AlanTuring Bornin1912 TuringMachines 1922:TroublesinSchool - - PDF document

SLIDE 1

1 TuringMachines

AModelofComputation

AlanTuring

• Bornin1912
• 1922:TroublesinSchool
• 1936:TuringMachine
• WWII:Bletchley Park
• 1954:Suicide?!

OriginalMotivation

Modeling“HumanComputers” Whatare“HumanComputers”? Rememberthe TuringTest!

OriginalMotivation

Modeling“HumanComputers” Whatare“HumanComputers”? (mightbeabiasedpicture)

Today’sPerspective

Modeling“HumanComputers” Whatare“HumanComputers”?

Today’sPerspective

Modeling“Computers” Whatare“Computers”?

SLIDE 2

2 Today’sPerspective

Modeling“Computers” Whatare“Computers”?

• StorageDevice
• read/writeaccess
• finitesize(conceptuallyarbitrarilylarge)
• ControlUnit
• defineswhichsteptodonext
• akaCPUs/Programs

Today’sPerspective

Modeling“Computers” Whatare“Computers”?

???

Today’sPerspective

Modeling“Computers” Whatare“Computers”? Modeling… Butwhat for???

Today’sPerspective

Modeling“Computers” Whatare“Computers”? Modeling… Butwhat for??? Reasoningabout

• algorithms
• computationalproblems

TreatingthemasMathematicalObjects!!!

Algorithmsas MathematicalObjects

• TMsaremeantforformulating&proving

generalstatementsaboutalgorithms: – WhatiscomputablebyTMs? – Howmuchtime/spacedoTMsneedto solveagivenproblem?

• TMsareNOTmeantfor

– programming – realcomputations – browsingtheweb Sowhat?! TuringMachines andrealcomputers?!

RelevanceofTMs: WhatisComputablebyTMs?

LCMs candoanythingthatcouldbedescribed as"ruleofthumb"or"purelymechanical". Thisissufficientlywellestablishedthatitis nowagreedamongstlogiciansthat"calculable bymeansofanLCM"isthecorrectaccurate renderingofsuchphrases.

(Turingin1948onhisLogicalComputingMachine)

SLIDE 3

3

RelevanceofTMs: Church-TuringThesis

LCMs candoanythingthatcouldbedescribed as"ruleofthumb"or"purelymechanical".

• Historically:AlotofmodelsareequivalenttoTMs

(i.e.,theydescribethesamesetofalgorithms)

• LambdaCalculus
• PartiallyRecursiveFunctions
• …
• Practically:Allknowncomputersystemsare

equivalenttoTMs.

RelevanceofTMs: HowefficientareTMs?

Allreasonablemodelsofcomputationare polynomially relatedtotheTMwrt.theirtime performance. Thisisestablishedbysimulationarguments…

RelevanceofTMs: HowefficientareTMs?

Allreasonablemodelsofcomputationare polynomially relatedtotheTMwrt.theirtime performance. Thisisestablishedbysimulationarguments…

. k nk

• fixed
• some
• for
• runtime
• with

TM

• a

by

• simulated
• be
• can
• 4
• Pentium

A

RelevanceofTMs: ExtendedChurch-TuringThesis

Allreasonablemodelsofcomputationare polynomially relatedtotheTMwrt.theirtime performance. Thisisestablishedbysimulationarguments… ….it’sathesis– notatheorem. DNA-Computing Quantum-Computing

RelevanceofTMs: ExtendedChurch-TuringThesis

TMscansimulaterealcomputersefficiently TMshaveamathematicallysimplestructure TMsaretheidealvehicletobuilda TheoryonEfficientComputability

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs

SLIDE 4

4 StorageDeviceofaTM

• Tape

– arbitrarilylongbutfinite stripdividedintocells – eachcellcontainsasingle symbol – finite alphabetofsymbols

• TapeHead

– accessesonecellatatime (activecell) – reads symbolfromactivecell – overwrites symboltoactivecell – moves left,rightorstays

# # 1 1 1 # # # # # # # # #

ControlUnitofaTM

• SetofStates
• finitesize
• TransitionFunction

givenastate andaninputsymbol,thecontrol decideswhich

• symboltowrite
• directiontheheadismoved
• newstatetoassume

# # 1 1 1 # # # # # # # # #

Control Unit

Example

T H B S 0:0/→ 1:1/→ #:#/← #:1/- 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ Alphabet:{0,1} BlankSymbol:#

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # S InputString

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # S

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # S ….

SLIDE 5

5 Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # S

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # T

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 # # # # # # # # # T

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # B ….

