Reducibility

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

2

Theorem 5.1 HALTTM is undecidable.

Assume that TM R decides HALT TM .Then, the following TM S will decide ATM , which is not true .

S=Oninput langle M,w rangle ,anencoding of aTM M∧astring w: {} # 1 . RunTM R on input langle M,w rangle . {} # 2 . If R rejects , reject . {} # 3 . If R accepts , simulate M onw untilhalts . {} # 4 . If Mhas accepted , bold accept ; if Mhas rejected , bold reject .

3

Theorem 5.2 ETM is undecidable.

." rejects, if ; accepts, If 3.

.input on Run 2.

above. as construct and ofn descriptio the Use1.

: string a and TM a of encodingan ,,input On "

not true. is which , decide will machine following The

does." ifaccept and input on run , If 2.

reject. , If 1.

:input On "

. decides TM that Assume

}.)( and machine Turing a is |{

TM

TM

TM

acceptRrejectR

MR

MM

wMwMS

AS

MwMwx

wx

xM

ER

MLMME

w

w

w

Proof.

Def.

4

Proof continued

decider. a is that means This

. doesneither input,any on loopnot does since that Notice

)( rejects , accepts

)( accepts , rejects

)( accept not does .2

}{)( accepts .1

:iespossibilit twoare There

S

SR

MLMRwMS

MLMRwMS

MLwM

wMLwM

ww

ww

w

w

5

Theorem 5.3 REGULARTM is undecidable.

wMML

wMML

rejectRacceptR

MR

wMaccept

wMx

acceptx

xM

M

wMwMS

AS

REGULARR

MLMMREGULAR

w

w

w

nn

w

w

accept not does regular not is )(

accepts regular is )(

." rejects, if ; accepts, If 3.

.input on Run 2.

." accepts if

and input on run form, thishavenot does If 2.

. ,10 form thehas If 1.

:input On "

where, TMConstruct 1.

:string a is and TM a is where,,input On "

not true. is which , decide will following The

. decides TM Assume

language.}regular a is )( and TM a is |{

TM

TM

TM

:Remark

Proof.

Def.

6

Continued Remark on Th. 5.3

cases. possible allexhaust cases twoThese

lemma. pumping by the be to

out ns which tur},0|10{)( then ,accept not does If

. triviallyis which ,)( have then we, accepts If

}.0|10{)( that see weall, ofFirst

1. and 0 contains alphabet input that theassumemay We

*

regular-non

regular

nMLwM

MLwM

nML

nnw

w

nnw

7

Theorem 5.4 EQTM is undecidable.

ion.contradictgot weThus e.undecidabl is that us tells5.2 Th.

." rejects, if ;accepts, If 2.

inputs. all rejects that TM a is where,,input on Run 1.

:TM a is where,input On "

. decide will TM following The . decide TMLet

TM

11

TMTM

E

rejectRacceptR

MMMR

MMS

ESEQR

Proof.

8

Definition 5.5 Computation History

.an ofnotion

theintroduce weLikewise, ion.configurat acceptingan is andion configuratstart

theis where,,,, ions,configurat of sequence a is on for

An string.input an and TM a be Let

121

history ncomputatiorejecting

history ncomputatioaccepting

l

l

C

CCCCwM

wM

9

Definition 5.6 Linear bounded automata

• A restricted type of Turing machine whererin the tape head is not permitted to move off the portion of the tape containing the input.

• If the machine tries to move its head off either end of the input, the head stays where it is, in the same way that the head will not move off the left-hand end of an ordinary Turing machine.

• The idea of changing the alphabet suggests that the definition be modified in such a way that the amount of memory allowed is linear in n.

10

Remark

work.some LBA takesan

by decided benot can that language decidable a with up Coming

LBA.an by decided becan CFLEvery

LBAs. all are and ,,,for deciders The CFGDFACFGDFA EEAA

11

Lemma 5.7 Let M be an LBA with q states and g symbols in the tape alphabet. There are exactly qngn distinct configurations of M for a tape of length n.

tapeon the strings possible ofnumber he t

head theof positions possible ofnumber the

control theof states ofnumber the

ng

n

q

Proof.

