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Preface

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----

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vi

Preface

Notations

Z

-

-

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Y

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R

--

S

---

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l

--

U

--

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-

--

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-

-

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G

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-

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dd

dd

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-

-

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viii

Notations

Contents

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--

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Contents

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Contents

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xii

Contents

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Contents

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Chap

ter

0

Introduction

0.1

BASICS

0.1.1

Sets

----

----

------

-

---

-

-

-Z

---

-----

-

-

---

--

---

-

----

--R

----

-

---

S

----

-

-

---

-

--

T

--

Z

----

(a)

Un

ion

--------

----

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-

--

X

---

X

R

R

X

(b)

Intersec

tion

---

Y---

--

Y

R

R

Y

Y

(c)

SetD

ifference

----

---

-

R

Q

(d)

Com

plemen

t

--

---

Properties

ofsetoperations

----

---

X

Y

X

X

Y

Y

--

X

X

X

X

Y

Y

Y

Y

2

TheoryofAutomata,FormalLanguagesandComputation

-

X

Y

Y

X

Y

Y

X

X

Y

X

-

X

Y

Y

X

--

X

Y

Y

X

Example0.1.1:

X

Y

Solution

R

X

B

R

Q

X

B

R

Q

Q

B

R

Q

R

Q

B

R

R

B

R

Y

X

S

Y

-

R

Y

B

R

R

B

R

Q

R

Q

B

R

Q

Q

B

R

Q

X

B

R

X

Y

S

X

X

Y

Example0.1.2:-------

Y

~

Y

Z

Z

S

Solution

R

R

B

R

Q

B

R

R

B

R

Y

~

S

Y

~

Introduction

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-

R

Y

~

R

Y

~B

R

R

B

R

Q

B

R

Y

~

Y

Z

X

Y

B

XZ

Y

Z

B

-

X

Y

B

Y

B

Y

Z

S

Y

Y

B

Y

B

Z

Z

Y

B

Y

Z

B

Y

B

S

Example0.1.3:--

X

X

X

X

Solution

--

X

X

T

X

X

R

X

X

B

R

R

X

B

R

R

R

B

R

R

R

4

TheoryofAutomata,FormalLanguagesandComputation

B

R

X

R

B

R

X

X

X

X

T

X

X

-

-

X

X

T

X

X

--

--

X

X

R

X

X

B

R

X

R

B

R

R

R

B

R

R

R

B

R

X

X

X

X

T

X

X

X

X

X

X

Example0.1.4:------

--

Y

Y

Y

Y

Solution

-

Y

Y

T

Y

Y

-

R

Y

Y

B

R

R

Y

B

R

R

R

B

R

R

R

B

R

Y

R

B

R

Y

Y

Y

Y

T

Y

Y

-

Y

Y

T

Y

Y

---

R

Y

Y

B

R

Y

R

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B

R

R

R

B

R

R

R

B

R

R

Y

B

R

Y

Y

Y

Y

T

Y

Y

Y

Y

Y

Y

Example0.1.5:----

X

~

~Y

~

Y

~

~X

~

Solution

R

X

~E

Q

X

E

Q

Q

E

R

~

R

~

E

R

~Y

~

R

Y

~E

Q

Y

E

Q

Q

E

R

~

R

~

E

R

~X

~

Y

~

~X

~

Example0.1.6:---

-

-

X

X

Y

Solution

X

X

X

Y

X

Y

Y

~

Y

~

X

Y

~X

~

Q

X

Y

~X

X

Y

~

Y

~X

Y

~X

Y

~

X

Y

~

6

TheoryofAutomata,FormalLanguagesandComputation

Y

~X

Y

~

Y

~

Y

~

Z

X

Q

Additiona

lTerm

ino

logy

--

-

Y

Z

--

--

--

--

-

-------

----

-

-

--

-

-

R

R

----

-

l

l

R

R

Example0.1.7:

-

Solution

--

---

-

Z

Example0.1.8:

-

Solution

-

-

-

-

Z

43 2

$# "

Example0.1.9:

l

X

l

l

X

lX

l

R

l

R

l

R

R

R

R

R

R

R

R

R

X

l

X

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Example0.1.10:

l

l

l

l

Solution

l

l

I

I

0.1.2

RelationsandFunctions

----

--

--

R

-

R

---

----

-

--

--

8

TheoryofAutomata,FormalLanguagesandComputation

Se

tS

Se

tT

a

b

c

d

e

w

xy z

--

y z x yx z

3 4 3 4 3 4 3 4 3 4 3 4

(1,,

3)

x

(1,,

4)

x

(1,,

3)

y

(1,,

4)

y

(1,,

3)

z

(1,,

4)

z

(2,,

3)

x

(2,,

4)

x

(2,,

3)

y

(2,,

4)

y

(2,,

3)

z

(2,,

4)

z

1 2

-

-----

Equ

iva

lence

Rela

tion

--

l

-

---

-

R

R

-

R

R

R

--

R

R

R

--

Part

ialOrdering

Re

lations

-

-

-

--

-

--

-

}---

-

-

}-

Part

ition

-

---

-

R

--

g

Y

Z

-

---

---

-

-

--

-

R

-

--

Introduction

9

a

b

c

d

e

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y

z

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----

-

R

-

--

Example0.1.11:

-

R

R

B

R

-

---

Solution

-

-

--

R

R

B

R

-

-

R

-

-

R

R

B

R

-

-

---

R

R

--

R

R

B

R

B

R

-

-

-

--

--

B

--

-----

-

R

R

B

R

B

R

-

--

-

--

-

B

R

R

B

R

B

-

--

-

Example0.1.12:

-

---

Solution

--

--

-

R

f

--

R

10

TheoryofAutomata,FormalLanguagesandComputation

R

-

f

R

-

-

f

B

B

B

f

--

f

f

B

B

-

B

B

f

-

--

--

---

-

Example0.1.13:--

--

--

Solution

----

Example0.1.14:

-

---

-

-

Solution

--

R

Q

---

R

Q

---

R

R

Q

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---g

R

R

-

R

Func

tions

-

---

--

-

--

-----

--

-

--

Func

tions

asre

lations

r

-

--

l-

R

-

KindsofF

unc

tions

r

--

---

-

~-

~

12

TheoryofAutomata,FormalLanguagesandComputation

a b c

wx

yz

a

b

c

d

e

w

x

y

z

r

--

--

r

-

-

-

-

-

-

-

--

r

--

-

-

r

--

Example0.1.15:

-

-

---

Solution

-

-

g

Example0.1.16:

-

r

-

Introduction

13

a

b

cd

w

x

y

z

a

b

c

d

e

w

x

y

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Solution

-

---

-

Example0.1.17:

---

Solution

I

I

I

g

I

--

-

Example0.1.18:

r

R

g

-

Solution

-

-

R

B

B

-

14

TheoryofAutomata,FormalLanguagesandComputation

a b c d

1 2 3

A

B

R

--

R

-

: 89* ()

: 89

* ()