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # B

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # # H OutputString

SLIDE 6

6 FormalTuringMachineDefinition

s K M , , , δ Σ =

K s∈

• state
• initial
• K
• states
• f
• set
• finite
• (alphabet)
• symboles
• f
• set
• finite
• Σ

} , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × K K δ

• function
• transition
• Example

T H B S 0:0/→ 1:1/→ #:#/← #:1/- 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ Alphabet:{0,1} BlankSymbol:#

s K M , , , δ Σ =

Example

T H B S 0:0/→ 1:1/→ #:#/← #:1/- 1:0/← 0:1/← 0:0/← 1:1/← #:#/→

s K M , , , δ Σ =

{ }

# , 1 , = Σ

#isalways included

Example

T H B S 0:0/→ 1:1/→ #:#/← #:1/- 1:0/← 0:1/← 0:0/← 1:1/← #:#/→

s K M , , , δ Σ =

{ }

# , 1 , = Σ

{ }

B T, S, = K

StateHis neverleft!

Example

T H B S 0:0/→ 1:1/→ #:#/← #:1/- 1:0/← 0:1/← 0:0/← 1:1/← #:#/→

s K M , , , δ Σ =

{ }

# , 1 , = Σ

{ }

B T, S, = K S s =

Example

T S 0:0/→ 1:1/→ #:#/←

s K M , , , δ Σ =

{ }

# , 1 , = Σ

{ }

B T, S, = K S s = .... , # , ) # , ( , , ) , ( , 1 , ) 1 , ( ←> =< →> =< →> =< T S S S S S δ δ δ

SLIDE 7

7

FormalTuringMachineDefinition

s K M , , , δ Σ =

K s ∈

• state
• initial
• K
• states
• f
• set
• finite
• (alphabet)
• symboles
• f
• set
• finite
• Σ

} , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × K K δ

• function
• transition
• ThisLecture
• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs

ExecutionofTMs

TheexecutionofaTMisdescribedformallyasa Sequenceof Configurations. AStepofTMisthetransitionfromone Configurationtothenextone. Twospecialconfigurations:

• InitialConfiguration
• HaltingConfiguration

Configuration

# # 1 1 1 # # # # # # # # # B

execution.

• some
• during
• f
• state
• entire
• the
• describes
• f
• ion

Configurat A M s K M C , , , δ Σ =

• cursorposition
• tapecontents
• state

Itmustcontain:

Configuration

# # 1 1 1 # # # # # # # # # B

execution.

• some
• during
• f
• state
• entire
• the
• describes
• f
• ion

Configurat A M s K M C , , , δ Σ =

*

, , , Σ ∈ ∈ > =< u w K q u w q C

• and
• with
• Triple

w u w isthestringupuntilthetapehead u containstherest #swhichhavenotbeenvisited areignored

Configuration

# # 1 1 1 # # # # # # # # # B

execution.

• some
• during
• f
• state
• entire
• the
• describes
• f
• ion

Configurat A M s K M C , , , δ Σ =

*

, , , Σ ∈ ∈ > =< u w K q u w q C

• and
• with
• Triple

w

> =< # 010 , 11 B, C

u

SLIDE 8

8 AComputationalStep

n n n n n

u u u u w w w w dir dir w q w q u w q C ... ' ' ... ' , ' , ' ) , ( , ,

2 1 1 1

= = =→ > =< > =<

−

• then
• If

.

• and
• δ

#sare padded

AComputationalStep

n n n n n

u u u u w w w w dir dir w q w q u w q C ... ' ' ... ' , ' , ' ) , ( , ,

2 1 1 1

= = =→ > =< > =<

−

• then
• If

.

• and
• δ

ThetransitionfromCtoC’isasinglestep.

y. analogousl

• and
• For

− = =→ dir dir . ' , ' , ' ' > =< u w q C C

• yield

to

• said
• is
• #sare

padded

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # S

> =< 1001 , 1 S, C Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # S

> =< 1001 , 1 S, C > =< 001 , 11 S, C

… …

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # S

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # T

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C

SLIDE 9

9

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 # # # # # # # # # T

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # B

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C > =< # 10 , 110 B, C

… …

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # B

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C > =< # 10 , 110 B, C

…

> =< # 11010 , # B, C Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

# # 1 1 1 # # # # # # # # # H

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C > =< # 10 , 110 B, C

…

> =< # 11010 , # B, C > =< # 1010 , 1 # H, C

InitialConfiguration

# # 1 1 1 # # # # # # # # S InputString

> =< 1001 , 1 , S C

Inourexample,westartedwiththeconfiguration

• input
• and
• Given

*

}) {# ( , , , − Σ ∈ Σ = x s K M δ > =<

n

x x x s C ... , ,

2 1

istheinitialconfiguration

HaltingConfiguration

> =< # 1010 , 1 # , H C

Inourexample,wehaltwiththeconfiguration # # 1 1 1 # # # # # # # # H OutputString } , , { , , R A H q u w q C

∈ > =< if

• ion,

configurat

• halting
• a
• is

SLIDE 10

10 HaltingConfiguration:Functions

# # 1 1 1 # # # # # # # # H OutputString Ifq=HthentheMcomputedafunction. Theresultisthestringbetweenthehead andthe first#totheright. 1 ) (

+ = x x M

• :

example

• ur
• In

Excursus:DecisionProblems

• RemembertheUNO-Problem.
• Givenasetofstates,youcanaskwhether

thereispeacefulseatingarrangement.

