Theory of automata
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Theory of Automata. By: Fasee Ullah MS(IT) from SZABIST~ Islamabad 4 International Publications. Background. Twentieth century has given the most incredible shocks and surprises e.g, The rise and fall of communism Nuclear war Television Moon walks Inherited engineering etc .

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Theory of Automata

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Theory of automata

Theory of Automata

By: Fasee Ullah

MS(IT) from SZABIST~ Islamabad

4 International Publications


Background

Background

  • Twentieth century has given the most incredible shocks and surprises e.g,

  • The rise and fall of communism

  • Nuclear war

  • Television

  • Moon walks

  • Inherited engineering

  • etc


Background1

Background

  • Due to these great novels

  • Our development moved towards calculating device(s)

  • In the 1930’s, A. Turing studied an “ abstract machine”

  • That had all the capabilities of today’s machines(computers)


Background2

Background

  • His abstract machine to define the upper bound and lower bound,

  • What could do and what couldn’t

  • was beneficial for turning machine

  • Also his conclusion is for today’s real machines

  • In the 1940’s and 1950’s, a machine invented


Background3

Background

  • Studied by a number of researchers

  • Currently called “Finite Automata”

  • Proposed to model the brain function

  • Also used for variety of other purposes

  • In late 1950’s, the Linguist N. Chomsky

  • Introduced new formal grammar

  • Not defined for machines at that time


Background4

Background

  • Has close relationship to abstract automata

  • Also important in development of software components and compilers

  • In 1969’s, S. Cook extended the theory of Turing “what could solve and what couldn’t”

  • S. Cook separated the solvable problems from those that can in principle be solved

  • Latter class of problems called “intractable or NP-hard”


Background5

Background

  • Moore’s Law

  • says that chip density doubles every eighteen months, This means that memory sizes, processor power increases,

  • Number of transistors on a chip will double every year


Year of introduction transistors

Year of IntroductionTransistors

  • 400419712,250

  • 800819722,500

  • 808019745,000

  • 8086197829,000

  • 2861982120,000

  • Intel386™ processor1985275,000

  • Intel486™ processor19891,180,000

  • Intel® Pentium® processor19933,100,000

  • Intel® Pentium® II processor19977,500,000

  • Intel® Pentium® III processor199924,000,000

  • Intel® Pentium® 4 processor200042,000,000

  • Intel® Itanium® processor2002220,000,000

  • Intel® Itanium® 2 processor2003410,000,000


Background6

Background

  • Due to theoretical approaches, what computer scientists do today, e.g

  • Finite Automata, formal grammars, turning machines etc

  • Helps in design and construction of different softwares and what we can expect from our softwares


What is theory

What is theory?

  • The word “theory” shows to study abstraction of computing system

  • In Abstraction, irrelevant complications dropped

  • In order to isolate important concepts


What does automata mean

What does automata mean?

  • It is the plural of automaton, and it means “something that works automatically”

  • Automata heavily used in compilers, text editors, circuits, AI etc

  • Shows how simple operations performed with help of set theoretic operations on language


Automaton

Automaton

  • It accepts input, produces output, may have some temporary storage and can make decisions in transforming the input into the output


What is automata theory

What is Automata Theory?

  • Study of mathematical models that describe with varying degrees of accuracy, parts of computers, types of mathematical computers and similar machines

  • The term Automata Theory, therefore, is used to refer to the study of such ‘Machines’ whose boundaries of capabilities could be predefined.


A simple computer

A simple computer

  • input: switch

  • output: light bulb

  • actions: flip switch

  • states: on, off

SWITCH

BATTERY


A simple computer1

SWITCH

BATTERY

A simple “computer”

  • input: switch

  • output: light bulb

  • actions:f for “flip switch”

  • states: on, off

f

on

start

off

f


Different kinds of automata

Different kinds of automata

  • This was only one example of a computational device, and there are others

  • We will look at different devices, and look at these kinds of questions:

    • What kinds of problems can a given type of device solve?

    • What things are impossible for this kind of device?

    • Is one type of device more powerful than another?


Some devices

Some devices

finite automataDevices with a finite amount of memory.Used to model “small” computers.

push-down automataDevices with infinite memory that can be accessed in a restricted way.

Used to model parsers, etc.

Turing MachinesDevices with infinite memory.

Used to model any computer.

time-bounded Turing MachinesInfinite memory, but bounded running time.Used to model any computer program that runs in a “reasonable” amount of time.


Introduction to languages

Introduction to languages

There are two types of languages

  • Formal Languages (Syntactic languages)

  • Informal Languages (Semantic languages)


Formal language

Formal Language

  • It is an abstraction of the general characteristics of programming languages

  • It consists of a set of symbols and some rules of formation of sentences

  • Sentences are formed by grouping the symbols


Formal language1

Formal Language

  • A formal language is the set of all strings permitted by the rules of formation

  • Study of formal languages is very useful in learning about the different programming languages


Central concepts of automata theory

Central Concepts of Automata Theory

  • Three basic concepts

    • Alphabet --- a set of symbols

    • Strings --- a list of symbols from an alphabet

    • Language--- a set of strings from the same alphabet


Central concepts of automata theory1

Central Concepts of Automata Theory

  • Alphabets

  • An ALPHABET is a nonempty set of symbols

  • It is denoted by S

  • Example:

    S = {a,b}

    where a and b are symbols


Central concepts of automata theory2

Central Concepts of Automata Theory

  • Strings are constructed from the individual symbols

  • Strings are finite sequences of symbols from the alphabet

  • Example : aabba, ababaaa, abbbaaa, etc are the strings formed by t he symbols of the alphabet


Central concepts of automata theory3

Central Concepts of Automata Theory

Assumptions

  • Lower case letters a,b,c,… are used for elements of the alphabet

  • Lower case letters u,v,w,… for string names eg w=aabbaba

  • This indicates that w is a string having specific value aabbaba


Central concepts of automata theory4

Central Concepts of Automata Theory

Concatenation of the strings

  • Two strings are concatenated by appending the symbols of one string to the end of the other string

  • Example u=aaabbb

    v=abbabba

    Concatenated string uv=aaabbbabbabba


Reverse of the string

Reverse of the string

  • The reverse of a string is obtained by writing the symbols in reverse order.

  • Example

    wR = anan-1an-2……………a0

    Where w=a0a1a2……………an


Length of the string

Length of the string

  • The length of the string is the number of symbols in the string

  • |w| = 5 if w = aabaa

  • Empty String has no symbols and is denoted by l

  • |l| = 0


S and s

S+andS*

  • S is an alphabet

  • S* is the set of allstrings obtained by concatenating zero or more symbols fromS

  • S ={a,b,c}

  • S*={e, a,b,c,aa,ab,ac,ba,bb,bc,ca,cb,cc,aaa,…}


Theory of automata

S+

  • S+ is the set of allstrings obtained by concatenating one or more symbols

  • S ={y}

  • S+ ={y,yy,yyy,yyyy,…}


Language

Language

  • A set of strings of characters from alphabet

    Grammar

  • Set of rules defining a language

  • Enable us to decide in a finite time

  • Given string of alphabet is or isn’t in the language.


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