R

-

-

-

0.1.3

Graphsan

dTrees

Grap

hs

---

----

--

-----

--

-

--

--

-

-

---

-----

-

-

---

Introduction

15

1

2

3

4

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--

-

---

--

---

--

----

---

-

--

---

---

--

-

---

----

16

TheoryofAutomata,FormalLanguagesandComputation

-

---

-

-

-----

-

-----

-

-----

----

--

--

--

----

-

Introduction

17

F

E

DB

C

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Solution

R

~-

-

-

-

-

-

---

----

--

-

--

Example0.1.20:--

--

Solution

-

-

--

------

i

--

20

TheoryofAutomata,FormalLanguagesandComputation

Example0.1.21:-----

R

R

--

R

Solution

---

I

I

Example0.1.22:

m

--

Solution

-

r-

r-

r------

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Example0.1.23:

m

Solution

m

m

m

--

-

-

m

Example0.1.24:

--

Solution

-

-

--

X

X

KK

-

--

X

KK

r---

r---

s

s

---

r---

Example0.1.25:

m

-

22

TheoryofAutomata,FormalLanguagesandComputation

Solution

X

X

KK

X

X

KK

m

KKK

KKK

X

X

X

X

KKK

K

-

Example0.1.26:

Solution

--

Example0.1.27:

-

--

-

-

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B

B

Example0.1.28:

Solution

--

-

--

-

-

-

----

Example0.1.29:

-

Solution

--

---

-

Example0.1.30:

-

Solution:

---

24

TheoryofAutomata,FormalLanguagesandComputation

---

Example0.1.31:

Solution

-

Example0.1.32:

-

m

Solution

-

Example0.1.33:

m

}

X

m

m

Y

Solution

X

m

X

}

g

m

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Y

m

Y

m

g

Example0.1.34:

--

-

X

Y

Solution

X

------

Y

---

Example0.1.35:

m

m

m

m

m

m

m

m

----

-

-

-

X

-

Y

-

X

-

Y

-

[

-

-

X

-

X

-

X

X

-

Y

Y

-

Y

-

X

Y

-

Y

Solution

--

X

Y

26

TheoryofAutomata,FormalLanguagesandComputation

-

-

-

-

-

0.1.5

BooleanL

ogic

----

---

-

---H

HH

--G

-

--F

----

G

F

-

--

--[

[

-

-u-

---I

ntroduction

27

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----r

--

-

-----

r-

0.1.6

Fun

damentalProofTechniques

(a)

Princ

ipleo

fMa

thema

tica

lInduc

tion

-----

----

--

--m---

---

-

--

-

--

-

-

-

m--

----

-

-

-

(b)

Pigeon-ho

lePrinc

iple

---

---

-

--

Example0.1.36:----

---

Solution

---

--

-

----

-

7'-----

--

Example0.1.37:

m

--

--

28

TheoryofAutomata,FormalLanguagesandComputation

I

: 89

* ()I

m

: 89

* ()I

m

: 89

* ()I

m

m

m

Example0.1.38:

m

-

-

-

--

-

-

I

-

-

-

Example0.1.39:

m

-

-

--

--

Introduction

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--

i

-

m

Example0.1.40:--m

L

--

Solution

: 89

* ()

--

--

L

L

7 56

' %&

L

--

: 89

* ()

7 56

' %&

L

-

30

TheoryofAutomata,FormalLanguagesandComputation

7 56

' %&

L

---

B

Example0.1.41:--

?

/m

m

: 89

* ()

: 89

* ()

7 56 6

' %& &

--

: 89

* ()

: 89

* ()

7 56 6

' %& &

7 56

' %&

--

: 89

* ()

: 89

* ()

7 56 6

' %& &

: 89

* ()

: 89

* ()

7 56 6

' %& &

: 89

* ()

: 89

* ()

7 56 6

' %& &

Introduction

31

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Example0.1.42:-

m-

-

---

--

--

---

-

B--

Example0.1.43:

m

;

L

--

-

--

L

-

L

-

L

L

Q

-

32

TheoryofAutomata,FormalLanguagesandComputation

B-

-

-

--

m

;

Example0.1.44:

R

-

-

-

-

-----

-

-

-

B

--

B

-----

-Q

--

B

-

--

-

B

R

Example0.1.45:--

--

-

--

Solution

Introduction

33

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--

KK

---

LLL

LLL

--

- -

;

;

;

-

-

Example0.1.46:

L

-

-

---

------

m

L

7 56

' %&

L

34

TheoryofAutomata,FormalLanguagesandComputation

--

R

Example0.1.47:

;

--

;

B

;-

--

;

-

-

;

;

B

--

Example0.1.48:

R

B

Introduction

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---

m

B

B

B

B

B

--

m

Example0.1.49:

--

;

: 89* ()

7 56

' %&

-

--

;

: 89* ()

7 56

' %&

: 89 9

* () )

:

89* ()

7 56

' %&

: 89* ()

7 56

' %&

: 89* ()

7 56 6

' %& &

-

B

Example0.1.50:

--

---

-

--

36

TheoryofAutomata,FormalLanguagesandComputation

Solution

-

--

KK

---

KK

-

------

--

-

-

--

Example0.1.51:--

-

-

---

-

Solution

--

---

-

-

7'

-

--

0.1.7

IntroductiontoGrammar

----

--

--

--

R

-

--

-

r

--

r

-

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------

R

B

R

-

B

B

B

B

B

KK

-

-

-

KK

--

-

-

Example0.1.52:

-

r r

--

Solution

B B B

B

r

--

-

--

r

B

--

-r

B

B

38

TheoryofAutomata,FormalLanguagesandComputation

m

Example0.1.53:-

m

Solution

m

-

r

r

m

-

--

r

-

-

rr r

Example0.1.54:

-

r r r r

Solution

-

-

--

----

R

-

R

---

R

-

R

Introduction

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------

r

-

-

--

-

--

Example0.1.55:

-

rr

-

m

--

-

-

r r

Solution

-

r r rr

r--

B B B

B B B B

-

m

-

r

r

r

--

-

--

40

TheoryofAutomata,FormalLanguagesandComputation

r

B

--

-

r

B

B

m

-

--

m

Example0.1.56:

-

rr r r r

R

-

Solution

B B B B B B B B

R

Example0.1.57:-

--

--

---

---

Introduction

41

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Solution

---

--

rr r

R

----

rr r

R

---

}

m

-

rr r r r

R

Example0.1.58:--

-

rrr

rrr

Soution

-

rrr

m

-

r

42

TheoryofAutomata,FormalLanguagesandComputation

r r

-

--

GLOSSARY

-

-

--

--

---

-

----

--

--

-

-

---

-

--

--

-

--

--

-

-

-

---

-

-

-

--

-

--

---

--

---

-----

---

-

-

-

-In

troduction

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-

---R

--

--

---

-

-----

-----

--

-

R

-

---

-

-----

-

I

R

I

-

R

R

-

-

-

-

-

----

R

K

-

m-

K

R

-

------

--H

HH

--G

-

--F

-

G

F

-

56

TheoryofAutomata,FormalLanguagesandComputation

[

-

--

----

---

--

m-

-

-

---

-

--

-

--

-

-

--

--

---

----

R

--

--

Introduction

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Chap

ter

1

DFAandNFA

1.1

DETERMINISTICFINITEAUTOMATA(DFA)

1.1.1AutomataWhatisit?

