Turing Machine

1. © 2020 Wael Badawy
1
Turing Machines
1
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2. © 2020 Wael Badawy
Copyright © 2020 Wael Badawy. All rights reserved
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2

3. © 2020 Wael Badawy

4. © 2020 Wael Badawy
The Language Hierarchy
4
*a
Regular Languages
Context-Free Languages
nn
ba R
ww
nnn
cba ww?
**ba
?

5. © 2020 Wael Badawy
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*a
Regular Languages
Context-Free Languages
nn
ba R
ww
nnn
cba ww
**ba
Languages accepted by
Turing Machines

6. © 2020 Wael Badawy
A Turing Machine
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............
Tape
Read-Write head
Control Unit

7. © 2020 Wael Badawy
The Tape
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............
Read-Write head
No boundaries -- infinite length
The head moves Left or Right

8. © 2020 Wael Badawy
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............
Read-Write head
The head at each transition (time step):
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right

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............
Example:
Time 0
............
Time 1
1. Reads
2. Writes
a a cb
a b k c
a
k
3. Moves Left

10. © 2020 Wael Badawy
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............
Time 1
a b k c
............
Time 2
a k cf
1. Reads
2. Writes
b
f
3. Moves Right

11. © 2020 Wael Badawy
The Input String
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............ à à à à
Blank symbol
head
àa b ca
Head starts at the leftmost position
of the input string
Input string

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States & Transitions
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1q 2qLba ,®
Read Write
Move Left
1q 2qRba ,®
Move Right

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Example:
1q 2qRba ,®
............ à à à ààa b ca
Time 1
1q
current state

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............ à à à ààa b ca
Time 1
1q 2qRba ,®
............ à à à ààa b cb
Time 2
1q
2q

15. © 2020 Wael Badawy
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............ à à à ààa b ca
Time 1
1q 2qLba ,®
............ à à à ààa b cb
Time 2
1q
2q
Example:

16. © 2020 Wael Badawy
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............ à à à ààa b ca
Time 1
1q 2qRg,®à
............ à à à àga b cb
Time 2
1q
2q
Example:

17. © 2020 Wael Badawy
Determinism
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1q
2qRba ,®
Allowed Not Allowed
3qLdb ,®
1q
2qRba ,®
3qLda ,®
Turing Machines are deterministic

18. © 2020 Wael Badawy
Partial Transition Function
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1q
2qRba ,®
3qLdb ,®
............ à à à ààa b ca
1q
Example:
No transition
for input symbol c
Allowed:

19. © 2020 Wael Badawy
Halting
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The machine halts in a state if there is
no transition to follow

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Halting Example 1:
............ à à à ààa b ca
1q
1q No transition from
HALT!!!
1q

21. © 2020 Wael Badawy
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Halting Example 2:
............ à à à ààa b ca
1q
1q
2qRba ,®
3qLdb ,®
No possible transition
from and symbol
HALT!!!
1q c

22. © 2020 Wael Badawy
Accepting States
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1q 2q Allowed
1q 2q Not Allowed
•Accepting states have no outgoing transitions
•The machine halts and accepts

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Acceptance
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Accept Input
If machine halts
in an accept state
Reject Input
If machine halts
in a non-accept state
or
If machine enters
an infinite loop
string
string

24. © 2020 Wael Badawy
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Observation:
In order to accept an input string,
it is not necessary to scan all the
symbols of the input string

25. © 2020 Wael Badawy
Turing Machine Example
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Accepts the language: *a
0q
Raa ,®
L,à®à
1q
Input alphabet },{ ba=S

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à à à àaaTime 0
0q
a
0q
Raa ,®
L,à®à
1q

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à à à àaaTime 1
0q
a
0q
Raa ,®
L,à®à
1q

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à à à àaaTime 2
0q
a
0q
Raa ,®
L,à®à
1q

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à à à àaaTime 3
0q
a
0q
Raa ,®
L,à®à
1q

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à à à àaaTime 4
1q
a
0q
Raa ,®
L,à®à
1q
Halt & Accept

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Rejection Example
0q
Raa ,®
L,à®à
1q
à à à àbaTime 0
0q
a

