1. Recursively Enumerable and Recursive Languages
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2. A language is recursively enumerable if some Turing machine accepts it
Let be a recursively enumerable language
and the Turing Machine that accepts it
For string :
then halts in a final state if if
then halts in a non-final state
or loops forever
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3. A language is recursive if some Turing machine accepts it
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
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4. Let be a recursive language
and the Turing Machine that accepts it
For string : if
then halts in a final state if
then halts in a non-final state
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5. 1. There is a specific language which is not recursively enumerable
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
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6. Non Recursively Enumerable
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7. We want to find a language that is not Recursively Enumerable
This language is not accepted by any
Turing Machine
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8. Consider Turing Machines that accept
Consider alphabet
Strings:
Consider Turing Machines that accept
languages over this alphabet. These TM’s are countable
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9. Example language accepted by
Alternative representation
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10. Costas Busch - RPI

11. The members of L consist of the strings with 1’s in the diagonal
Consider the language
The members of L consist of the
strings with 1’s in the diagonal
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12. Costas Busch - RPI

13. consists of the 0’s in the diagonal
Consider the language
consists of the 0’s in the diagonal
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14. Costas Busch - RPI

15. Theorem: The Language is not recursively Enumerable
Proof: Assume for contradiction that
is recursively enumerable
There must exist some machine
that accepts
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16. Question:
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17. Answer:
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18. Question:
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19. Answer:
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20. Question:
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21. Answer:
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22. Similarly:
for any
Because either: or
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23. Therefore, the machine cannot exist
Therefore, the language
is not recursively enumerable
End of Proof
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24. We want to find a language which
Is recursively
enumerable
But not
recursive
There is a
Turing Machine
that accepts
the language
The machine
doesn’t halt
on some input
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25. We will prove that the language
Is recursively enumerable
but not recursive
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26. Costas Busch - RPI

27. is recursively enumerable
Theorem:
The language
is recursively enumerable
Proof: We will give a Turing Machine that accepts
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28. Turing Machine that accepts
For any input string
Compute , for which
Find Turing machine
(using an enumeration procedure
for Turing Machines)
Simulate on input
If accepts, then accept
End of Proof
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29. Recursively enumerable
Observation:
Recursively enumerable
Not recursively enumerable
(Thus, also not recursive)
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30. Non Recursively Enumerable
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