MELJUN CORTES Automata Lecture Intro to Finite Automata 2

8/21/2019 MELJUN CORTES Automata Lecture Intro to Finite Automata 2

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Theory of Computation (With Automata Theory)

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Introduction to Finite Automata

Introduction to Finite

Automata

Automata Theory

What is Automata Theory?

• Au tomata theo ry is the study

of abstract machines.

• Abstract machines are

theoretical representations of real-world machines (such as

a computer) but without the

unnecessary details

associated with a particular

instance of that machine.

• By removing these details, it is

easier to analyze the operation

of the machine being studied.


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• For example, assume that the

machine is a simple light bulbconnected to a power source

via a single toggle switch.

If the lever of the toggle switch

is pushed or flipped, the lightbulb turns on. If the lever is

pushed again, it turns off.

The operation of this machinecan be described by the

following graph:

Light

Bulb is

On

push lever of

switch

Light

Bulb is

Off

push lever of

switch


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Each node in the graphrepresents the “state” of the

light bulb. The bulb can only

be in one of two states, o f f or

on .

The edges of the graph show

when the machine moves

between the two states.

Assume initially that the bulb is

turned off (it is in the off state).

If the lever of the toggle switch

is pushed, it moves from the

off state to the on state.

Now the bulb is in the on state.

If the lever is pushed again,

the light bulb moves from the

on state to the off state.


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This is now an abstract model

of the light bulb machine. With

this model, the operation of the

machine can be analyzed

without having the actual

physical machine present.

For example, it is easy to see

from the graph that if the lever

of the switch is pushed 6 timesin succession, the light bulb

will end up in its current state.

This abstract model of the lightbulb machine is called an

automaton and the graph that

represents it is called its state

diagram .


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• Automata theory therefore

uses abstract or mathematical

models to represent machines

and computers.

Like the light bulb machine, a

computer can also be in one

state or another depending

upon the contents of its main

memory, processor registers,etc.

When input data arrives or an

event occurs (like the clickingof the mouse or the typing of a

character on the keyboard),

the computer moves from one

state to another.


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The following are just some of

the things that can be done

once models have been

established:

1. Determine the capabilities

and limitations of each

machine.

2. Determine which machine

is more powerful.

3. Determine what each

machine can compute.


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• An automaton is not only used

to represent machines

(hardware) but they can also

be used to represent software

(such as operating systems

and compilers).

• The different types of

automata that will be

encountered in this course are

finite automata (deterministic

and nondeterministic), push-down automata, and Turing

machines.


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Finite Automata

Deterministic Finite Automata

• A f in i te au tomaton is a

machine that has a limited

(finite) number of states.

• Because of this limitation,

these are normally used to

model computers with a limited

amount of memory or programs that process only a

small amount of data.

• Discussions here will focus on

hypothetical machines whoseonly task is to determine

whether an input string or a

series of input symbols is

acceptable or not based on

some predefined rule.


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• Given the following state

diagram of a finite automatonM 1.

Take note of the followingfacts about M 1:

1. It has four states labeled

q0, q1, q2, and q3.

2. It has an in i t ial s tate which is q0.

3. It has one f inal or accept

state which is q2.

4. It has edges connecting

the states. These edges

represent t ransi t ions .

q 1

0

qq 2

q 3

1

0, 1

0

1

0

1

q 0


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• Consider the following process

flow for M 1:

1. The machine is currently at

the initial state, q0.

2. The system receives a 0

as its input. So the system

moves from q0 to q1.

3. The system receives

another 0 as its second

input. So the system stays

at state q1.

4. The system receives a 1

as its third input. So the

system moves from state

q1 to q2.


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5. The system receives a 0as its fourth input. So the

system moves from state

q2 to q1.

6. The system receives a 1as its fifth and last input.

So the system moves from

state q1 to q2.

Since the system is now at

state q2 and it is a final state,

then it is said that the system

accepts the input string 00101.

To generalize, M 1 accepts any

string that starts with a 0,

followed by any number of 0s

and 1s, as long as the last

input is a 1.


