Turing Machines Wen-Guey Tzeng Computer Science Department - - PowerPoint PPT Presentation

Turing Machines

Wen-Guey Tzeng Computer Science Department National Chiao Tung University

Alan Turing

• One of the first to conceive a machine

that can run computation mechanically without human

• intervention. We call “algorithmic

computation” now.

• The first to conceive a universal

machine – that can run programs

• Break German Enigma Machine

(encryption) in WWII

• Turing Award: the most prestigious

award in CS

• Bio movie: The Imitation Game, 2015

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1912-1954

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TM Model

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Definition

• A Turing machine (TM) M is defined by

M=(Q, , , , q0, ฀, F)

– Q: the set of internal states – : the input alphabet – : the tape alphabet – : QQ{L,R}, the transition function – ฀: blank symbol – q0Q: the initial state – FQ: the set of final states

• Note: this TM is deterministic.

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Transition function 

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• (q0, a)=(q1, d, R)

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Example:

• M=({q0, q1}, {a,b}, {a, b, ฀}, , q0, ฀, {q1})

– (q0, a)=(q0, b, R) – (q0, b)=(q0, b, R) – (q0, ฀ )=(q1, ฀, L)

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Transition graph

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– (q0, a)=(q0, b, R) – (q0, b)=(q0, b, R) – (q0, ฀ )=(q1, ฀, L)

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TM runs forever

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• Check TM on input aab

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Instantaneous description

• x1qx2 = a1a2…ak-1 q akak+1…an
• Also, called “configuration”

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Example:

• M=({q0, q1}, {a,b}, {a, b, ฀}, , q0, ฀, {q1})

– (q0, a)=(q0, b, R) – (q0, b)=(q0, b, R) – (q0, ฀ )=(q1, ฀, L)

• ID: q0aa, bq0a, bbq0฀, bq1b
• Move:

– q0aa ├ bq0a ├ bbq0฀├ bq1b (halt)

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TM halts

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• A TM on input w “halts” if q0w reaches a

configuration that  is not defined.

– A TM on an input w may not halt. (Run forever)

• Halting configuration: x1qx2 (halt)

– Halting does not imply acceptance

• Accepting configuration: x1qfx2 (halt)

– Must halt and enter a final state qf – Acceptance implies halting

• Infinite loop (non-stop): x1qx2 ├* 

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A computation

• A computation is a sequence of configurations

that leads to a halt state.

• q0aa ├ bq1b ├ bbq1฀├ bq1b (halt)

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Standard TM

• It is deterministic.
• There is only one tape.
• The tape is unbounded in both directions.
• All other cells are filled with blanks.
• No output devices.
• The input is on the tape in the beginning.
• The left string on the tape can be viewed as
• utput.

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TM as

• Language acceptors
• Transducers

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TM as language acceptors

• A string w is accepted by M if

– M on input w enters a final state and halts. – Note: no need to read the entire input

• The language accepted by M

L(M)={w+: q0w├* x1qfx2 (halt), qfF, x1, x2* }

– x1qfx2 (halt) is a halt configuration.

• A language accepted by some TM is called a

“recursively enumerable (r.e.) ” set.

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Example:

• Design a TM to accept the strings of form 00*
• The transition function

– (q0, 0)=(q1, 0, R) – (q1, 0)=(q1, 0, R) – (q1, ฀)=(q2, ฀, R)

• F={q2}

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Example:

• Design a TM to accept L={anbn : n1}
• Idea:

– Loop

• Mark leftmost ‘a’ as ‘x’ in the tape
• Move all the way right to mark ‘b’ as ‘y’

– After marking all a’s, move all the way right to find ฀

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• Example, w=aabb
• Mark leftmost ‘a’ as ‘x’ in the tape

– (q0, a)=(q1, x, R)

• Move all the way right to mark ‘b’ as ‘y’

– (q1, a)=(q1, a, R) – (q1, y)=(q1, y, R) – (q1, b)=(q2, y, L)

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. . . ฀ a a b b ฀ . . .

