Turing Machines

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Turing Machines

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The Language Hierarchy
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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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Input string
Blank symbol
......
......
head
Remark the input string is never empty
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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 lambda transitions allowed
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Partial Transition Function
Example
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......
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 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
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
A Turing machine for language
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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
Infinite loop
Time 4
Time 5
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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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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
A Move
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Time 4
Time 5
Time 6
Time 7
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Equivalent notation
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Initial configuration
Input string
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The Accepted Language
For any Turing Machine
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