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Formal Definition of a Deterministic

Finite Automaton (DFA)

• A deterministic finite

automaton (DFA) is composed

of

1. A finite set of states

designated as Q.

2. A finite set of symbols

(alphabet) used as inputsdesignated as .

3. A t rans i t ion func t ion

designated as .

4. A start state designated

as q0.

5. A set of f inal states or

accept states designated

as F .


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• The transition function is a

table that specifies when statetransitions (movement from

one state to another) are

carried out.

For the DFA M 1, its transitionfunction is

For each input symbol, theautomaton can move to only

one state from any given

current state. This is what is

meant by the term

“deterministic.”

0 1

q0 q1 q3

q1 q1 q2q2 q1 q2

q3 q3 q3


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• Using the formal definition of a

DFA, M 1 can now be described

as a 5-tuple:

M 1 = {Q, , , q0, F }

where:

1. Q = {q0, q1, q2, q3}

2. = {0, 1}3. :

4. Start State = q0.

5. F = {q2}

0 1

q0 q1 q3

q1 q1 q2

q2 q1 q2

q3 q3 q3


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• More examples of DFAs:

1. DFA M 2

2. DFA M 3

q 1

0, 1

q 0

0, 1

q 1

1

q 0

1

00


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3. DFA M 4

4. DFA M 5

0, 1q 1

0, 1

0, 1

q 2q 0

0 0

q 2q 1q 0

1 1 0, 1


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5. DFA M 6

6. DFA M 7

1

0

01

q 0

q 1

q 3

q 2

q 4

0

1

0

0 1

1

1

0

1

0, 1

q 0 q 1 q 2

0


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• For DFA M 2:

1. Q = {q0, q1}

2. = {0, 1}

3. :


5. F = {q0}

M 2 accepts all strings

(including the empty string ε )

whose length is even.

0 1

q0 q1 q1

q1

q0

q0


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• For DFA M 3:

1. Q = {q0, q1}

2. = {0, 1}

3. :


5. F = {q0}


(including the empty string ε)

that contain an even number

of 1’s.

0 1

q0 q0 q1

q1 q1 q0


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• For DFA M 4:

1. Q = {q0, q1, q2}

2. = {0, 1}

3. :


5. F = {q1}

M 4 accepts all strings that only

has exactly one 0.

0 1

q0 q1 q0

q1 q2 q1q2 q2 q2


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• For DFA M 5:

1. Q = {q0, q1, q2}

2. = {0, 1}

3. :


5. F = {q0}


(including the empty string ε)

whose length is divisible by 3.

0 1

q0 q1 q1

q1 q2 q2q2 q0 q0


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• For DFA M 6:

1. Q = {q0, q1, q2, q3, q4)

2. = {0, 1}

3. :


5. F = {q1, q3}

• M 6 accepts all strings that start

and end with the same

symbol.

0 1

q0 q1 q3

q1 q1 q2

q2 q1 q2

q3 q4 q3

q4 q4 q3


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• For DFA M 7:

1. Q = {q0, q1, q2}

2. = {0, 1}

3. :


5. F = {q1}

M 7 accepts all strings that

have at least one 1 and the

last 1 is always followed by an

even number of consecutive

0s.

0 1

q0 q0 q1

q1 q2 q1

q2 q1 q1


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• The set of strings that anautomaton accepts is called

the language o f the

automaton .

For example, the language of DFA M 2 is the set of all strings

over the alphabet = {0, 1}

whose length is even.

If L is the language recognized

by a DFA M , then it is

designated as L(M ).

So for the language of M 2, itcan be mathematically

described as:

L(M 2) = {w the length of w

is even}


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• A language that is recognized

by a DFA is called a regular

language .

For example, the language

composed of all strings over

the alphabet = {0, 1} that

starts and ends with the samesymbol is a regular language

because there is a DFA

(specifically, M 6) that

recognizes it.

Therefore, L(M 6) is a regular

language.

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MELJUN CORTES Automata Lecture Intro to Finite Automata 2