Turing Machines

1
Turing Machines

2
The Language Hierarchy
?
?
Context-Free Languages
Regular Languages
3
Languages accepted by Turing Machines
Context-Free Languages
Regular Languages
4
A Turing Machine
Tape
......
......
Read-Write head
Control Unit
5
The Tape
No boundaries -- infinite length
......
......
Read-Write head
The head moves Left or Right
6
......
......
Read-Write head
The head at each time step 1.
Reads a symbol 2. Writes a
symbol 3. Moves Left or Right
7
Example
Time 0
......
......
Time 1
......
......
1. Reads
2. Writes
3. Moves Left
8
Time 1
......
......
Time 2
......
......
1. Reads
2. Writes
3. Moves Right
9
The Input String
Input string
Blank symbol
......
......
head
Head starts at the leftmost position of the input
string
10
Input string
Blank symbol
......
......
head
Remark the input string is never empty
11
States Transitions
Write
Read
Move Left
Move Right
12
Example
Time 1
......
......
current state
13
Time 1
......
......
Time 2
......
......
14
Example
Time 1
......
......
Time 2
......
......
15
Example
Time 1
......
......
Time 2
......
......
16
Determinism
Turing Machines are deterministic
Not Allowed
Allowed
No lambda transitions allowed
17
Partial Transition Function
Example
......
......
Allowed
No transition for input symbol
18
Halting
The machine halts if there are no possible
transitions to follow
19
Example
......
......
No possible transition
HALT!!!
20
Final States
Allowed
Not Allowed

• Final states have no outgoing transitions
• In a final state the machine halts

21
Acceptance
If machine halts in a final state
Accept Input
If machine halts in a non-final state
or If machine enters an infinite loop
Reject Input
22
Turing Machine Example
A Turing machine that accepts the language
23
Time 0
24
Time 1
25
Time 2
26
Time 3
27
Time 4
Halt Accept
28
Rejection Example
Time 0
29
Time 1
No possible Transition
Halt Reject
30
Infinite Loop Example
A Turing machine for language
31
Time 0
32
Time 1
33
Time 2
34
Time 2
Time 3
Infinite loop
Time 4
Time 5
35

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

36
Formal Definitionsfor Turing Machines

37
Transition Function
38
Transition Function
39
Turing Machine
Input alphabet
Tape alphabet
States
Transition function
Final states
Initial state
blank
40
Configuration
Instantaneous description
41
Time 4
Time 5
A Move
42
Time 4
Time 5
Time 6
Time 7
43
Equivalent notation
44
Initial configuration
Input string
45
The Accepted Language
For any Turing Machine
Initial state
Final state
46
Standard Turing Machine
The machine we described is the standard

• Deterministic
• Infinite tape in both directions
• Tape is the input/output file

47
Computing FunctionswithTuring Machines

48
A function
has
Result Region
Domain
49
A function may have many parameters
Example
Addition function
50
Integer Domain
Decimal
5
Binary
101
Unary
11111
51
Definition
A function is computable if there is
a Turing Machine such that
Initial configuration
Final configuration
final state
initial state
For all
Domain
52
In other words
A function is computable if there is
a Turing Machine such that
Initial Configuration
Final Configuration
For all
Domain
53
Example
is computable
The function
are integers
Turing Machine
Input string
unary
Output string
unary
54
Start
initial state
The 0 is the delimiter that separates the two
numbers
55
Start
initial state
Finish
final state
56
The 0 helps when we use the result for other
operations
Finish
final state
57
Turing machine for function
58
Execution Example
Time 0
(2)
(2)
Final Result
59
Time 0
60
Time 1
61
Time 2
62
Time 3
63
Time 4
64
Time 5
65
Time 6
66
Time 7
67
Time 8
68
Time 9
69
Time 10
70
Time 11
71
Time 12
HALT accept
72
Another Example
is computable
The function
is integer
Turing Machine
Input string
unary
Output string
unary
73
Start
initial state
Finish
final state
74
Turing Machine Pseudocode for

• Replace every 1 with

• Repeat

• Find rightmost , replace it with 1
• Go to right end, insert 1

Until no more remain
75
Turing Machine for
76
Example
Start
Finish
77
Another Example
if
The function
if
is computable
78
Turing Machine for
if
if
Input
or
Output
79
Turing Machine Pseudocode

• Repeat

Match a 1 from with a 1 from
Until all of or is matched

• If a 1 from is not matched
• erase tape, write 1
• else
• erase tape, write 0

80
Combining Turing Machines

81
Block Diagram
Turing Machine
input
output
82
Example
if
if
Adder
Comparer
Eraser
83
Turings Thesis

84
Turings thesis
Any computation carried out by mechanical
means can be performed by a Turing Machine
(1930)
85
Computer Science Law
A computation is mechanical if and only if it
can be performed by a Turing Machine
There is no known model of computation more
powerful than Turing Machines
86
Definition of Algorithm
An algorithm for function is a Turing Machine
which computes
87
Algorithms are Turing Machines
When we say
There exists an algorithm
We mean
There exists a Turing Machine that executes the
algorithm