There is something

7/23/2019 There is something

http://slidepdf.com/reader/full/there-is-something 1/46

Finite Automata and

Non Determinism

http://cis.k.hosei.ac.jp/~yukita/



2

Definition 1.1: Finite Automaton

.theis 5.and,theis 4.

,theis :3.

,thecalledsetfiniteais2.,thecalledsetfiniteais1.

where),,,,,(tupple-5aisA

0

0

statesaccept of set statestart

functiontransition

alphabet states

automaton finite

Q F Qq

QQ

Q

F qQ

⊆∈

→Σ×

Σ

Σ

δ

δ



3

State Diaram for M 1

!3!1 !2

" 1

1"

"# 1



$

Data %epresentation for &1

.}{5.

andstate,starttheis 4.

,

10

asdescriedis3.

{0,1}2.},,{1.

2

1

223

232

211

321

q F

q

qqq

qqq

qqq

qqqQ

=

=Σ=

δ



'

(ask "1

DFA

1. )mp*ement &1 +ith your fa,orite

prorammin *anuae.

2. -) (+o 0uttons for input " and 1

State chart +ith the current state hih*ihted



anuae of &1

}1lastthef!ll!w0s!f nu"ere#enan

and1!neleastatc!ntains${)(

.sa% thatcan&eaccepts."achinethatstrin'sall!f settheis

),(aswritten,he

1 ww M L

A M M

M L M

=

recognizes

machineof language



State Diaram for &'

!2

!1

!"

"

1

12

2# 4 % 5 S

5 ( 6

1# 4%5S5(6

2

"# 4%5S5(6 "



7

Data %epresentation for &'

.}{5.

andstate,starttheis 4.

,

210

asdescriedis3.

,0,1,2}*+*{2.

},,{1.

0

0

10202

02101

21000

210

q F

qqqqqq

qqqqq

qqqqq R

qqqQ

=

=Σ

=

δ



8

)nforma* Description of M '

M ' keeps a runnin count of the sum of

the numerica* sym0o*s it reads# modu*o 3.

5,ery time it recei,es the 4%5S5(6sym0o* it resets the count to ".

M ' accepts if the sum is "# modu*o 3.



1"

Definition 1.: %eu*ar anuae

A *anuae is ca**ed a regular language if

some finite automaton reconi9es it.



11

5amp*e 1.8: A finite automaton E 2

E 2 reconi9es the reu*ar *anuae of a**

strins that contain the strin ""1 as a

su0strin.

""1"# 1""1# ""1# and 111111""1111" are

a** accepted#

0ut 11 and """" are not.



12

Find a set of states of E 2

;ou

1. ha,en<t just seen any sym0o*s of the

pattern#

2. ha,e just seen a "#

3. ha,e just seen "" or#

$. ha,e just seen the entire pattern ""1. Assin the states q#q"#q""# and q""1 to these

possi0i*ities.



13

Dra+ a State Diaram for E 2

!"" !""1

" "# 1

1

1

"!"

"

!

1



1$

%eu*ar =perations on anuaes

!perati!n.a!nelasttheand

,!perati!nsare!perati!nsfirst tw!he

}.eachand0${:

}.and${:

}.!r${ :

lan'ua'es. eandet

21

unary

binary

A xk x x x A

B y A x xy B A

B x A x x B A

B A

ik ∈≥=

∈∈=

∈∈=∪

Star

ionConcatenat

Union



1'

5amp*e 1.11

}.,'!!dadad

d,'!!dad'!!'!!dad,'!!d!d,'!!d'!!d'!

adad,'!!dad,'!!d'!!d, ad,'!!d,,{

and}, ad'irl ad!%,'!!d'irl,'!!d!%,{

},'irl !%, ad,'!!d,{

ha#ewehen,}.'irl !%,{ and} ad'!!d,{et

}.,, a,{ ealphaetet the

ε =

==∪

==

Σ

A

B A

B A

B A



1

(heorem 1.12 >*osedness for nion

.iss!

lan'ua'es,re'ularareandif s,!ther w!rd/n

!perati!n.uni!nthe

under cl!sedislan'ua'esre'ular!f classhe

21

21

A A

A A

∪



?roof of (h 1.12 1

?roof of (heorem 1.12

!n.c!nstructithe!f sc!rrectneschec thehen,

.),,,,(where

,rec!'niet!!nstruct

.),,,,(where,rec!'nieand

),,,,,(where,rec!'nieet

21

2222222

1111111

F qQ M

A A M

F qQ M A M

F qQ M A M

δ

δ

δ

Σ=

∪

Σ=

Σ=



?roof of (h 1.12 17

>onstruction of M

}.!r$),{()()( 5.).,( 4.