---

-

--

--

---

--

-

-

--

----

-

1.1.2

Typ

esofAutomaton

-

-

---

-

-

----

--

---

----

Con

tro

lUn

it

Ou

tpu

t

Inpu

File

Storage

------

1.1.3

Definition

ofDeterministicFiniteAutomaton

--

---

---

l

r

-

R

-

S

--

--

--

-

-

-

---

--

Example1.1.1:-

-

R

--

-

-

-

--

---

-

-

-

-

DFAandNFA

59

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Solution

------

-

--

-

--

-

---

-

-

-

---

--

-

-

-

---

-

-

---

---

-----

-

-

-------

Example1.1.2:-

---

-

-

-

-

-

-

60

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

q2

q3

a

b

b

b

b

a

--

Solution

---

-

-

-

----

------

-

-

-

--

--

Example1.1.3:

-

-

Solution

-----

--

-

-

-

-

-

DFAandNFA

61

q1

q2

q3

0

1

1

00

,1

--

q1

q2

0

1

1

0

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-----

-

Example1.1.4:-

m

Solution

m

--

K

-----

----

Example1.1.5:----

-

--

-

---

Solution

---

--

-

-

62

TheoryofAutomata,FormalLanguagesandComputation

q0

q2

q1

b

a,b

a,b

Example1.1.6:--

R

-

-----

Solution

------

--

--

-

-

---

---

Example1.1.7:

--

DFAandNFA

63

q0

q1

q2

q3

q4

a

b

a

c

Inpu

q0

q1

q2

q3

b bbb

a

a

a

a

q0

q1

q2

q3

a

q4

b,c

c

a

b

a

c

b,c

b

a,b,c

a

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Solution

--

---

-

-

-

-

-

--

--

Example1.1.8:

-

Solution

--

-

-

-

-

-

-

---

-

-

64

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

q2

q3

q4 a

b

b

a,b

a

a

a

b

q0

q1

q2

q3

a

b

a,b

a

a

b

b

Example1.1.9:-----

-

Solution

--

-

----

----

-

-

-

-

--

Example1.1.10:--

----

-

Solution

-

DFAandNFA

65

q0

q1

q2

q3

b

a,b

a

b

aa,b

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-

-

-

--

---

----

--

-

------

-

-

--

Example1.1.11:---

-

Solution

-

---

66

TheoryofAutomata,FormalLanguagesandComputation

q2

q1

q0

b

a

a b

b

a

q1

q0

q20 1

0

1

Example1.1.12:---

-

--

Solution

Example1.1.13:

R

--

Solution

-

DFAandNFA

67

q1

q0

q20

0

0

1

1

1

1

q0

q1

q3

q2

1 11 1

0

0

0

0

q0

q3

q1

0

0

0,1

q2

1

0 1

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Example1.1.14:-

-

-

Solution

R--

R-

Example1.1.15:

m

R

m

R

Solution

m

--

-

-

-

68

TheoryofAutomata,FormalLanguagesandComputation

q2

q0

q1

b

a

b

a

b

q1

b

q0

a

b

(b)

q1

a

b (a

)

m

R

-

g-

-

--

-

--

Example1.1.16:

--

--

----

-

--

Solution

--

-----

--

DFAandNFA

69

q3

q0

q1 a

b

a

b

q2

b

ab

q3

q0

a

b

a

b

q1

q2

b

a

b a

q2

q0

a

b

a

b

q1

ab

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----

-

K

--

1.2

NON

-DETERMINISTICFINITEAUTOMATA(NFA)

Definition

--

--

---

---

l

X

r

R

---

S

----

-

-

-

-

------

-

--

Example1.2.1:----

-

Solution

--

70

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

0-9

1-9

q0

q1

0-9

q2

+,

1-9

q0

q1

q2

q3

a

a

a

b

b

b

b

a

--

----

Example1.2.2:

R

-

X

Solution

Example1.2.3:--

---

Solution

---

-

-

--

Example1.2.4:-

R

m

-

-

DFAandNFA

71

q0

q1

abac

a,b,c

a,b,c

q0

q1

q2

a b

q0

q1

q2

1 1

0

1

q1

q2

q3

q4

1

0,1

0,1

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Solution

-

---

--

----

--

Example1.2.5:-

--

Z

Z

Solution

--

---

72

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

q2

q3

a,b

a

a,b

a,b

q1

q2

q0

q3

10

0,1

0,1

Example1.2.6:

-

-

X

Solution

-

X

-

X

X

-

--------

-

Example1.2.7:--

m

X

m

Solution

m

X

m

----

-

m

-----

DFAandNFA

73

q0

q1

q2

q3

q4

q5

a

a a

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Example1.3.1:-

-

-

Z

Solution

~

~

~

~

~

-

~

Z

~

~

-

----

~

~

Z

Z

Z

Z

-

~

-

~

X

XZ

~

-

X

X

76

TheoryofAutomata,FormalLanguagesandComputation

----

----

Example1.3.2:--

--

Z

Z

Z

Z

Solution

-------

Z

Z

Z

Z

Z

DFAandNFA

77

q0

q1

a

q2

b

a

b

[

]q0

[

]q1

[

]

q0,q

1

a

b

b

Za

b

a

ab

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---

-

--

--

Example1.3.3:--

Solution

-------

---

Z

Z

Z

Z

Z

Z

-----

78

TheoryofAutomata,FormalLanguagesandComputation

Z

~--

r

--

--

Example1.3.4:-

Solution

----

Z

Z

Z

Z

--

--

--

DFAandNFA

79

[

]

q0,q

3

[

]

q0,q1

[

]

q0,q1

2,q

[

]

q0,q

3

4,q

0

01

1

01

0

1

[

]q0

q0

q1

q2

q3

a

a

b

b

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-

--

-

---

-

-

---

-

-

---

--

1.4.4

Lan

guagesdefinedbyRegularExpressions

-----

-

-

--

--

R

Z

X

1.4.5

Reg

ularExpressionstoNFA

---

------

---

--

--

-

82

TheoryofAutomata,FormalLanguagesandComputation

x

NFAforx

NFAfor

--

Z-

Z--

--

--

-

--

---

----

-

---

----

1.4.6

NFAstoR

egularExpression

-------

--

---

---

DFAandNFA

83

NFAforZ

NFAforrr12

Mr(

)1

Mr(

)2

NFAfor

+r

r

1

2

Mr(

)1

Mr(

)2

Mr(

)1

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----

------

Solution

--

---

-

--

--

--

-

--

--

-

-

----

---

---

-

-

-

--

--

--

--

86

TheoryofAutomata,FormalLanguagesandComputation

---

-

-

-

--

-

-

Example1.4.6:---

--

Solution

-

---

-

---

----

----

-

---

-----

-

Example1.4.7:--

--

Solution

Example1.4.8:----

DFAandNFA

87

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7/23/2019 TC Reference

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-

-

--

--

-

-

-

-

--

--

-------

-

l

1.6.3

MooreMachine

----

-

--

-

---

Example1.6.1:---

--

-

--

--

Solution

----

----

90

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

q2

q3

1,1

0,1

0,0 0

,0

1,0

0,1

1,0

Example1.6.2:--

-

Solution

-

--

--

r

r

1.7

PROPERTIESOFREGULARSETS(LANGUAGES)

-

--

1.7.1

Closure

---

-

------

-----

-

-

--

-

-

DFAandNFA

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---

-

-

--

--

-

---

---

---

-

EXERCISES

-----

--

---

---

----

g

---

-

--

m

----

-

-

Z

100

TheoryofAutomata,FormalLanguagesandComputation

--

--

Z

Z

Z

-

-

---

DFAandNFA

101

a

a

a

b

b

a

a

a

b

b

b

a

b

b

b

a

q3

q1

q2

q4

q0

0 1

0,1

1

01

0

0

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-

--

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

--

-

-

--

--

----

------

--

---

----

-

-

102

TheoryofAutomata,FormalLanguagesandComputation

s2

s1 s3

s4

s0

Start

1,1

1,0

0,00

,0

0,0

1,0

1,0

s1

s0

s2

s3

Start

1,1

1,0

0,00

,01

,1

1,1

0,0

0,0

--

--

-

----

-

--

----

------

-

--

-

-

-

-

-

-

KK

-

-

--

-

KK

-

-

KK

-

---

-

--

-

-

-

-

- --

- -

--

-

DFAandNFA

103

s0

s1

s2

Start

1

0

0

1

1

010

s0

s1 s2

Start

1,1

0,1

1,0

0,0

0,0

1,0

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--

--

-

KK

--

---

-

-

-------

----

-----

-

---

---

----

--

- --

--

-

-

-

--

X

--

X

---

m

m

---

------

--

106

TheoryofAutomata,FormalLanguagesandComputation

3

1

2b

a

a,b

a

s0

s1 s2

Start

1

0

0

0

1

1

--

---

---

----

---

--

R

-

R

--

--

-

--

------

---

---

---

--

DFAandNFA

107

s0

s1

s2

s3

a,b

a

a

b

s0

s1

s2

s3

a,b

a

a

b

a,b

q0

q1

q2

0

0

1

0

1

q0

q1

a b

a

b

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R

R

R

R

R

----

--

--

-

SHORTQUESTIONSANDANSWERS

-

---

---

-

-

-

-

108

TheoryofAutomata,FormalLanguagesandComputation

q0

q1

q2

10

1

0

1

q0

q1

q2

00

0

1

1

q0

q1

q2

0

0

1 1

1

q0

q1

q2

00

1

0

1

-

-

---

-

--

---

-

-

---

-

--

-

--

-

--

---

---

l

r

-

R

-

S

--

-

--

--

-

-

--

--

---

---

l

X

r

R

--

S-

---

---

--

-

-

--

-------

-

--

DFAandNFA

109

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--

f

-

-

------

---

---

---

-

-

-

114

TheoryofAutomata,FormalLanguagesandComputation

Chap

ter

2

Con

text-FreeGrammars

2.1

INTRODUCTION

2.1.1

Definition

ofCFG

-

---

---

-

---

-

--

R--

----

r

-

--

B

2.1.2

Exampleo

fCFG

-

--

r r r

R

------

-----

---

------

2.1.3

Right-Line

arGrammar

-

X

r

X

--

r

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r

----

-----

-

2.1.4

Right-LinearGrammarsandNFAs

-

----

-

-

--

--

------

2.1.5

Left-LinearGrammar

--

r

r

-----

-

----

-

116

TheoryofAutomata,FormalLanguagesandComputation

A

B

x

A

x

y

B

z

A

B

A

x

A

xB

rA

xyzB

rA

B

rA

x

r

1 11 1

0

0

0

0

S

A

B

C

S

r

S

B

r0

S

A

r0

A

C

r0

A

S

r1

B

S

r0

B

C

r1

2.1.6

Conversio

nofLeft-linearGrammarinto

Right-Line

arGrammar

-

-

r

r

r

r

-

-

-

-

-

-

-

-

--

--

--

-

-

-

---

--

Example2.1.1:--

Solution

-

r

r

r

-

B

B

B

B

-

R

-

-

-

rr r r

-C

ontext-freeGrammars

117

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-

m

Example2.1.2:--

m

m

Solution

r rr r

r rr r

2.2

DERIVATIONTREES

--

--

-

--

---

2.2.1

DefinitionofaDerivationTree

-

--

-

-

X

--

-

R

-

KK

-

r

K

K

---

-

118

TheoryofAutomata,FormalLanguagesandComputation

2.2.2

SententialForm

-

r

r

r

r

---

B

B

B

B

--

---

2.2.3

PartialDerivationTree

-

X

X--

2.2.4

RightMos

t/LeftMost/MixedDerivation

-

rr

4 3 2

$ # "

B B B B B B B

-

-

--

Context-freeGrammars

119

a

b

S

A

B

B

a

a

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B B B B B B B

---

B B B B B B B

-

-

---

Example2.2.1:

-

-

-

rr r r

---

B

r

B

B

B

120

TheoryofAutomata,FormalLanguagesandComputation

Solution

Example2.2.2:--

r rr

Solution

-

-

-

B

B

B

BB B B

--

Context-freeGrammars

121

a

S A

S

b

S

a

S

a

a

A

b

S a

a

S A

B

(a)S

a

AB

r

(b)A

Bba

r

a

S A

B

B

b

a

(c)Br

a

S A

B

B

b

a

c

(d)B

bB

r

a

S A

B

B

b

a

c

b

B

(e)B

c

r

a

S A

B

B

b

a

b

cc

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Example2.2.3:

rr

4 3 2

$ # "

Solution

B B B B B B B

R

-

R R

---

m

Example2.2.4:

rr r

4 31 21

$ #1 "1

Solution

B

R

B

B

R

122

TheoryofAutomata,FormalLanguagesandComputation

a

SS

a

S

bb

a

b

S C a

S S c

a

B

B

R

B B B

R

-

-

R

-

KK

KK

Example2.2.5:

r

r rr

Solution

Context-freeGrammars

123

a

S S S

a

b

b

c

E

E

*

E

E

E+

id

id

id

B

id*id+id

E

E

E

E

(

)*

id

id

id

E

E

+

B

id+id)*id

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Example2.2.6:

rr

4 3 2

$ # "

Solution

B

B

B

B

B

-

-

m

Example2.2.7:

-

r r r

R

r r

R

r r r

R

rr r

R

r r r

-

r r r r

Solution

-

-

-

BR

B

B

B

B

B

-

-

m

124

TheoryofAutomata,FormalLanguagesandComputation

S

S S

a

a

S

SS

aS

aa

BB B

S a

S

a

B

BRB B

R

r r

rR

-

r r

r

r r

r r

Example2.2.8:

rrrr

4 31 21

$ #1 "1

Solution

B

B

B

B

B

B

B

B

B

B

-

--

-

m

m

Context-freeGrammars

125

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Example2.2.9:

rr r

4 31 21

$ #1 "1

Solution

B

B

B

B

B

B

K

K

---

m

Example2.2.10:-

-

R

R

R

Solution

r rR

r

r r r r

R

126

TheoryofAutomata,FormalLanguagesandComputation

a

S

bA

a

S

A

b

S

a

A

b

r r r r rR

Example2.2.11:

