Languages and Finite Automata

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Formal Languages Turing Machines Hinrich
Schütze IMS, Uni Stuttgart, WS 2006/07 Slides
based on RPI CSCI 2400 Thanks to Costas Busch
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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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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
......
......
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
Language?
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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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Language?
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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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Another Turing Machine Example
Language?
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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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Machine for L vvv in a,b ?

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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

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Computing FunctionswithTuring Machines

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A function
has
Range
Domain
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A function may have many parameters
Example
Addition function
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Integer Domain
Decimal
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Binary
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Unary
11111
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Definition
A function is computable if there is
a Turing Machine such that
Initial configuration
Final configuration
final state
initial state
For all
Domain
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In other words
A function is computable if there is
a Turing Machine such that
Initial Configuration
Final Configuration
For all
Domain
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Example
is computable
The function
are integers
Turing Machine
Input string
unary
Output string
unary
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Start
initial state
The 0 is the delimiter that separates the two
numbers
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Start
initial state
Finish
final state
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The 0 helps when we use the result for other
operations
Finish
final state
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Turing machine for function
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Execution Example
Time 0
(2)
(2)
Final Result
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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
HALT accept
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Another Example
is computable
The function
is integer
Turing Machine
Input string
unary
Output string
unary
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Start
initial state
Finish
final state
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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
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Turing Machine for
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Example
Start
Finish
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Another Example
if
The function
if
is computable
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Turing Machine for
if
if
Input
or
Output
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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

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Combining Turing Machines

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Block Diagram
Turing Machine
input
output
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Example
if
if
Adder
Comparer
Eraser