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• Move back to find leftmost ‘a’

– (q2, y)=(q2, y, L) – (q2, a)=(q2, a, L) – (q2, x)=(q0, x, R)

• Repeat until all a’s are marked as x’s

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. . . ฀ x a y b ฀ . . .

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• After marking all a’s as x’s, move all the way

right to find ฀

– (q0, y)=(q3, y, R) – (q3, y)=(q3, y, R) – (q3, ฀)=(q4, ฀, R)

– If ‘b’ is encountered during the action of moving right, TM halts on a non-final state.

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. . . ฀ x x y y ฀ . . .

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Example:

• Design a TM to accept L={w{a,b}+ : na(w)=nb(w)}
• Idea:

– Loop

• Pose at the beginning of the string
• Mark leftmost ‘a’ as ‘x’ in the tape
• Move all the way right to find the first b and mark it as

– After marking all symbols as X, enter the final state

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TM as transducers

• To compute a function f(w)=w’, where w, w’

are strings

– q0w├* qf w’ (halt)

• A function f:DR is TM-computable

if there is a TM M=(Q, , , , q0, ฀, F) such that q0w├* qf f(w) (halt) for all wD

• We will show in the following that TM can

compute complicated functions

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Example: (integer adder)

• Given two positive integer x and y, compute

x+y

– x and y are unary-represented

• Eg. X=3, w(x)=111

– Input: w(x) 0 w(y) – Output: w(x+y) 0 – That is, q0 w(x)0w(y)├qf w(x+y)0

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• The transition function
• (q0, 1)=(q0, 1, R)
• (q0, 0)=(q1, 1, R)
• (q1, 1)=(q1, 1, R)
• (q1, ฀)=(q2, ฀, L)
• (q2, 1)=(q3, 0, L)
• (q3, 1)=(q3, 1, L)
• (q3, ฀ )=(q4, ฀, R)
• Run q0111011

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Example: (string copier)

• Design a TM to compute q0w├* qf ww, w=1+
• Steps
• 1. Replace every 1 by x
• 2. Find the rightmost x and replace it with 1
• 3. Travel to the right end and create 1 (in replace
• f ฀)
• 4. Repeat 2 and 3 until there are no x’s

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Run q011

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Examples: (comparer)

• Design a TM to compute

q0w(x)0w(y)├* qY w(x)0w(y) if xy q0w(x)0w(y)├* qN w(x)0w(y) if x<y

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Combining TM for complicated tasks

• Design a TM to compute

f(x,y)=x+y if xy, and f(x,y)=0 if x<y

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• qC,0w(x)0w(y)

├* qA,0 w(x)0w(y) if xy ├* qE,0 w(x)0w(y) if x<y

• qA,0w(x)0w(y) ├* qA,f w(x+y)0
• qE,0w(x)0w(y) ├* qE,f 0

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Example: (if-then-else)

• if a then qj else qk

– (qi, a)=(qj0, a, R), for qiQ – (qj0, c)=(qj, c, L), for all c – (qi, b)=(qk0, b, R) for b-{a} – (qk0, c)=(qk, c, L), for all c

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Example: (integer multiplier)

• Design a TM for q0w(x)0w(y)├* qf w(xy)
• Steps

– Loop until x contains no more 1’s

• Find a 1 in x and replace it with another a
• Replace leftmost 0 by 0y

– Replace all a’s with 1’s

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Macroinstructionssubprogram

• A calls B

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Turing’s Thesis

• Mechanical computation

– Computation without human intervention

• Turing Thesis: any computation that can be

carried out by mechanical means can be carried out by some Turing machine.

• Also called “Church-Turing Thesis”

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• Reasoning:

– Anything that can be done by an existing digital computer can be done by a TM – No one has yet been able to suggest a problem, solvable by algorithms, but cannot be solved by some TM – Other models are proposed. But, they are no more powerful than TM’s.

• Even quantum computers are no more powerful than

TM’s in terms of the power language acceptance

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Algorithms

• An algorithm for a functin f:DR

is a TM such that

q0d├* qff(d) for all dD

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