)),(),,(()),,((,),( 3.

(&h%) .alphaetsa"etheha#eandthatassu"ecan&e2.

}.and$),{( 1.

2211212121

210

22112121

21

22112121

F r F r r r F QQ F F qqq

ar ar ar r aQr r

M M

Qr Qr r r QQQ

∈∈=×∪×==

=Σ∈∀∈∀

Σ

∈∈=×=

δ δ δ



?roof of (h 1.12 18

>orrectness

;ou shou*d check the fo**o+in.

1. For any strin reconi9ed 0y M 1 is

reconi9ed 0yM

.2. For any strin reconi9ed 0y M 2 is

reconi9ed 0y M .

3. For any strin reconi9ed 0y M isreconi9ed 0y M 1 or M 2.



2"

(heorem 1.13 >*osedness for

concatenation

.iss!

lan'ua'es,re'ularareandif s,!ther w!rd/n!perati!n.

i!nc!ncatenatundercl!sedislan'ua'esre'ular!f classhe

21

21

A A

A A



21

Nondeterminism

(o pro,e (heorem 1.13# +e need

nondeterminism.

Nondeterminism is a enera*i9ation of

determinism. So# e,ery deterministic

automaton is automatica**y a

nondeterministic automaton.



22

Nondetermistic Finite Automata

A nondeterministic finite automaton can 0e

different from a deterministic one in that

@ for any input sym0o*# nondeterministic one

can transit to more than one states.

@ epsi*on transition

NFA and DFA stand for nondeterministic

finite automaton and deterministic finiteautomaton# respecti,e*y.



23

NFA N 1

!3!1 !2

"#1

1 "#ε !$

"#1

1



2$

?ara**e* +or*d and NFA

... ...

accept



2'

5amp*e 1.1$ NFA N 2

!3!1 !2

"#1

1 "#1 !$"#1

et *anuae A consist of a** strins o,er "#1B containin a 1 in the thirdposition from the end. N 2 reconi9es A.



2

A DFA e!ui,a*ent to N 2

!"1"!""" !1""

"

" " !11"

!"11!""1 !1"1

1 !111

1

"

1

""

1

"1

1"1

1



2

5amp*e 1.1' NFA N 3

ε

"

et *anuae A consist of a** strins "k # +here k is a mu*tip*e of 2 or 3. N 3

reconi9es A.

ε

"

"

"

"



27

A DFA e!ui,a*ent to N 3

"!２

!１

!3!'

"!"

" " " !$

"

!C1

1 11 1 1

1

"# 1



28

5amp*e 1.1 NFA N $

!1

!2 !3

a

0

a#0

εa

N $ accepts , a, baba# and baa. N $ does not accept b, nor babba.



3"

Definition 1.1: NFA

.theis 5.

and,theis 4.

,theis 2:3.

,thecalledsetfiniteais2.,thecalledsetfiniteais1.

where),,,,,(tupple-5aisA

0

0

statesaccept of set

statestart

functiontransition

alphabet states

automaton finitenisticnondetermi

Q F

Qq

Q

Q

F qQ

Q

⊆

∈

→Σ×

Σ

Σ

ε δ

δ



31

5amp*e 1.17 NFA N 1

}.{ 5.

state.starttheis 4.

}{}{

}{

}{}{

},{}{10

as'i#enis 3.

}1,0{ 2.

},,,,{ 1.

4

1

444

43

332

2111

4321

q F

q

qqq

qq

qqq

qqqq

qqqqQ

=

∅

∅∅

∅

∅

=Σ

=

ε

δ

!3!1 !2

"#1

1 "#ε!$

"#1

1



32

)n +hat situation is Non

Determinism re*e,ant

Eon Neumann machines are deterministic.

o+e,er# there are many cases +here

machine specification is a** +e need.



33

(heorem 1.18

5,ery nondeterministic finite automaton

has an e!ui,a*ent deterministic finite

automaton.

@ Def. (he t+o machines are equivalent is they

reconi9e the same *anuae.



?roof of (h 1.18 3$

?roof of (h. 1.18

}.${ 4.

and },{ 3.

,),(),( 2.

