reducibility yukita/. 2 theorem 5.1 halt tm is undecidable
TRANSCRIPT
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Reducibility
http://cis.k.hosei.ac.jp/~yukita/
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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 .
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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
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15
Remark on Th. 5.9
ion.specificat its is need weAll
.LBA run not need we5.9, Th. of proof In the B
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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
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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
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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
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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.
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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
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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 #
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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 #
…
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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 #
…
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24
Demonstration by example(7)
# qaccept # #
# qaccept # #
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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
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26
Definition 5.12 Computable Function
tape.itson )(just with halts ,input every on
, TM some iffunction computable a is :function A **
wfw
Mf
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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
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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.
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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
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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
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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
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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
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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
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34
Remark
BABA mm means
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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.
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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