• Thisadecisionproblem:Theanswerisasinglebit.
• Theirsimplestructureishelpfulwithincomplexity.
• FormalLanguage:Thesetofpositiveinstances.
• Functionalproblemscanbe“reduced”

todecisionproblems.

HaltingConfiguration:Decision

# # 1 1 1 # # # # # # # # A Ifq=A(q=R)thenMaccepted(rejected)theinputx.

{ }

A ) ( ) ( , , ,

*

= Σ ∈ = > Σ =< x M x M L s K M

• :
• :

language

• a
• decides
• an
• Such

δ

RuntimeofaTM

. }) {# ( , , ,

*

− Σ ∈ Σ = x s K M

• and
• Let

δ Thethenumberofstepsbetweeninitial andhaltingconfigurationistheruntimeof Monx. IfMdoesnotreachahaltingstate(H,A,R),then Mdoesnotterminate(runsforever). ). ( }) {# ( |) (|

*

n f M x x f M

• time
• in
• runs
• then
• all
• for
• less
• r
• steps
• within

halts

• If

− Σ ∈

Example

# # 1 1 1 # # # # # # # # # H

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C > =< # 10 , 110 B, C

…

> =< # 11010 , # B, C > =< # 1010 , 1 # H, C

RuntimeofMon110110: 12steps

Example

T H B S #:#/← #:1/- 0:0/→ 1:1/→ 1:0/← 0:1/← 0:0/← 1:1/← #:#/→ # # 1 1 1 # # # # # # # # S

Example

nsteps n+1steps 1step Mrunsin2n+2

SLIDE 11

11 SpaceusedbyaTM

. }) {# ( , , ,

*

− Σ ∈ Σ = x s K M

• and
• Let

δ Thenumberofsymbolsinthelargestconfiguration isthespacerequiredbyMoninputx. ). ( }) {# ( |) (|

*

n f M x x f M

• space
• in
• runs
• then
• all
• for
• less
• r
• space
• within

runs

• If

− Σ ∈

Example

# # 1 1 1 # # # # # # # # # H

> =< 1001 , 1 S, C > =< 001 , 11 S, C

…

> =< ε , # 11001 S, C > =< # , 11001 T, C > =< # , 1100 T, C > =< # 10 , 110 B, C

…

> =< # 11010 , # B, C > =< # 1010 , 1 # H, C

SpacerequirementofM

• ninput110110:

7cells Maximalconfigurations

Example

# # 1 1 1 # # # # # # # # # H Mrunsinspacen+2

Configuration

# # 1 1 1 # # # # # # # # # B execution.

• some
• during
• f
• state
• entire
• the
• describes
• f
• ion

Configurat A M s K M C , , , δ Σ =

*

, , , Σ ∈ ∈ > =< u w K q u w q C

• and
• with
• Triple

w u w isthestringupuntilthetapehead u containstherest #swhichhavenotbeenvisited areignored

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs
• Encoding
• Constantsdonotmatter

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs
• Encoding
• Constantsdonotmatter

Multi-TapeTMs

• Insteadofasingletape,weuseseveraltapes
• Theyarededicated:

– Input Tape (readonly) – WorkTapes (read/write) – Output Tape (writeonly)

SLIDE 12

12 Multi-TapeTMs:Definition

: , , , s K M δ Σ =

• definition
• the
• adapting

k k k

K K } , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × δ restisthesame. Readandwriteksymbols, moveonktapes IfMisak tapeTM,then

Multi-TapeTMs:Configuration

* 1 1

, , ,... , , Σ ∈ ∈ > =<

i i k k

u w K q u w u w q C

• and
• with

IfMisak tapeTM,then : , , ,

*

Σ ∈ ∈ > =< u w K q u w q C

• and
• with
• adpating

….justktapes

Multi-TapeTMs:SpaceBound

* 1 1

, , ,... , , Σ ∈ ∈ > =<

i i k k

u w K q u w u w q C

• and
• with

Thenumberofsymbolsinthelargestconfiguration isthespacerequiredbyMoninputx. Butonlythecontentsoftheworktapesarecounted! I.e.,inputandoutputarenotconsideredfor spacebounds.

Multi-TapeTMs:Stronger??