,2 1.

such that),,,,(!nstruct

arr!ws.n!hasfirst thatassu"euset

.lan'ua'es!"e'rec!'niin Athe e),,,,(et

00

0

0

∅≠∩′∈=′=′

=′

=′′′′Σ′=

Σ=

∈

F RQ R F

qq

ar a R

Q

F qQ M

N

A F qQ N

Rr

Q

δ δ

δ

ε

δ



?roof of (h 1.18 3'

)ncorporate ε arro+s

0

0 0

( ) { $ can e reached fr!" % tra#elin' al!n' er! !r

"!re arr!ws.}

&e "!dif% as f!ll!ws.( , ) { $ ( ( , )) f!r s!"e }.

&e "!dif% as f!ll!ws.

({ }).

&e !"it the c!rrectness p

E R q q R

R a q Q q E r a r R

q

q E q

ε

δ δ δ

=

′

′ = ∈ ∈ ∈

′

′ =

r!!f.



3

>oro**ary 1.2"

A *anuae is reu*ar if and on*y if some

nondeterministic finite automaton

reconi9es it.



3

5amp*e 1.21 NFA N $ to DFA

1

2 3a

0

a#0

εa

}}.3,2,1{},3,2{},3,1{},2,1{},3{},2{},1{,{2

astaes e"a%setstates6he .7Ae8ui#alentan

c!nstructwant t! we,}}3{,1,},,{},3,2,1{{9i#en

}3,2,1{

4

∅=

=

D D

ba N δ



37

(ask "2

?ara**e* Gor*d

1. Grite a proram that simu*ates N$.

2. -)

@ (hree 0uttons for input "# 1# and epsion.

@ State chart that ref*ect the 0ranchin of the

+or*d.



38

Start and Accept states

f!ll!ws.as'i#enisdia'ra"statehe

}}.3,2,1{},3,2{},3,1{},3{{isstatesaccept!f sethe}.3,1{})1({ isstatestarthe = E



$"

(he state diaram of D

φ 1B 2B 1#2B

3B 1#3B 2#3B 1#2#3B

a#0

a#0

a

a

a

a

a

a

0

0 00

0

0



$1

(heorem 1.22 (he c*ass of reu*ar *anuaes is

c*osed under the union operation.

N 1

N 2

N

ε

ε



$2

?roof of (h. 1.22

≠=∅

==

∈∈=

=

=

Σ=

Σ=

Σ=

.and f!r

and f!r},{

f!r),(f!r),(

),( 4.

3.

!f statestarttheis statehe 2.

}{ 1.

.rec!'niet!),,,,{!nstruct

.rec!'nie ),,,,(

and,rec!'nie ),,,,(et

0

021

22

11

21

0

210

210

222222

111111

ε

ε

δ δ

δ

δ

δ

δ

aqq

aqqqq

QqaqQqaq

aq

F F F

N.q

QQqQ

A A F qQ N

A F qQ N

A F qQ N



$3

(heorem 1.23 (he c*ass of reu*ar *anuaes is

c*osed under the concatenation operation.

N 1N 2

N ε

ε



$$

?roof of (h. 1.23

∈

=∈

≠∈

∉∈

=

=

Σ=

Σ=

Σ=

.f!r),(

and f!r}{),(

and f!r),( and f!r),(

),( 4.

!f thatassa"etheisstatesaccept!f sethe 3.

.!f thatassa"etheis statestarthe 2.

1.


.rec!'nie ),,,,(

and,rec!'nie ),,,,(et

22

121

11

111

2

11

21

2121

222222

111111

Qqaq

a F qqaq

a F qaq F qQqaq

aq

N

N q

QQQ

A A F qQ N

A F qQ N

A F qQ N

δ

ε δ

ε δ δ

δ

δ

δ

δ



$'

(heorem 1.2$ (he c*ass of reu*ar *anuaes is

c*osed under the star operation.

N 1

N

ε ε

ε



?roof of (h. 1.2$

≠=∅

==

=∈

≠∈

∉∈

=

=

=

Σ=

Σ=

.and f!r

and f!r}{

and f!r}{),( and f!r),(

and f!r),(

),( 4.

}{ 3.state.startnewtheis statestarthe 2.

}{ 1.


.rec!'nie ),,,,(et

01

111

11

111

10

0

10

10

111111

ε

ε

ε δ ε δ

δ

δ

δ

δ

aqq

aqqq

a F qqaqa F qaq

F qQqaq

aq

F q F q

QqQ

A F qQ N

A F qQ N

There is something

Documents

Transcript of There is something