-

r rr r r r r r

--

-

Solution

-

B

B

B

B

B

B

B

B

B

B

-

B

B

B

B

B

B

B

B

B

2.3

PARSINGA

NDAMBIGUITY

2.3.1

Parsing

--

-

--

-

--

---

- ----

-

---

Context-freeGrammars

127

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2.3.2

Exh

austiveSearchParsing

-----

-

---

--

-

--

-

----

-

-------

--

-------

--

---

-

R

---

-----

-----

---

-

----

2.3.3

Top

down/BottomupParsing

--

-

-

--

---

-

-

rr

----

r---

--

--

128

TheoryofAutomata,FormalLanguagesandComputation

a

S S

S

a

S

S

b

a

S

Sb

b

b

-

-

----

-

r---

--

--

2.3.4

Ambiguity

r

-----

-

--

---

-

----

B

B

B

B

B

B

B

B

-----

B

B

B

B

B

B

B

B

-

-

--

-

---

C

ontext-freeGrammars

129

S

S

S b

Sa

a

a

b

b

b

S

S

S

-

a

S

b

b

S

a

S

S

S

S

a

b

S

a

b

S

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2.3.5

AmbiguousGrammars/AmbiguousLanguages

----

-

-

-

-

R

---

--

---

--

-----

--

----

--

-

--

---

Example2.3.1:

rr r

--

Solution

--

-----

rrrrr r

-

-

B B

B

$ #1 "1

Example2.3.2:

r

r

-

-

Solution

-

--

R

-

-

130

TheoryofAutomata,FormalLanguagesandComputation

--

---

--

Example2.3.3:

rr

--

Solution

B

r

B

r

Q Q

B

r

B

r

B

QQ

--

--

2.4

SIMPLIFICA

TIONOFCFG

2.4.1

Simplifica

tionofCFG-Introduction

---

--

X

---

-----

-

-

-

-

r

Context-freeGrammars

131

S

S

S

S

S

b

b

S

S

S

S

S

b

a

S

S

b

a

a

a

S

S

b

a

a

a

a

b

(a

)

(b)

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--

$

-

Proce

dure

tofin

dCFGw

ithou

t-Pro

duc

tions

-

r

---

-

r

KK

KK

$---

r

m

KK

R

X

-

$--

----

-

r-$

-----

(b)Un

itPr

oduc

tions

Remova

l

r

R

-

-----

----

-

--

$

$$

$

---

--

Procedure

toremovetheunitproductions:

-

-

B

--

-

134

TheoryofAutomata,FormalLanguagesandComputation

-

r

B

-

-

$-$

-

--

B

$

r

KK

r

KK

---$

(c)Le

ftRecurs

io

nRemova

l

---

r

R

X

-----

--

-

Example2.4.1:

-

r r

$-$-

Solution

-

r

r

-

$-

r r

$-

--

--

B

B

B

Context-freeGrammars

135

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B

B

$

-

$

Example2.4.2:-

rr r r r

$

---

--

Solution

---

B

B

B

$

~

~--

rr r

---

-

r

-

r

--

Example2.4.3:

rr r r

------

136

TheoryofAutomata,FormalLanguagesandComputation

Solution

---

-

r

r

-

---

---

-

B

B

-

---

--

-

r

-

-----

rr r

--

-

--

-

--

--

----

$

$

$

$

$

$

rr

Example2.4.4:

rr

---

Solution

rr

-

Context-freeGrammars

137

S

B

A

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m

r

-

---

-

rr

--

--

m

Example2.4.5:

-

r

r

r

r

r

Solution

-

-

-

rr r rr

---~

-

Example2.4.6:-

-

rr r rrr

138

TheoryofAutomata,FormalLanguagesandComputation

Solution

r

r

r

$

-

-

B

r

-

r

$

B

B

r

r

$

--

$

$

$

$

rr r r rr

Example2.4.7:

rr r rr

-

Solution

r r

--

r r

r

r

Context-freeGrammars

139

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$

rr r

Example2.4.8:

rr r

Solution

---

--

B B B B

---

rr r

-

rr r

rr r

-$

140

TheoryofAutomata,FormalLanguagesandComputation

S

B

A

Example2.4.9:

rr

-----

Solution

rr

-----

-

-

r

-

rr

r

-$

Example2.4.10:

rr r

-$

Solution

--

-

--

--

r

-

--

r

-

r

r

-

r

r

-

r

Context-freeGrammars

141

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--$

-

r rr

2.5

NOR

MALFORMS

-

-

-

-

2.5.1

Cho

mskyNormalForm(CNF)

-

--

r

r

R

R

Proce

dure

tofin

dEqu

iva

len

tGram

mar

inCNF

-

-

--

--

-

-

---

---

----

--

-----

-

r

m

KK

--

r

-

~---

142

TheoryofAutomata,FormalLanguagesandComputation

r

~ --

~-

-

~

~

-

~---

-

-

r

KK

--

m

r

-

r

KK

--

r r r

LL

LL

LL

LL

~

-

---

Example2.5.1:-

-

rr r

Solution

--

--

r r

r

rrr

r

r

r

r

Context-freeGrammars

143

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~

~--

r r r r r rr

~

r

---

-

r r r

--

-

r r r r r r r r

r

Example2.5.2:-

-

rr r

144

TheoryofAutomata,FormalLanguagesandComputation

Solution

---

-

--

r

r

r

r

r

r

r

r

r

---

~

r r r r r r

~

rr

---

-

r r r r

-

--

-

r r r r

Context-freeGrammars

145

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r r r

r

Example2.5.3:

r

r

-

Solution

--

-

r

r

r

r

r

r

r

r

r

r

r

~

-

~

r r r r r r

r

~

r r

146

TheoryofAutomata,FormalLanguagesandComputation

r

---

--

r

r

r

r

r

r r

r

r r r

--

r r r r r r r r

r r r

r

Example2.5.4:

-

r

Solution

--

-

r

r

Context-freeGrammars

147

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r

r

r

~

-~

r r r r

~

r

-

r r

--

-

r r r r

r

2.5.2

Gre

ibachNormalForm

----

--

--

----

--------

--

--

-

r

R

R

148

TheoryofAutomata,FormalLanguagesandComputation

--

-

-

--

-

GLOSSARY

-

r

r

-

r

r

-

-

-

--

---

-

-----

----

---

---

r

R

-

-

--

-

r

r

R

R

REVIEW

QUESTIONS

---

-

-

-

--

-

Context-freeGrammars

149

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-

r

r

r

--

-

-

-----

-

-

r

-

-

r

--

-

r r r r

r r r

----

rr r rr

-

rr

152

TheoryofAutomata,FormalLanguagesandComputation

q0

q4

q3

q5

a

aba

rr r

-

rr r

-

rr

rr

r

-

-

rr r

-

R--

R

-

r r

R

---

-

m

m

m

m

S

HORT-QUESTIONSANDANSWERS

-

---

---

-

---

-

--

R

--

Context-freeGrammars

153

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--

--

---

----

-

---

r

R

R

158

TheoryofAutomata,FormalLanguagesandComputation

Chap

ter

3

PushdownAutomata

3.1

DEFINITION

S

---

m

-

---

---

-

----

--

---

-

-

m

-

--

-

m

---

-

----

--

---

---

-

---

---

m--

---

3.1.1

Nondeterm

inisticPDA(Definition)

-

---

---

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3.1.7

Acc

eptingStringswithNPDA(FormalVersion)

---

-

---

----

-

-

R

R

R

Example3.1.1:--

m-

Solution

m

-

-

-----

--

--

-

----

-

---

-

R

Example3.1.2:-

m

-

164

TheoryofAutomata,FormalLanguagesandComputation

Solution

m

-

-

Example3.1.3:

m

-

Solution

m--

-

----

Example3.1.4:-

m

-

Solution

THEOREM:

-

-

~

~

~

~

~

-

-

~

~

~

~

-

~

~

~

PushdownAutomata

165

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--

--

-

-

Example3.3.1:

}

}

-

Solution

-

-

}

}

--

-------

-

-

-

Q

-

-

Example3.3.2:-

S

--

--

-

--

R

-

-

--

-

gZ

-

-

-

-

--

--

-

-

~

~-

---

R

~

R

Q

~

-

Example3.3.3:

m

g

-

174

TheoryofAutomata,FormalLanguagesandComputation

---

-

-

-

--

m

Y

--

Y

-

-

--

--

m

g

-

Example3.3.4:

R

-

---

Y

m

--

-

-

R

-

Example3.3.5:

m

-

Solution

---

-

-

}

}

-

-

}

PushdownAutomata

175

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Solution

---

---

--

R

--

R

m

-

R

--

--

-

Example3.4.3:---

---

Solution

-

-----

---

-

-

Table:TransitionTableforDPDA.

-

-

-

-

-

-

-------

------

---

--

178

TheoryofAutomata,FormalLanguagesandComputation

-

--

---

-

R

--

-

-

--

--

-

Q

GLOSSARY

--

-

-

l

X

l

----

l

PushdownAutomata

179

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---

-

Example4.1.1:---

-

Solution

-----

-

--------

---

-

-

--

-

---

----

--

-

---

-------

-

r

Example4.1.2:

--

--

Solution

--

--

-

-

---

-

---

-

--

-

190

TheoryofAutomata,FormalLanguagesandComputation

---

oo

---

-

--

-

Example4.1.3:--

m

Solution

----

------

-

-

----

-

---

q0:

Move

al

l

the

way

to

the

le

ft

q1:

Erase

a

n

a

.

TuringMachines

191

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Chap

ter

5

ChomskyHierarchy

5.1

CON

TEXTSENSITIVEGRAMMARSANDLANGUAGES

---

--

----

-

r

R

R

X

-----

r

----

-

r

R

X

}

-----

--

-

---

-

---

---

-

-----

--

-

X

Example5.1.1:

m

-

--

Solution

-

----

-

r r r r r

--

--

B

B

B

B

B

B

B

B

B

B

B

---

-

-

-

---

5.2

LINEARBO

UNDEDAUTOMATA

--

-----

-

-----

THEOREM(I):--

--

-

--

THEOREM(II):

-

--

5.3

RELATIONSHIPOFOTHERGRAMMARS

THEOREM(I):

---

-

r

-

--

r

-

THEOREM(II):-----

ChomskyHierarchy

211

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--

-

----

---

---

----

--

-----

--

--

Example6.3.1:-

--

Solution

--

KK

Example6.3.2:-

-

-

Solution

-

-

Example6.3.3:-

-

----

-

220

TheoryofAutomata,FormalLanguagesandComputation

Solution

-

KK

Example6.3.4:-

-

-

-

--

Solution

-

-

Example6.3.5:-

-

---

-

Solution

---

-

m

--

Computability

221

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Example:

-

Solution

--

Example6.4.2:

-

-

Solution

-----

-

--

--

KK

KK

KK

KK

KK

--

-

--

-

-

~

-

-

-

-

-

---

-

224

TheoryofAutomata,FormalLanguagesandComputation

~

--

-

-

---

-

---

--

-

--

Example6.4.3:

-

-

Solution

--

-

I~

-

--

--

Example6.4.4:-

-

4 3 2

--

Solution

-

-

4 3 2

-

-

-

-

---

Example6.4.5:

--

Computability

225

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Solution

i

i ---

-

$ # "

--

Example6.4.6:

-

-

Solution

---

---

-

-

-

i

-----

i

---

-

----

-

-

-

i

---

---

Example6.4.7:-

g

4 3 2

-

-

Solution

-

226

TheoryofAutomata,FormalLanguagesandComputation

g

4 3 2

-

--

-

g

4 3 2

--

-

--

Example6.4.8:--

g

4 3 2

--

Solution

--

g

4 3 2

----

--

-

--

---

-

Example6.4.9:

--

Computability

227

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Solution

I

---

--

Example6.4.10:

-

m

4 3 2

-

-

Solution

--

-

m

4 3 2

-

i

i

----

--

-

-

Example6.4.11:-

--

-

-

Solution

--

g

4 3 2

-

-

228

TheoryofAutomata,FormalLanguagesandComputation

-

--

-

g

4 3 2

--

K

LL

-

--

------

-

--

--

6.5

ACKERMANNSFUNCTION

----

----

---

---

-

----

Example6.5.1:

-

-

-

Solution

-

Computability

229

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-

-

-

-

-

-

-

-

-

-

GLOSSARY

----

-----

-----

--

----

--

-------

---

----

230

TheoryofAutomata,FormalLanguagesandComputation

---

-

----

-

----

-

---

--------

-

-

REVIEW

QUESTIONS

---

------

---

--

---

--

--

--

-

-

-

EXERCISES

-

-

-

-

Computability

231

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----

F

--

--

F

---

--

F

-

-

Illus

tra

tion

-

F

-

---F

--

-

F

F

G

B-

--

B

248

TheoryofAutomata,FormalLanguagesandComputation

B-

B

-

-

----

-

Illus

tra

tion

B

-

---

-

----

-

--

B

--

--

--

----

-

--

B

--

E

-

-

--

E

-

E

-

-----

-

E

--

B

G

B

--

E-

-

E

-

---

-

Illus

tra

tion

-

-

-

---

E

-

PropositionsandPredicates

249

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---

E

-

--

-

----

-----

-----

-

Example8.1.1:-----

-

-

--

-

Solution

GH

G

H

GH

H

G

Example8.1.2:-----

-----

-

--

-

Solution

---

G

250

TheoryofAutomata,FormalLanguagesandComputation

--

---

--

---

G

B

-

-

B

---

-

-

F

----

Example8.1.3:

----

-

-

-

--

E

H

E

H

E

G

B

Solution

E

-----

H

E

H

-----

E

--

GH

-----

B-----

PropositionsandPredicates

251

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Example8.1.4:

-

-

---

-

---

--

--

-

Solution

E

E

E

GH

BH

Example8.1.5:

-

--

--

G

H

GH

H

G

F

H

HH

FH

Solution

G

---

H

GH

---

H

G

----

FH

---

HH

FH

---

-

Example8.1.6:

--

--

-

-

252

TheoryofAutomata,FormalLanguagesandComputation

Solution

Example8.1.7:--

-

--

-

F

G

F

G

GH

GHG

H

B

-

B

E

B

Solution

F

F

G

G

-

F

G

F

F

G

-

GH

H

GH

GH

GH

H

GH

H

GH

GH

-

B

B

-

PropositionsandPredicates

253

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-

B

E

B

-

-B

-

E

E

-

B

B

B

Example8.1.8:-

H

F

GH

F

Solution

H

H

H

F

H

F

H

F

GH

F

Example8.1.9:HH

GH

H

H

H

GH

HH

GH

Example8.1.9:---

E

-

B

B

--

B

B

E

-

Solution

E

B

B

-

254

TheoryofAutomata,FormalLanguagesandComputation

-

--

E

-

B

B

-

-

--

B

B

-

E

-

8.1.2

Tautology

,ContradictionandContingency

----

------

F

H

---

---

G

H

--

-

Example8.1.10:

B

---HF

-

-

B

E

H

F

-

Solution

H

B

H

F

B

E

H

F

-

B

---H

F

---

B

E

H

F

--

-

Example8.1.11:----

-

--

FH

GH

BH

H

HG

E

H

FH

PropositionsandPredicates

255

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Solution

FH

H

FH

-

-

GH

H

GH

--

-

BH

H

H

HH

BHH

-

-

H

G

E

H

FH

G

H

G

HH

H

FH

H

G

EH

FH

-

-

Example8.1.12:----

---

256

TheoryofAutomata,FormalLanguagesandComputation

HF

E

H

GH

B

E

H

BH

B

E

H

BH

G

F

B

G

F

G

Solution

HF

E

H

GH

B

E

H

BH

B

G

B

B

B

B

G

B

---

-

G

F

B

G

F

G

-

Example8.1.13:

---

-

-

-

----

--

--

--

---

--

E

GH

G

B

G

B

H

G

PropositionsandPredicates

257

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Solution

H

G

B

B

H

GH

E

GH

-

-

G

B

G

B

H

F

-

Example8.1.14:-

-

GH

E

H

G

B

Solution

-

---

-

Example8.1.15:

KK

--

Z

KK

---

}

}

--

Z

Solution

KK

----

-

8.1.3

Log

icalIdentities

----

--

--

--

--

-

-----

-

-

-

--

-

258

TheoryofAutomata,FormalLanguagesandComputation

Table.LogicalIdentities

E

F

F

E

G

G

F

E

F

F

G

E

G

G

F

F

E

F

F

--F

G

G

E

G

G

--G

H

F

E

H

GH

H

G

E

H

FH

$ # "

--

G

G

E

G

F

G

-GF

F

G

E

F

G

F

-FG

F

E

G

E

F

E

G

E

FH

E

GH

E

EHH

B

E

H

F

E

E

B

G

B

G

B

E

B

B

B

G

BH

EH

-

B

E

H

BH

-

---

----

Table.LogicalImplications

B

F

G

B

G

B

B

--

B

GH

BH

--

H

G

F

B

--

B

G

B

B

B

-

B

B

B

B

B

B

G

B

B

G

B

G

E

G

E

B

E

PropositionsandPredicates

259

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Example8.1.15:--

--

-

-

-

-

Solution

--

-

-

--

--

--

-

-

--

--

Example8.1.16:----

-

----GH-

--

--

F

FH

F

H

G

B

B

B

Solution

F

FH

E

HH

GH

FH

E

H

H

GH

G

E

HH

GH

G

F

H

G

B

E

H

H

GH

H

G

B

E

H

H

GHHH

G

F

E

H

H

G

H

G

GH

B

B

E

B

H

F

E

H

FH

F

E

H

F

FH

E

FH

E

Example8.1.17:----

-

----FH-

--

--

260

TheoryofAutomata,FormalLanguagesandComputation

G

GH

B

FH

GH

G

Solution

G

GH

E

HH

G

F

E

HH

FH

F

B

FH

GH

G

E

H

F

FH

GH

G

E

H

F

FH

GH

G

E

H

GH

G

F

GH

G

FH

GH

G

E

H

G

F

GH

F

H

G

GH

F

GH

GH

E

H

FH

Example8.1.18:----

-

-

G

B

E

HH

F

BH

E

B

H

GH

B

E

Solution

G

B

E

H

G

F

E

H

FH

F

E

FH

FH

E

FH

E

HH

F

BH

E

H

F

FH

E

H

FH

F

E

H

F

E

H

E

B

H

GH

BE

H

FH

G

F

E

H

G

E

Example8.1.19:-B

--B----

PropositionsandPredicates

261

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Solution

B

-

--

B

--

B

-

-

--

Example8.1.20:--

-

-H

G

EH

E

F

E

G

Solution

G

H

G

E

H

G

262

TheoryofAutomata,FormalLanguagesandComputation

E

H

E

H

G

E

H

G

E

H

E

F

E

H

E

H

E

H

H

E

HH

GH

E

F

E

G

HG

E

H

G

H

G

E

H

H

G

GH

G

E

H

F

G

F

E

H

F

E

G

Example8.1.21:---

H

B

B

H

B

B

HG

B

BH

Solution

H

B

B

H

B

H

B

B

-

PropositionsandPredicates

263

7/23/2019 TC Reference

140/180

H

B

B

B

H

B

H

B

B

-

H

G

B

BH

H

B

H

H

G

B

H

G

B

BH

-

Example8.1.22:

F

GHH

GH

FH

FH

GH

FH

GH

-

Solution

F

GH

GH

FH

FH

GH

FH

GH

E

F

GH

H

GH

G

FH

F

FH

F

E

F

G

F

G

FH

F

FH

F

E

F

G

F

FH

F

G

F

E

F

G

F

FH

F

G

E

F

G

FH

F

G

E

-

Example8.1.23:

r

G

r

F

r

264

TheoryofAutomata,FormalLanguagesandComputation

Solution

r

G

r

E

H

F

GH

F

E

H

GH

F

E

H

F

F

E

F

r

Example8.1.24:

H

G

r

H

FH

F

B

H

F

H

F

r

H

FH

F

B

G

FH

FH

F

B

G

FH

F

B

G

FH

F

B

G

FH

F

B

FH

G

FH

F

B

G

FH

F

B

FH

F

B

FH

BH

F

Example8.1.25:

F

-

F

Gr

G

r

Solution

--

F

G

r

G

r

-

F

r

r

-

F

-

-

-

-

-

r

-

-

--

-

-

r

-

-

-

-

-

F

-

F

G

r

G

r

-

F

8.2

LOGICALINFERENCE

-

-

PropositionsandPredicates

265

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---

r

---

Ru

lesofIn

ference

-

G

B

G

B

$ # "

B

F

B

F

$ # "

H

B

r

B

r

H

r

B

H

r

H

B

G

H

F

B

---

r

B

--

H

r

BH

--

r

r

B

r

-

F

r

r

B

-

HH

E

G

E

G

F

E

F

$ # "

-

G

G

E

G

G

F

F

E

F

F

$ # "

---

F

G

E

F

G

F

G

G

E

G

F

G

$ # "

-

H

G

E

H

FH

H

F

E

H

GH

$ # "

--

F

E

G

E

F

GH

E

G

FH

E

F

FH

E

266

TheoryofAutomata,FormalLanguagesandComputation

G

GH

E

r

EH

F

H

r

E

GH

r

EH

rH

r

r

E

G

r

H

E

E

EH

E

E

r

G

r

E

E

G

FH

GH

Example8.2.1:

F

--

-

F

F

r

H

H

r

GH

GH

r

F

Solution

F

rH

H

r

GH

F

r

GH

-

GH

r

F

F

r

F

-

F

F

--

Example8.2.2:

F

-

F

G

r

G

r

Solution

F

H

r

Q

r

E

H

F

r

H

r

H

r

-

r

EH

rH

r

H

r

F

Example8.2.3:

G

F

--

--

F

r

r

H

PropositionsandPredicates

267

7/23/2019 TC Reference

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Solution

r

H

H

--

F

r

--

G

F

Q

B

G

Example8.2.4:-

-

--

r

H

F

H

r

H

Solution:

F

H

-

r

H

H

---

H

r

--

Example8.2.5:H

r

BH

Solution

r

H

rH

-

r

E

H

r

H

H

H

--

r

B

Example8.2.5:

F

--

--

F

F

rH

H

r

GH

GH

r

F

268

TheoryofAutomata,FormalLanguagesandComputation

Solution

F

F

rH

H

--

H

r

GH

GH

--

GH

r

F

F

--

Example8.2.6:-

--

-

-

-

-

-

-

Solution

-

-

-

-

rH

r

H

r

r

r

-

H

rH

-

r

-

r

r

-

r

Example8.2.7:-

-----

--

----

PropositionsandPredicates

269

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Solution

-

----

--

H

ru

H

r

$ #1 "1

--

H

r

G

u

B

H

r

u

H

r

G

u

Hr

G

u

B

-

-

Example8.2.8:-----

----

-

----

-------

270

TheoryofAutomata,FormalLanguagesandComputation

Solution

F H

FH

--

F

B

-

rH H

rH

H

H

r

BH

-

-

-

r

---

--

--

--

r

F

--

r

F

--

Example8.2.9:---

-

----

-

PropositionsandPredicates

271

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144/180

-

-

--

-

--

---

Solution

-

--

-

-

-

H

rrGH

$ #1 "1

-

r

r

r

GH

H

G

H

H

--

H

--

H

GH

--

-

-

--

rr

r

-

---

Example8.2.10:-

-

-----

-

--

-

-

--

-

272

TheoryofAutomata,FormalLanguagesandComputation

-------

-

---

-----

----

Solution

-

H

Fr

H

H

F

r

H

GH

-

H

H

--

----

--

-

H

FH

G

F

G

F

G

H

FH

G

F

G

F

-

H

FH

GH

F

-

F

H

F

FH

F

G

FH

F

GH

F

PropositionsandPredicates

273

7/23/2019 TC Reference

145/180

Example8.2.11:

-

--

G B

G

F B

F

B B

B

B

F

BH

BH

GH

Solution

G B

G

G

r

--

-

F B

F

F

B

H

r

H

r

F

-

B B

B

274

TheoryofAutomata,FormalLanguagesandComputation

r

r

r

G

r

r

r

G

rB

r

B

B-

r

F

rH

rH

GH

r

F

F

--

r

H

H

--

r

H

HH

rH

rH

H

--

GH

B-

PropositionsandPredicates

275

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7/23/2019 TC Reference

147/180

--

---

--

I

Solution:

---

F

F

_

F

F

_

}

F

H

H_

F

F

F

_

F

Example8.3.2:

__

__

-----

__

__

__

_

F_

F_

K

K

F

F

F

F

F

F

KK

KK

F

F

F

F

F

F

F

KK

KK

KK

F

F

F

F

F

F

FKKKK

F

_

F_

F_

F

__

KK

KK

Example8.3.3:-

-

----I-

F_

I

F_

I

278

TheoryofAutomata,FormalLanguagesandComputation

_F

I

_F

I

Solution

F

_

I

-

F

_

I

-

_F

I

I

I

Q

-

_F

I

I

I

-

Example8.3.4:--

---

--

F_

B_

F

_

F

BF

_

Solution

F_

B_

F

_

_

-

PropositionsandPredicates

279

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_

F

BF

_

K

-

Example8.3.5:---

-----

------

F

F

F_

_

F

Solution

----

-----

---

---

Example8.3.6:-------

-----

-

-

F

F

F

F_

_

F

__

Solution

F

G

F

F

G

G

G

F_

F

G

F

_

F

G

F

G

__

F

F

F

280

TheoryofAutomata,FormalLanguagesandComputation

Example8.3.7:---

----

-F

_

---

-F

_

-

--

--

-F

_

_

Solution

g

-

g

----

F_

_

Example8.3.8:---

----

---

---

--

----

--

Solution

FF

_

FF

_

_F

8.4

QUANTIFIE

RSANDLOGICALOPERATORS

-

----

-

-

------

PropositionsandPredicates

281

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-

F

F

-

F

G

B

-

HF

B

_

GH

-

-

F

H

B

--

--

--

--

---

-

-

F

BF

F

F

B

-----

---

-

----

-----

F

F

_

--

---

----

---

- H

F

F_

-

__

F

H

__

F

g

-

--

-

-

282

TheoryofAutomata,FormalLanguagesandComputation

Log

ica

lRe

lations

F

B

--

B

_

--

FH

EH

_

F

EH

_H

EHF

F

G

EF

G

F

F

EF

F

F

GF

EF

G

F

FF

EF

F

_

G

E_

G

_

F

E_

F

_

G

E

_

G_

_

F_

E

_

F

Example8.4.1:

-----

g

g

g

--

--

m

---

}

}-

-

--

--

Solution

F

BF

g

g

g

F

F

H

BH

GH

F

F

B

G

PropositionsandPredicates

283

7/23/2019 TC Reference

150/180

F

E

--

--

m

F

H

E

H

F

---

F

F

F

H

GH

BH

}

}-

-

F

F

H

GH

B

F

F

F

G

BF

F

G

B

HF

F

G

--

F

F

G_

BF

F

G

B

-

Example8.4.2:----

------

--

--

----

---

F

-------

F

B

F

FH

_

GH

F

F

F

G

B

F

F

BF

F

B

Solution

F

B

-

-

284

TheoryofAutomata,FormalLanguagesandComputation

FH

----

_

GH

---

FF

F

G

B

F

--F---

--

F

BF

F

B

---

--

F

-

Example8.4.3:--

------

-

-

----

------

-

-

Solution

-

-

_

G

-

-

H_

G

-

-

--

F

B_

G

F

_

GH

G__

G

-

B_

Example8.4.4:---

F

--

_H

--

_

-FH

--

_

-F

H

PropositionsandPredicates

285

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151/180

Solution

F

E

HHF

E

H_H

E

H_H

_

E

HH

_

E

HFH

E

HF

H

Example8.4.5:---

_

---F_

-

Solution

_

E_

GF

B

Example8.4.6:--

F

B

B

F

BF

-

Solution:

F

B

F

BF

---

Example8.4.7:--

_

G

_

G_

--

K

-

-

--

286

TheoryofAutomata,FormalLanguagesandComputation

_

G

B

_

G_

Solution

_

G

E

G

F

G

E

G

F

G

G

F

E

F

G

F

G

F

G

F

E

F

G

F

E

_

G_

--

Example8.4.8:----

-

----

-

---

---

-

----

--

---

Solution

--

-

F

r

_

r

-

--

r

--

-

-

--

r

r

r

r

PropositionsandPredicates

287

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152/180

--

--

G

-

--

Example8.4.9:-

-

--

--

-

--

-

-

Solution

-

-

-

-

H_

r

H

FH

r

_

r

_

r

H

F

r

H

_H

r

FH

r

H

r

_

r

r

H

r

_H

r

B---

Example8.4.10:

F

F

r

H

F

-

Solution

F

F

F

_

_

F

288

TheoryofAut