12

Theorem 5.8 ALBA is decidable.

." halted,not hasit If

. rejected, hasit and halted has If

. accepted, hasit and halted has If 2.

halts.it untilor steps for on Simulate 1.

:string a is andLBA an is where,,input On "

follows. as is decides that algorithm The LBA

reject

rejectM

acceptM

qngwM

wMwML

A

n

Proof.

13

Theorem 5.9 ELBA is undecidable.

.

and of statestart theis where, Here,

.on for history n computatio

acceptingan is sequence thisif checks Then, .,,, strings into

delimiters the toaccording up breaks first, ,input an receives When

# # # # # #

.on for history n computatio acceptingan is if input

its accepts that LBA an construct we,input an and TM aFor

decidable. is that assume We

21

02101

21

LBA

321

n

n

l

CCCC

wwww

MqwwwqC

wM

BCCC

xBxB

x

wMxx

BwM

E

l

Proof.

14

Proof (continued)

." accepts, if ; rejects, If 3.

.input on Run 2.

above. as and from LBA Construct 1.

:string a is and TM a is where,,input On "

ion.contradict toleads which follows, as

decides that TMConstruct . decides TM that Suppose

TM

LBA

rejectRacceptR

BR

wMB

wMwMS

A

SER

15

Remark on Th. 5.9

ion.specificat its is need weAll

.LBA run not need we5.9, Th. of proof In the B

16

Theorem 5.10 ALLCFG is undecidable.

. accepts )(

.accept not does )( that Note

decidable. is for problem emptiness that theus tellsassumption The

y.effectivelstack its uses step, 3rd In the

." then , yieldnot does If 3.

. then , ofion configurat acceptingan not is If 2.

. then , ofion configuratstart not the is If 1.

:#####input On "

.input on TM of

history comutation acceptingan represent not does #### #

such that , ##### strings theall accepts

thatPDA aconstruct Wedecidable. is that Assume

})( andCFG a is |{ .

*

*

1

1

)1(121

21

)1(121

CFG

*CFG

1

1

wMDL

wMDL

D

D

acceptCC

acceptMC

acceptMC

CCCD

wM

CCC

CCC

DALL

GLGGALL

ii

l

l

l

l

l

l

Proof.

Def

17

The Post Correspondence Problem

match. no has ,, collection The

set. above for thematch of examplean is

dominos. of collection a is ,,,

ba

acc

a

ca

ab

abc

c

abc

ab

a

a

ca

ca

b

ab

a

c

abc

a

ca

ab

a

ca

b

18

Definitions of PCP and MPCP

domino}.first with thestarts that

match a with PCP theof instancean is |{

match}. a with PCP theof instancean is |{

. where,,,, sequence a ismatch a and

,,,,

:dominos of collection a is PCP theof instanceAn

212121

2

2

1

1

PPMPCP

PPPCP

bbbtttiii

b

t

b

t

b

tP

P

ll iiiiiil

k

k

19

Theorem 5.11 PCP is undecidable.

. ),L,,(),( if

,,every and ,,every For (3)

. ),R,,(),( if

,,every and ,every For (2)

.for dominofirst theis ##

# )1(

dominos. of collection a be Let

).,,,,,,(Let

. deciding construct WePCP. thedecide TMLet

210

rejectaccept0

TM

Prcb

cqabraq

Qrqcba

Pbr

qabraq

Qrqba

Pwwwq

P

qqqQM

ASR

n

Proof.

20

Proof continued

.#

## (7)

. and

, every For (6)

._#

# and

#

# (5)

.

, every For (4)

accept

accept

accept

accept

accept

Pq

Pq

aq

q

aq

a

P

Pa

a

a

21

Demonstration by example(1), (2)+(4)

#

# q0 0 1 0 0 #

# q0 0 1 0 0 #

# q0 0 1 0 0 # 2 q7 1 0 0 #

22

Demonstration by example(2)+(4), (3)+(4)

# 2 q7 1 0 0 #

# 2 q7 1 0 0 # 2 0 q5 0 0 #

…

# 2 0 q5 0 0 #

# 2 0 q5 0 0 # 2 q9 0 2 0 #

…

23

Demonstration by example(4)+(6)

#

# 2 1 qaccept 0 2 #

…

# 2 1 qaccept 0 2 # ... #

# 2 1 qaccept 0 0 # 2 1 qaccept 2 # ... # qaccept #

…

24

Demonstration by example(7)

# qaccept # #

# qaccept # #

25

MPCP to PCP

*,

*

*,,

*

*,

*

*,

**

*

withPCP toequivalent is

,,,,

withMPCP

*******

*****

*****

:Notation

3

3

2

2

1

1

3

3

2

2

1

1

21

21

21

k

k

k

k

n

n

n

b

t

b

t

b

t

b

t

b

t

b

t

b

t

b

t

uuuu

uuuu

uuuu

26

Definition 5.12 Computable Function

tape.itson )(just with halts ,input every on

, TM some iffunction computable a is :function A **

wfw

Mf

27

Definition 5.15 Mapping Reducibility

. to ofreduction thecalled is function The

.)(

,every for where,:function computable a is thereif

, written , language to is Language**

m

BAf

BwfAw

wf

BABA

reduciblemapping

28

Theorem 5.16

e.undecidabl

is then e,undecidabl is and If

outputs." tever output wha and )(input on Run 2.

).( Compute 1.

:input On "

. decides following theThen, . to

fromreduction thebe let and ,for decider a be Let

decidable. is then decidable, is and If

m

m

BABA

MwfM

wf

wN

ANBA

fBM

ABBA

5.17 Corrollary

Proof.

29

Example 5.18 Reduction from ATM to HALTTM

.",Output 2.

loop." aenter rejects, If 3.

accepts, If 2.

.on Run 1.

:input On "

. machine following theConstruct 1.

:,input On "

.

for reduction a computes machine following The

TMmTM

wM

M

accept.M

xM

xM

M

wMF

HALTA

F

30

Example 5.19 ATM MPCP PCP

e. transitivisty reducibili mapping that Note

.reductions mapping following the

shown have we5.11, Th. of proof In the

m

mTM

PCPMPCP

MPCPA

31

Example 5.20 ETM EQTM

inputs. all rejects where

,,output the toinput themapsthat

reduction shown the have we5.4, Th. of proof In the

1

1

M

MMM

f

32

Example 5.21

ation.complementby affectednot isty decidabili since

eundecidabl is that shows still which , to

fromreduction shown the have we5.2, Th. of proof In the

TMTMTM EEA

33

Theorem 5.22 Turing recognizability

le.recognizab-Turingnot is then

le,recognizab-Turingnot is and If

5.23

outputs." tever output wha and )(input on Run 2.

).( Compute 1.

:input On "

. recognizes following The . to

fromreduction thebe and for recognizer thebe Let .

le.recognizab-Turing is then

le,recognizab-Turing is and If

m

m

B

ABA

MwfM

wf

wN

ANBA

fBM

A

BBA

Corrollary

proof

34

Remark

BABA mm means

35

Theorem 5.24 EQTM is neither Turing-recognizable nor co-Turing-recognizable.

.",Output 2.

." accepts,it If .on Run 1.

:inputany On "

." 1.

:inputany On "

. and TMs twofollowing theConstruct 1.

:string a and TM a is where,input On "

follows. as obtained becan to ofreduction The

le]recognizab-Turingnot is [

21

2

1

21

TMTM

TM

MM

acceptwM

M

Reject

M

MM

wMwMF

EQA

EQ

proof.

36

Proof (continued)

.",Output 2.

." accepts,it If .on Run 1.

:inputany On "

." 1.

:inputany On "

. and TMs twofollowing theConstruct 1.

:string a and TM a is where,input On "

follows. as obtained becan to ofreduction The

le]recognizab-Turingnot is [

21

2

1

21

TMTM

TM

MM

acceptwM

M

Accept

M

MM

wMwMG

EQA

EQ