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Human-aware Robotics 1 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

Mapping Reducibility Human-aware Robotics › ~yzhan442 › teaching › CSE355 › Lectures › mappi… · qSolve S using M R by constructing M S using M R qA contradiction since

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  • Human-awareRobotics

    1

    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

  • Human-awareRobotics

    2

    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

  • Human-awareRobotics

    3

    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

  • Human-awareRobotics

    4

    The EQTM problem

    LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    SS’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

  • Human-awareRobotics

    5

    The EQTM problem

    LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    S’ssolver R’ssolver

    convertStoR

    ?reject

    accept

    Macceptwornot

  • Human-awareRobotics

    6

    LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    6

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M1:accept

    M

    w

    The EQTM problem

    M2:1) ifx!=waccept2) otherwise

    Macceptwornot

  • Human-awareRobotics

    7

    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

  • Human-awareRobotics

    8

    The EQTM problem

    LetususeS=ATMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    SS’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

    WecanchangeStobeanyunsolvableproblem!

  • Human-awareRobotics

    9

    The EQTM problem

    LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    SS’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

    ?

    Maccepttheemptylanguageornot

  • Human-awareRobotics

    10

    LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    10

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M1

    The EQTM problem

    Maccepttheemptylanguageornot

    M

  • Human-awareRobotics

    11

    LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    11

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M1reject

    The EQTM problem

    Maccepttheemptylanguageornot

    M

  • Human-awareRobotics

    12

    LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    12

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M1reject

    The EQTM problem

    Asimplersolution!

    Maccepttheemptylanguageornot

    M

  • Human-awareRobotics

    13

    The EQTM problem

    LetususeS=ETMq AssumethatMR solvesR=EQTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

  • Human-awareRobotics

    14

    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

  • Human-awareRobotics

    15

    Summary

    Decidablelanguageslearned:ADFA,ANFA,GRE,ERE,EQDFA,ACFG,ECFG

    Undecidablelanguageslearnedsofar:ATM,HALTTM,ETM,REGULARTM,EQTM,ALLCFG(onthebook)

  • Human-awareRobotics

    16

    Rice’s theorem

    ConsiderATM

  • Human-awareRobotics

    17

    Rice’s theorem

    Proofintextbook(5.28).YoucannotuseRice’stheoreminexamsforproofsunlesstoldtoL

    ConsiderATM

  • Human-awareRobotics

    18

    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

  • Human-awareRobotics

    19

    LBA

    EquivalenttoaFA?

  • Human-awareRobotics

    20

    LBA

    EquivalenttoaFA?NO!Thetapesizeisdependentontheinput

  • Human-awareRobotics

    21

    ALBA

    ALBAisdecidable!

  • Human-awareRobotics

    22

    ALBA

    ALBAisdecidable!Finitedistinctconfigurationssoweknowwhenitloops!

  • Human-awareRobotics

    23

    ALBA

    ALBAisdecidable!Finitedistinctconfigurationssoweknowwhenitloops!

    qstates,nisthetapesize,gistapealphabetsize

  • Human-awareRobotics

    24

    The ELBA problem

    LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    SS’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

  • Human-awareRobotics

    25

    The ELBA problem

    LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    S’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

  • Human-awareRobotics

    2626

    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

  • Human-awareRobotics

    2727

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    B(LBA)acceptonlyifxisacomputationhistoryofMacceptingw

    The ELBA problem

    LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

  • Human-awareRobotics

    2828

    S’ssolver R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    B(LBA)acceptonlyifxisacomputationhistoryofMacceptingw

    The ELBA problem

    LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    TheconstructionofthisLBAisonthebook

    Thatisthemaximumtapesizeweneed!

  • Human-awareRobotics

    2929

    The ELBA problem

    LetususeS=ATMq AssumethatMR solvesR=ELBAq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    AsimilarproofshowsthatALLCFGisundecidable(Theorem5.13)

  • Human-awareRobotics

    30

    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

  • Human-awareRobotics

    31

    Mapping Reducibility M

    f(w)

    w

  • Human-awareRobotics

    32

    Mapping Reducibility

    Thechallengeofdesignf(s)liesinthebi-directionmapping

  • Human-awareRobotics

    33

    Mapping Reducibility

    SS’ssolver R’ssolver

    convertStoR MapthesolutionofRtoasolutiontoS

    Thechallengeofdesignf(s)liesinthebi-directionmappingReductionwetalkedabout:

    f(S):everythinginsidetheredblock(bothsteps)exceptfortheswitchingbetweenacceptandrejectintheend

  • Human-awareRobotics

    34

    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!

  • Human-awareRobotics

    35

    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

  • Human-awareRobotics

    36

    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?

  • Human-awareRobotics

    37

    R’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M’M

    S’ssolver

    Macceptwornot

    The halting problem

    f()

    LetususeS=ATMq AssumethatMR solvesR=HALTTMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    WhatisMintheinputisnotaproperTM?

  • Human-awareRobotics

    38

    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.

  • Human-awareRobotics

    39

    ETM

    S’ssolverR’ssolver

    convertStoR

    MapthesolutionofRtoasolutiontoS

    M1

    1)x!=wreject2)x==w

    M

    LetususeS=ATMq AssumethatMR solvesR=ETMq SolveSusingMR byconstructingMS usingMRq AcontradictionsinceSisnotsolvable!

    Macceptwornotreductionfrom!

  • Human-awareRobotics

    40

    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)!

  • Human-awareRobotics

    41

    Mapping reducibility

    ToprovesomeproblemBisnotTMrecognizable,weonlyneedtoshowthat

    Interpretingthenotationbasedontheproblems’“difficultylevels”;easytoremember!

  • Human-awareRobotics

    42

    ATM mEGTMATM mEGTM

    Mapping reducibility

  • Human-awareRobotics

    43

    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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