제10장. other models of tm’s - yonsei

제10장. Other Models of TM’s

학습목표

TM의다양한변형모델을이해, Universal TM과 Linear Bounded Automata를

통하여 TM의가능성확인

개요

• TM의다양한변형들

- Stay-option, Semi-infinite tape, Off-line tape

•보다강력한 Tape구조를갖는 TM들

- Multi-tape, Multidimensional tape

• Nondeterministic TM

• Universal TM

- Reprogrammable (general) TM

• Linear Bounded Automata

- Tape사용에제약을가한 TM

그래봐야모두같다!

뛰어봐야벼룩!

이건좀이상해보이지만그래도같다

Minor 변형 TM들

LL

LL

nMMM

nMMM

ddd

ddd

ˆˆˆ *

ˆ

*

ˆ1*

ˆ0

10

├├├

├├├

)()(in,in language same eaccept th automata of classes of Equvalence Def.

212211 MLMLCMCM =∋∃∀≡⋅

•그냥적용하기는좀애매하니택한방법이 Simulation

itSimulatingclass2ndtheinmachinea,classoneinmachineevery ∋∃∀⋅

먼저두 TM이같음을 formal하게정의합시다

자, 이제 TM의변형과그것이 power는같음을확인합시다

TM with a Stay-Option},,{: SRLQQ ×Γ×→Γ×δ

cf) TM with multiple tracks= each tape symbol can be a composite of characters

abc

Track 1Track 2Track 3

변형 TM-1

),,ˆ(),ˆ(

),,ˆ(),ˆ(ˆ),,(),(

)or,,ˆ(),ˆ(ˆ)or,,(),(

Lcqcq

Rbqaq

Sbqaq

RLbqaq

RLbqaq

jjs

jsi

ji

ji

ji

=

=→

=

=→

=

⋅

⋅

δ

δ

δ

δ

δ

이거앞에서한번본것같죠!

Move할때마다헤드를옮겨야하는게좀…

TM with Semi-Infinite Tape

b a

Reference pointiq

# a

# b

iq̂

),,(),( Lcqaq ji =δ

)),#,(#,ˆ())#,(#,ˆ(ˆ)),,(,ˆ()),(,ˆ(ˆ

Rpq

Lbcqbaq

jj

ji

=

=

δ

δ

• Idea : partitioning the states into two parts → QU and QL

변형 TM-2 테이프의한쪽끝으로만무한히쓸수있다면…

Off-Line TM

• Idea : using input file

CU

e f Tape

a b c dRead-onlyinput file

g

CU

a b c d

e f g1

10 0

0 0 0

변형 TM-3 입력용테이프가따로있다면…

Homework : Exercises 10.1

•여러가지변형에대한연습문제가주로인데, 모두할필요는없고,

• 5번

• 9번

•정도를그냥해봅시다.

Multitape TM’s

nnn RLQQ },{: ×Γ×→Γ×⋅δ

q0

a b c d e f

Tape 1 Tape 2

movementforthandbackrepeatedwithout}{Ex. −−nnba

TM with More Complex Storage

multi-track tape을사용하면간단히해결!

왔다리갔다리하지않고도쉽게해결이가능하네요

Multidimensional TM’s

tapetracktwobySimulation},,,{:

−⇒×Γ×→Γ× DURLQQδ

a

21 # #

b

1 0 # - 3 #

TM with More Complex Storage

2D address scheme1, -1 1, 1 1, 2

-1, 1


-전체적으로 detail한유도가까다로운문제들이므로기본정의로부터

새로운정의를유도할수있는지확인하는정도에서끝냅시다.

Nondeterministic TM (1)

},{2:

withTM:TMnisticNondetermiDef.

RLQQδ ×Γ×→Γ×

caaqaaaqaabqaaaq

LcqRbqaq

□├

├

20

10

210 )},,(),,,{(),(.Ex =δ

이제까지의경험에의하면이런식의 nondeterminism이해당오토메타의 power에영향을끼칠수도있을텐데…

gbookkeepinwithngbacktrackiticdeterminis:IdeaTMnisticnondetermiTMticdeterminisThm

⋅≡

caaqaabqaaaq □├ 210 orEx

# # # # ########

## a a aq0

# # # # ####

###

###

##

##

a a

a a

b

cq1

q2

#

#

Nondeterministic TM (2)실제로 TM의경우에는 power에영향을못미친다


• 3 : Nondeterministic solution이훨씬간단하게얻어질수있음을확인

• 4 : 세부분으로나누고매칭을시도

Universal TM• reprogrammable TM for general purpose machines

– Mu simulates the computation of M on w

• Standard way of describing TM

L

L

L

,111,11,1blank:},,,{

statefinal,stateinitial:},,,{

121

2121

==Γ===

aaaaqqqqqQ

m

n

LL 10101011011011),,(),(.Ex 3221 Laqaq =δ

CU(Mu)

Description of M

Tape contents of M

Internal states of M

범용컴퓨터로서의 TM으로확장 인코딩약속필요

Set Theory : Countable

• countable : 1-1 correspondence with positive integers

– ordering

?/ofquotientsall.Ex qp

LL ?32,4

1,31,2

1,11

– if we can write the elements in some sequence

→ enumeration procedure

32

22

12

L41

31

21

11 →

13

M

모든 TM이 0과 1의스트링으로표현된다는사실 일반적특성유도

Countable이아닌가?

아! 순서를바꾸니 countable이네!

L22

*

11

*

0 ##

withTM:procedurenenumeratioDef.

sxqsxqq ss ├├□

⋅

orderproper,,,,,,,,,,,,,

firstatstringtheoflengthconsiderdictionaryainorderingalphabetic},,,{.Ex

=

=Σ

⋅Laaacccbcabcbbbaacabaacba

cba

1steptoreturn3.itignoreNo

tapetheonitwriteYesTM.adefinesitifseetocheck.2

orderproperinstringnextthegenerate.1.pfcountable.isinfinite,althoughTM,allofsetTheThm

→→

TM : Countable


•이절의내용도따로연습을할정도는아니지만

• 7번

• 8번

•정도의문제를해봅시다.

Linear Bounded Automata

• restriction : only a finite part of tape → FA

only the part of tape occupied by input → LBA

),],()],(

),[,()[,(TMticdeterminisTMnisticnondetermiLBADef.

Lqq

Rqq

ji

ji

=

=≠=⋅

δ

δ

Fqxqxwq ff ∈∋⋅

][][acceptanceDef.

21

*

0 ├

}1|{.Ex ≥= ncbaL nnn

:}0|{.Ex ! ≥= naL n

rejectremaindernonzero)iiacceptleftsingle)i

4,3,2bys'ofno.divide

→→a

a L

onebydivisorngincrementidivisortheofmultiplesatthoseexceptsymbolsallremoving

−−

a a a a a a ]a a a

[[ ]

a’sdivisor (scratch space)

LBA : 예

LBA > PDA

LBA 로 accept할수없는 L ?

TM으로 accept할수없는 L ?


- 4(d)

- 4(e) 정도를한번해봅시다.

제10장. other models of tm’s - yonsei

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