PM} Σ1 PM clocking

system

{1, 2, …, 12}

Σ1 or Σ

2 12 a specific hour in the day


Strings - 32.2 Empty String is Λ (Lamda) is of

length zero

Σ0 = Λ

2.3 Reverse(xyz) =zyx2.4 Palindrome is a string whose

reverse is identical to itself.If Σ = {a, b} thenPALINDROME ={Λ and all strings x such thatreverse(x) = x }radar, level, reviver, racecar,madam, pop and noon.Dr. Hussien M. Sharaf

Strings - 42.5 Kleene star * or Kleene closureis similar to cross product of a set/string over itself.

If Σ = {x}, then Σ* = {Λ x xx xxx ….}

If Σ = {x}, then Σ+ = {x xx xxx ….}

If S = {w1 , w2 , w3 } then

S* ={Λ, w1 , w2 , w3 , w1w1 , w1w2 , w1w3 , ….}

S+ ={w1 , w2 , w3 , w1w1 , w1w2 , w1w3 , ….}

Note1: if w3 =Λ, then Λ S+

Note2: S* = S

**


Part ThreeExercises


Exercise 1 Assume Σ={0, 1}1. How many elements are there in Σ2?Length(Σ) X Length(Σ) = 2 X 2 = 23 =42. How many combinations of in Σ3?2 X 2 X 2 = 23 3. How many elements are there in Σn?(Length(Σ))n= 2n


Exercise 2 Assume A1={AM, PM}, A2 = {1, 2, …59}, A3 = {1, 2, …12}1. How many elements are there in A1 A3?Length(A1) X Length(A3) = 2 X 12 =24

2. How many elements are there in A1 A2 A3)?

Length(A1) X Length(A3) = 2 X 59 X 12 = 1416


Exercise 3 Assume L1={Add, Subtract},

DecimalDigits = {0,1, 2, …9} 1. Construct integer numbers out of L2.DecimalDigits +

-A number of any length of digits.2. Construct a language for assembly

commands from L1 and DecimalDigits .L1 DecimalDigits + {,} DecimalDigits + - Commands in form of Add 1000, 555


Exercise 4 Assume HexaDecimal = {0,1, …9,A,B,C,D,E,F}1. How many HexaDecimals of length 4?

164 2. How many HexaDecimals of length n?

16n 3. How many elements are there in

{0, 1}8 ?28 = 256


End of Lecture 1


Theory of computation Lec1

Education