1. Turing Machines

2. Invented by Alan Turing in 1936.
A simple mathematical model of a general
purpose computer.
It is capable of performing any calculation
which can be performed by any computing
machine.

3. The Language Hierarchy
n n n ?
a b c ww ?
Context-Free Languages
n n R
a b ww
Regular Languages
a* a *b *

4. Languages accepted by
Turing Machines
n n n
a b c ww
Context-Free Languages
n n R
a b ww NDPA
Regular Languages
Finite
a * a *b * Automata

5. A Turing Machine
Tape
...... ......
Read-Write head
Control Unit

6. The Tape
No boundaries -- infinite length
...... ......
Read-Write head
The head moves Left or Right

7. ...... ......
Read-Write head
The head at each time step:
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right

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

9. Time 1
...... a b k c ......
Time 2
...... a f k c ......
1. Reads b
2. Writes f
3. Moves Right

10. The Input String
Input string Blank symbol
...... a b a c ......
head
Head starts at the leftmost position
of the input string
Are treated as left and right brackets for the
input written on the tape.

11. States & Transitions
Read Write
Move Left
q1 a b, L q2
Move Right
q1 a b, R q2

12. Example:
Time 1
...... a b a c ......
q1
current state
q1 a b, R q2

13. Time 1
...... a b a c ......
q1
Time 2
...... a b b c ......
q2
q1 a b, R q2

14. Example:
Time 1
...... a b a c ......
q1
Time 2
...... a b b c ......
q2
q1 a b, L q2

15. Example:
Time 1
...... a b a c ......
q1
Time 2
...... a b b c g ......
q2
q1 g, R q2

16. Determinism
Turing Machines are deterministic
Allowed Not Allowed
a b, R q2 a b, R q2
q1 q1
q3 a d, L q3
b d, L
No lambda transitions allowed

17. Partial Transition Function
Example:
...... a b a c ......
q1
a b, R q2 Allowed:
q1 No transition
for input symbol c
b d, L q3

18. Halting
The machine halts if there are
no possible transitions to follow

19. Example:
...... a b a c ......
q1
a b, R q2
No possible transition
q1
HALT!!!
b d, L q3

20. Final States
q1 q2 Allowed
q1 q2 Not Allowed
• Final states have no outgoing transitions
• In a final state the machine halts

21. Acceptance
If machine halts
Accept Input
in a final state
If machine halts
in a non-final state
Reject Input or
If machine enters
an infinite loop

22. Turing Machine Example
A Turing machine that accepts the language:
aa *
a a, R
,L
q0 q1

23. Time 0 a a a
q0
a a, R
,L
q0 q1

24. Time 1 a a a
q0
a a, R
,L
q0 q1

25. Time 2 a a a
q0
a a, R
,L
q0 q1

26. Time 3 a a a
q0
a a, R
,L
q0 q1

27. Time 4 a a a
q1
a a, R Halt & Accept
,L
q0 q1

28. Rejection Example
Time 0 a b a
q0
a a, R
,L
q0 q1

29. Time 1 a b a
q0
No possible Transition
a a, R Halt & Reject
,L
q0 q1

30. Infinite Loop Example
b b, L
a a, R
,L
q0 q1

31. Time 0 a b a
q0
b b, L
a a, R
,L
q0 q1

32. Time 1 a b a
q0
b b, L
a a, R
,L
q0 q1

33. Time 2 a b a
q0
b b, L
a a, R
,L
q0 q1

34. Time 2 a b a
q0
Time 3 a b a
q0
Time 4 a b a
q0
Time 5 a b a
q0
... Infinite Loop

35. Because of the infinite loop:
•The final state cannot be reached
•The machine never halts
•The input is not accepted

36. Another Turing Machine
Example n n
Turing machine for the language {a b }
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