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0q
Raa ,®
L,à®à
1q
à à à àbaTime 1
0q
a
No possible Transition
Halt & Reject

33. © 2020 Wael Badawy
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Accepts the language: *a
0q
but for input alphabet }{a=S
A simpler machine for same language

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à à à àaaTime 0
0q
a
0q
Halt & Accept
Not necessary to scan input

35. © 2020 Wael Badawy
Infinite Loop Example
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0q
Raa ,®
L,à®à
1q
Lbb ,®
A Turing machine
for language *)(* baba ++

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à à à àbaTime 0
0q
a
0q
Raa ,®
L,à®à
1q
Lbb ,®

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à à à àbaTime 1
0q
a
0q
Raa ,®
L,à®à
1q
Lbb ,®

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à à à àbaTime 2
0q
a
0q
Raa ,®
L,à®à
1q
Lbb ,®

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à à à àbaTime 2
0q
a
à à à àbaTime 3
0q
a
à à à àbaTime 4
0q
a
à à à àbaTime 5
0q
a
Infiniteloop

40. © 2020 Wael Badawy
40
Because of the infinite loop:
•The accepting state cannot be reached
•The machine never halts
•The input string is rejected

41. © 2020 Wael Badawy
Another Turing Machine
Example
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Turing machine for the language }{ nn
ba
0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
1³n

42. © 2020 Wael Badawy
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Match a’s with b’s:
Repeat:
replace leftmost a with x
find leftmost b and replace it with y
Until there are no more a’s or b’s
If there is a remaining a or b reject
Basic Idea:

43. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àba
0q
a bTime 0 à

44. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àbx
1q
a b àTime 1

45. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àbx
1q
a b àTime 2

46. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
2q
a b àTime 3

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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
2q
a b àTime 4

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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
0q
a b àTime 5

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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
1q
x b àTime 6

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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
1q
x b àTime 7

51. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx x y
2q
àTime 8

52. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx x y
2q
àTime 9

53. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
0q
x y àTime 10

54. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
3q
x y àTime 11

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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
3q
x y àTime 12

56. © 2020 Wael Badawy
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0q 1q 2q3q
Rxa ,®
Raa ,®
Ryy ,®
Lyb ,®
Laa ,®
Lyy ,®
Rxx ,®
Ryy ,®
Ryy ,®
4q
L,à®à
à àyx
4q
x y à
Halt & Accept
Time 13

57. © 2020 Wael Badawy
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If we modify the
machine for the language }{ nn
ba
we can easily construct
a machine for the language }{ nnn
cba
Observation:

58. 58
Formal Definitions
for
Turing Machines
58

59. © 2020 Wael Badawy
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Transition Function
1q 2qRba ,®
),,(),( 21 Rbqaq =d

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1q 2qLdc ,®
),,(),( 21 Ldqcq =d
Transition Function

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Turing Machine:
),,,,,,( 0 FqQM àGS= d
States
Input
alphabet
Tape
alphabet
Transition
function
Initial
state
blank
Accept
states

62. © 2020 Wael Badawy
Configuration
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à à àba
1q
a
Instantaneous description:
cà
baqca 1

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à àyx
2q
a b
Time 4
à àyx
0q
a b
Time 5
à à
A Move: aybqxxaybq 02 !
(yields in one move)

64. © 2020 Wael Badawy
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à àyx
2q
a b
Time 4
à àyx
0q
a b
Time 5
à à
bqxxyybqxxaybqxxaybq 1102 !!!
à àyx
1q
x b
Time 6
à àyx
1q
x b
Time 7
à à
A computation

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bqxxyybqxxaybqxxaybq 1102 !!!
bqxxyxaybq 12
*
!Equivalent notation:

66. © 2020 Wael Badawy
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Initial configuration: wq0
à àba
0q
a b à
w
Input string

67. © 2020 Wael Badawy
The Accepted Language
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For any Turing Machine M
}:{)( 210 xqxwqwML f
*
= !
Initial state Accept state

68. © 2020 Wael Badawy
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If a language is accepted
by a Turing machine
then we say that is:
•Turing Acceptable
•Recursively Enumerable
M
L
L
•Turing Recognizable
Other names used: