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Turing Machines
Wen-Guey Tzeng
Computer Science Department
National Chiao Tung University
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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
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 
• (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
– (q0, a)=(q0, b, R)
– (q0, b)=(q0, b, R)
– (q0,  )=(q1, , L)
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TM runs forever
• 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 ├ bq0b ├ bbq0├ bq1b (halt)
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TM halts
• 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 mean acceptance
• Accepting configuration: x1qfx2 (halt)
– Must halt and enter a final state qf
• 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
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•
•
•
•
•
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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
output.
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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.
• 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, a)=(q0, 0, R)
– (q0, )=(q1, , R)
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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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• 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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• Move back to mark leftmost ‘a’ as ‘x’
– (q2, y)=(q2, y, L)
– (q2, a)=(q2, a, L)
– (q2, x)=(q0, x, R)
• After marking all a’s, move all the way right to
find 
– (q0, y)=(q3, y, R)
– (q3, y)=(q3, y, R)
– (q3, )=(q4, , R)
– If in the middle ‘b’ is encountered, TM stops.
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• Run q0aabb
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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
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Example:
• 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)0w(y)
– Output: w(x+y)0
– That is, q0w(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, , R)
(q2, 1)=(q3, 0, R)
(q3, 1)=(q3, 1, L)
(q3,  )=(q4, , R)
• Run q0111011
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Example:
• 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
of )
4. Repeat 2 and 3 until there are no x’s
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Run q011
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Examples:
• 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
• Design a TM to compute
q0w(x)0w(y)├* qf w(xy)
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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 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:
• 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├* qf f(d) for all dD
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