37. Time 0 a a b b
q0
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

38. Time 1 x a b b
q1
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

39. Time 2 x a b b
q1
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

40. Time 3 x a y b
q2
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

41. Time 4 x a y b
q2
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

42. Time 5 x a y b
q0
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

43. Time 6 x x y b
q1
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

44. Time 7 x x y b
q1
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

45. Time 8 x x y y
q2
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

46. Time 9 x x y y
q2
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

47. Time 10 x x y y
q0
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

48. Time 11 x x y y
q3
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

49. Time 12 x x y y
q3
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

50. Time 13 x x y y
q4
Halt & Accept
q4 y y, R y y, L
y y, R a a, R a a, L
,L
y y, R a x, R b y, L
q3 q0 q1 q2
x x, R

51. Observation:
If we modify the
machine for the language n n
{a b }
we can easily construct
n n n
a machine for the language {a b c }

52. Formal Definitions
for
Turing Machines

53. Transition Function
q1 c d, L q2
(q1, c) (q2 , d , L)

54. Turing Machine:
Input Tape
alphabet alphabet
States
M (Q, , , , q0 , , F )
A partial Final
Transition states
function Initial Blank : a special symbol
state Of

55. Configuration
c a b a
q1
Instantaneous description: ca q1 ba

56. Time 4 Time 5
x a y b x a y b
q2 q0
A Move: q2 xayb  x q0 ayb

57. Time 4 Time 5
x a y b x a y b
q2 q0
Time 6 Time 7
x x y b x x y b
q1 q1
q2 xayb  x q0 ayb  xx q1 yb  xxy q1 b

58. q2 xayb  x q0 ayb  xx q1 yb  xxy q1 b
Equivalent notation: q2 xayb  xxy q1 b

59. Initial configuration: q0 w
Input string
w
a a b b
q0

60. The Accepted Language
For any Turing Machine M
L( M ) {w : q0 w  x1 q f x2 }
Initial state Final state

61. Standard Turing Machine
The machine we described is the standard:
• Deterministic
• Infinite tape in both directions
•Tape is the input/output file

62. Design a Turing machine to recognize all
strings in which 010 is present as a
substring. 0,0,R
0,0,R 1,1,R 0,0,R
q0 q1 q2 H
1,1,R
1,1, R

63. DFA for the previous language
0
0 1 0
q0 q1 q2
1
1
0,1

64. Turing machine for odd no of 1’s
1, 1 , R
1, b , R
1, b , R

65. Recursively Enumerable
and
Recursive
Languages

66. Definition:
A language is recursively enumerable
if some Turing machine accepts it

67. Let L be a recursively enumerable language
and M the Turing Machine that accepts it
For string w:
if w L then M halts in a final state
if w L then M halts in a non-final state
or loops forever

68. Definition:
A language is recursive
if some Turing machine accepts it
and halts on any input string
In other words:
A language is recursive if there is
a membership algorithm for it

69. Let L be a recursive language
and M the Turing Machine that accepts it
For string w:
if w L then M halts in a final state
if w L then M halts in a non-final state

70. We will prove:
1. There is a specific language
which is not recursively enumerable
(not accepted by any Turing Machine)
2. There is a specific language
which is recursively enumerable
but not recursive

71. Non Recursively Enumerable
Recursively Enumerable
Recursive

72. We will first prove:
• If a language is recursive then
there is an enumeration procedure for it
• A language is recursively enumerable
if and only if
there is an enumeration procedure for it

73. The Chomsky Hierarchy

74. Unrestricted Grammars:
Productions
u v
String of variables String of variables
and terminals and terminals

75. Example unrestricted grammar:
S aBc
aB cA
Ac d

76. Theorem:
A language L is recursively enumerable
if and only if L is generated by an
unrestricted grammar

77. Context-Sensitive Grammars:
Productions
u v
String of variables String of variables
and terminals and terminals
and: |u| |v|

78. The language n n n
{a b c }
is context-sensitive:
S abc | aAbc
Ab bA
Ac Bbcc
bB Bb
aB aa | aaA

79. The language n n n
{a b c }
is context-sensitive:
S abc | aAbc
Ab bA
Ac Bbcc
bB Bb
aB aa | aaA

80. Theorem:
A language L is context sensistive
if and only if
L is accepted by a Linear-Bounded automaton

81. Observation:
There is a language which is context-sensitive
but not recursive

82. The Chomsky Hierarchy
Non-recursively enumerable
Recursively-enumerable
Recursive
Context-sensitive
Context-free
Regular