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Human-awareRobotics
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Mapping Reducibility
• 2017/11/16
• Chapter 5.3 in Sipser
Ø Announcement:q Slides for this lecture are here:
http://www.public.asu.edu/~yzhan442/teaching/CSE355/Lectures/mapping.pdf
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Human-awareRobotics
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Last time• Reducibility
q Reducibility
q Using reducibility
Ø Goals:
o Learn about the use of reducibility
o Learn to use reducibility to prove that a problem is unsolvable
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Human-awareRobotics
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Outline for today• Reducibility
q Selecting problem S
q LBA
• Mapping reducibility
q Formal definition of reducibility
q Use mapping reducibility
Ø Goals:
o Learn a formal definition of reducibility
o Learn to use mapping reducibility
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Human-awareRobotics
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The EQTM problem
LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
SS’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
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Human-awareRobotics
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The EQTM problem
LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
S’ssolver R’ssolver
convertStoR
?reject
accept
Macceptwornot
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LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1:accept
M
w
The EQTM problem
M2:1) ifx!=waccept2) otherwise
Macceptwornot
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Human-awareRobotics
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LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
7
S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1:accept
M
w
The EQTM problem
M2:1) ifx!=waccept2) otherwise
Macceptwornot
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Human-awareRobotics
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The EQTM problem
LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
SS’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
WecanchangeStobeanyunsolvableproblem!
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Human-awareRobotics
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The EQTM problem
LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
SS’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
?
Maccepttheemptylanguageornot
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Human-awareRobotics
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LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1
The EQTM problem
Maccepttheemptylanguageornot
M
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Human-awareRobotics
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LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
11
S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1reject
The EQTM problem
Maccepttheemptylanguageornot
M
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LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1reject
The EQTM problem
Asimplersolution!
Maccepttheemptylanguageornot
M
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The EQTM problem
LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
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Steps to use reducibility• Using reducibility (S reduces to R) to prove unsolvability
q Select an unsolvable (undecidable problem) S (key)
q Assume R is solvable, solve S to derive a contradiction
q Check the input for R and S
q Think about how to use outputs for R to reason about outputs for S (key)
q Convert S input to R’s input
q Convert R’s output to S’s output
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Summary
Decidablelanguageslearned:ADFA,ANFA,GRE,ERE,EQDFA,ACFG,ECFG
Undecidablelanguageslearnedsofar:ATM,HALTTM,ETM,REGULARTM,EQTM,ALLCFG(onthebook)
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Rice’s theorem
ConsiderATM
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Rice’s theorem
Proofintextbook(5.28).YoucannotuseRice’stheoreminexamsforproofsunlesstoldtoL
ConsiderATM
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Outline for today• Reducibility
q Selecting problem S
q LBA
• Mapping reducibility
q Formal definition of reducibility
q Use mapping reducibility
Ø Goals:
o Learn a formal definition of reducibility
o Learn to use mapping reducibility
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Human-awareRobotics
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LBA
EquivalenttoaFA?
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Human-awareRobotics
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LBA
EquivalenttoaFA?NO!Thetapesizeisdependentontheinput
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Human-awareRobotics
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ALBA
ALBAisdecidable!
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Human-awareRobotics
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ALBA
ALBAisdecidable!Finitedistinctconfigurationssoweknowwhenitloops!
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ALBA
ALBAisdecidable!Finitedistinctconfigurationssoweknowwhenitloops!
qstates,nisthetapesize,gistapealphabetsize
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The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
SS’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
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Human-awareRobotics
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The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
S’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
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Human-awareRobotics
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1
The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
Whycan’twedothis?
M11)x!=wreject2)x==w
M
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
B(LBA)acceptonlyifxisacomputationhistoryofMacceptingw
The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
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Human-awareRobotics
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S’ssolver R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
B(LBA)acceptonlyifxisacomputationhistoryofMacceptingw
The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
TheconstructionofthisLBAisonthebook
Thatisthemaximumtapesizeweneed!
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The ELBA problem
LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
AsimilarproofshowsthatALLCFGisundecidable(Theorem5.13)
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Human-awareRobotics
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Outline for today• Reducibility
q Selecting problem S
q LBA
• Mapping reducibility
q Formal definition of reducibility
q Use mapping reducibility
Ø Goals:
o Learn a formal definition of reducibility
o Learn to use mapping reducibility
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Human-awareRobotics
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Mapping Reducibility M
f(w)
w
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Mapping Reducibility
Thechallengeofdesignf(s)liesinthebi-directionmapping
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Mapping Reducibility
SS’ssolver R’ssolver
convertStoR MapthesolutionofRtoasolutiontoS
Thechallengeofdesignf(s)liesinthebi-directionmappingReductionwetalkedabout:
f(S):everythinginsidetheredblock(bothsteps)exceptfortheswitchingbetweenacceptandrejectintheend
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Mapping Reducibility• S is reduced to R (R is more difficult than S)
q If R is decidable, S is also decidable!
q If S is undecidable, R is also undecidable!
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Human-awareRobotics
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Outline for today• Reducibility
q Example for selecting your unsolvabe problem S
q LBA
• Mapping reducibility
q Formal definition of reducibility
q Use mapping reducibility
Ø Goals:
o Learn about a formal definition of reducibility
o Learn to use mapping reducibility
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Human-awareRobotics
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is undecidable!
The halting problem
S’ssolver R’ssolver
convertStoR
accept
reject
Maccept
reject
w
LetususeS=ATMq AssumethatMR solvesR=HALTTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
MapthesolutionofRtoasolutiontoS
Macceptwornot
Whatisf(S)?
Howdowecombinethetwoparts?
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R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M’M
S’ssolver
Macceptwornot
The halting problem
f()
LetususeS=ATMq AssumethatMR solvesR=HALTTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
WhatisMintheinputisnotaproperTM?
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R’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M’M
S’ssolver
Macceptwornot
The halting problem
f()
WhatisMintheinputisnotaproperTM?
LetususeS=ATMq AssumethatMR solvesR=HALTTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
Ø Ingeneral,whenwedescribeaTuringmachinethatcomputesareductionfromAtoB,improperlyformedinputsareassumedtomaptostringsoutsideofB.
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ETM
S’ssolverR’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1
1)x!=wreject2)x==w
M
LetususeS=ATMq AssumethatMR solvesR=ETMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
Macceptwornotreductionfrom!
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ETM
S’ssolverR’ssolver
convertStoR
MapthesolutionofRtoasolutiontoS
M1
1)x!=wreject2)x==w
M
LetususeS=ATMq AssumethatMR solvesR=ETMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!
Macceptwornotreductionfrom!
reductionfromATM toETM,infact,doesnotexist(exercise5.5)!
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Mapping reducibility
ToprovesomeproblemBisnotTMrecognizable,weonlyneedtoshowthat
Interpretingthenotationbasedontheproblems’“difficultylevels”;easytoremember!
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ATM mEGTMATM mEGTM
Mapping reducibility
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Human-awareRobotics
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Outline for today• Reducibility
q Example for selecting your unsolvabe problem S
q LBA
• Mapping reducibility
q Formal definition of reducibility
q Use mapping reducibility
Ø Goals:
o Learn about a formal definition of reducibility
o Learn to use mapping reducibility
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