)). ( ( ) ( ' ) ( ' )) ( (

2 n

f O x M x M M n f O k M

• time
• in
• runs
• which
• with
• TM
• tape

1-

• a
• is
• there
• Then
• .
• time
• in
• running
• TM
• tape
• a
• be
• Let

= −

(Ontheotherhand:Palindroms canbedecidedbya

• 2-tapeTMwithintimeO(n)
• 1-tapeTMrequiresO(n2).)

Multi-TapeTMs

• Insteadofasingletape,weuseseveraltapes
• Theyarededicated:

– Input Tape (readonly) – WorkTapes (read/write) – Output Tape (writeonly)

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs

ThisLecture

• DefinitionofTMs
• ExecutionofTMs
• Multi-TapeTMs
• Non-DeterministicTMs

SLIDE 13

13 DeterministicTMs

TheTMswesawsofarweredeterministic. I.e.,theinputdeterminedtheoutcomeof thecomputation.

} , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × K K δ

I.e.,weusedatransitionfunction: That’stheway,ourrealcomputerswork….

Non-DeterministicTMs

Non-DeterministicTMsareaformalismto expresscertainalgorithms. ….butyoucannotsimulateanondet.TMdirectly byarealcomputer… Westartwithanexample…

Example:UNO

?

• in
• nodes
• all
• includes
• which

circle

• a
• there
• Is
• graph
• undirected
• an
• Given

V E V G . , > =<

1 2 3 4 Note:Weareonlylookingatthedecisionproblem…

Example:UNO

?

• in
• nodes
• all
• includes
• which

circle

• a
• there
• Is
• graph
• undirected
• an
• Given

V E V G . , > =<

1 2 3 4 Sure:

Example:UNO

?

• in
• nodes
• all
• includes
• which

circle

• a
• there
• Is
• graph
• undirected
• an
• Given

V E V G . , > =<

Ifyoutrytosolvethisproblem,youwillendup enumeratingthepossiblesolutions…

reject

• 3.

accept

• then
• in
• path
• a
• is
• if
• 2.
• f
• n

permutatio

• each
• for
• 1.

G V

• π

reject

• 3.

accept

• then
• in
• path
• a
• is
• if
• 2.
• f
• n

permutatio

• a
• guess
• 1.

G V

• π

Example:UNO

reject

• 3.

accept

• then
• in
• path
• a
• is
• if
• 2.
• f
• n

permutatio

• each
• for
• 1.

G V

✁

π

Almostequivalently,youcanwrite: Itmightmake awrong guess?!?

SLIDE 14

14

reject

• 3.

accept

• then
• in
• path
• a
• is
• if
• 2.
• f
• n

permutatio

• a
• guess
• 1.

G V

✁

π

Example:UNO

…itmightmakeawrongguess,but ifthereexistsasolution,atleastoneguesswillfindit! Awayofcapturesuch“algorithms”: Non-deterministicTMs.

DeterministicTMs

} , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × K K δ

Weemphasizedthefactthatdet.TMsusea transitionfunction.

Non-Deterministicand DeterministicTMs

} , , { R}) A, {H, ( : − → ← × Σ × ∪ → Σ × K K δ

Weemphasizedthefactthatdet.TMsusea transitionfunction.

} , , { R}) A, {H, ( − → ← × Σ × ∪ × Σ × ⊆ K K δ

Forareason.Non-det.TMsusea transitionrelation.

Non-DeterministicTMs

} , , { R}) A, {H, ( − → ← × Σ × ∪ × Σ × ⊆ K K δ

Forareason.Non-det.TMsusea transitionrelation.

*

, , , Σ ∈ ∈ > =< u w K q u w q C

• and
• with

: same

• the
• still
• are
• ions

Configurat

Buthowdoesitrun???

AComputationalStep

n n n n n

u u u u w w w w dir dir w q w q u w q C ... ' ' ... ' , ' , ' ) , ( , ,

2 1 1 1

= = =→ > =< > =<

−

• then
• If

.

• and
• δ

ThetransitionfromCtoC’isasinglestep.

y. analogousl

• and
• For

− = =→ dir dir . ' , ' , ' ' > =< u w q C C

• yield

to

• said
• is
• ANondeterministic

ComputationalStep

n n n n n

u u u u w w w w dir dir w q w q u w q C ... ' ' ... ' , ' , ' , , , ,

2 1 1 1

= = =→ >∈ < > =<

−

• then
• If

.

• and
• δ

ThetransitionfromCtoC’isasinglestep.

y. analogousl

• and
• For

− = =→ dir dir . ' , ' , ' ' > =< u w q C C

• yield

to

• said
• is

SLIDE 15

15 Example

H S #:#/- #:0/→ #:1/→

> =< ε , # S, C

Example

H S #:#/- #:0/→ #:1/→

> =< ε , # S, C > =< ε , # 1 S, C > =< ε , # S, C > =< ε , # H, C