Turing Machines

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Turing Machines

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The Language Hierarchy
?
?
Context-Free Languages
Regular Languages
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Languages accepted by Turing Machines
Context-Free Languages
Regular Languages
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A Turing Machine
Tape
......
......
Read-Write head
Control Unit
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The Tape
No boundaries -- infinite length
......
......
Read-Write head
The head moves Left or Right
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......
......
Read-Write head
The head at each 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
3. Moves Left
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Time 1
......
......
Time 2
......
......
1. Reads
2. Writes
3. Moves Right
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The Input String
Input string
Blank symbol
......
......
head
Head starts at the leftmost position of the input
string
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States Transitions
Write
Read
Move Left
Move Right
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Example
Time 1
......
......
current state
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Time 1
......
......
Time 2
......
......
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Example
Time 1
......
......
Time 2
......
......
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Example
Time 1
......
......
Time 2
......
......
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Determinism
Turing Machines are deterministic
Not Allowed
Allowed
No epsilon transitions allowed
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Partial Transition Function
Example
......
......
Allowed
No transition for input symbol
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Halting
The machine halts if there are no possible
transitions to follow
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Example
......
......
No possible transition
HALT!!!
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Final States
Allowed
Not Allowed

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

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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
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Turing Machine Example
A Turing machine that accepts language a
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Time 0
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Time 1
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Time 2
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Time 3
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Time 4
Halt Accept
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Rejection Example
Time 0
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Time 1
No possible Transition
Halt Reject
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Infinite Loop Example
Another Turing machine for language aand is
this one correct???
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Time 0
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Time 1
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Time 2
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Time 2
Time 3
Time 4
Time 5
... Infinite Loop
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• Because of the infinite loop
• The final state cannot be reached
• The machine never halts
• The input is not accepted

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Another Turing Machine Example
Turing machine for the language
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Time 0
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Time 1
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Time 2
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Time 3
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Time 4
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Time 5
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Time 6
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Time 7
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Time 8
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Time 9
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Time 10
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Time 11
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Time 12
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Time 13
Halt Accept
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Observation
If we modify the machine for the language
we can easily construct a machine for the
language
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Formal Definitionsfor Turing Machines

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Transition Function
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Transition Function
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Turing Machine
Input alphabet
Tape alphabet
States
Transition function
Final states
Initial state
blank
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Configuration
Instantaneous description
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Time 4
Time 5
ayb
q
x
xayb
q
A Move
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0
2
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Time 4
Time 5
Time 6
Time 7
b
q
xxy
yb
q
xx
ayb
q
x
xayb
q
?
?
?
1
1
0
2
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b
q
xxy
yb
q
xx
ayb
q
x
xayb
q
?
?
?
1
1
0
2

b
q
xxy
xayb
q
Equivalent notation
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1
2
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Initial configuration
Input string
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The Accepted Language
For any Turing Machine




)
(
x
q
x
w
q
w
M
L

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2
1
0
f
Initial state
Final state
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Standard Turing Machine
The machine we described is the standard

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

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Implementation-level descriptions
ww contains an equal number of 0s and 1s
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Implementation-level descriptions
ww contains an equal number of 0s and 1s
On input string w
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Implementation-level descriptions
ww contains an equal number of 0s and 1s

• On input string w
• Scan the tape mark the 1st 0 which is unmarked.
  If none is found, go to 4. Otherwise, move the
  head back to the front of the tape.

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Implementation-level descriptions
ww contains an equal number of 0s and 1s

• On input string w
• Scan the tape mark the 1st 0 which is unmarked.
  If none is found, go to 4. Otherwise, move the
  head back to the front of the tape.
• Scan the tape mark the 1st 1 which is unmarked.
  If none is found, reject.

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Implementation-level descriptions
ww contains an equal number of 0s and 1s

• On input string w
• Scan the tape mark the 1st 0 which is unmarked.
  If none is found, go to 4. Otherwise, move the
  head back to the front of the tape.
• Scan the tape mark the 1st 1 which is unmarked.
  If none is found, reject.
• Move the head back to the front of the tape go
  to 1.

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Implementation-level descriptions
ww contains an equal number of 0s and 1s

• On input string w
• Scan the tape mark the 1st 0 which is unmarked.
  If none is found, go to 4. Otherwise, move the
  head back to the front of the tape.
• Scan the tape mark the 1st 1 which is unmarked.
  If none is found, reject.
• Move the head back to the front of the tape go
  to 1.
• Move the head back to the front of the tape.
  Scan the tape to see if any unmarked 1s remain.
  If none are found, accept otherwise, reject.

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Try this one
ww contains twice as many 0s as 1s

• On input string w
• Scan the tape ...
• (Hint, you can mark and double mark.)

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Try this one
ww anbn, n ? 0

• On input string w
• Scan the tape ...