1. Busch Complexity Lectures: Undecidable Problems (unsolvable problems)
Prof. Busch - LSU

2. Decidable Languages Recall that: A language is decidable,
if there is a Turing machine (decider)
that accepts the language and
halts on every input string
Decision
On Halt:
Turing Machine
YES
Accept
Input
string
Decider for NO
Reject
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3. A computational problem is decidable
if the corresponding language is decidable
We also say that the problem is solvable
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4. Corresponding Language: (Decidable)
Problem:
Does DFA accept
the empty language ?
Corresponding Language:
(Decidable)
Description of DFA as a string
(For example, we can represent as a
binary string, as we did for Turing machines)
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5. Determine whether there is a path from
Decider for :
On input :
Determine whether there is a path from
the initial state to any accepting state
DFA
DFA
Decision:
Accept
Reject
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6. Corresponding Language: (Decidable)
Problem:
Does DFA accept
a finite language?
Corresponding Language:
(Decidable)
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7. Check if there is a walk with a cycle
Decider for :
On input :
Check if there is a walk with a cycle
from the initial state to an accepting state
DFA
DFA
infinite
finite
Decision:
Accept
Reject
(NO)
(YES)
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8. Corresponding Language: (Decidable)
Problem:
Does DFA accept string ?
Corresponding Language:
(Decidable)
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9. Decider for : On input string : Run DFA on input string If accepts
Then accept (and halt)
Else reject (and halt)
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10. accept the same language?
Problem:
Do DFAs and
accept the same language?
Corresponding Language:
(Decidable)
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11. Let be the language of DFA
Decider for :
On input :
Let be the language of DFA
Construct DFA such that:
(combination of DFAs)
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12. and
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13. or
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14. which is a solvable problem for DFAs:
Therefore, we only need
to determine whether
which is a solvable problem for DFAs:
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15. Undecidable Languages
undecidable language = not decidable language
There is no decider:
there is no Turing Machine
which accepts the language
and makes a decision (halts)
for every input string
(machine may make decision for some input strings)
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16. For an undecidable language, the corresponding problem is
undecidable (unsolvable):
there is no Turing Machine (Algorithm)
that gives an answer (yes or no)
for every input instance
(answer may be given for some input instances)
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17. We have shown before that there are undecidable languages:
Turing-Acceptable
Decidable
is Turing-Acceptable and undecidable
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18. We will prove that two particular problems are unsolvable:
Membership problem
Halting problem
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19. Corresponding language:
Membership Problem
Input:
Turing Machine
String
Question:
Does accept ?
Corresponding language:
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20. We will assume that is decidable; We will then prove that
Theorem:
is undecidable
(The membership problem is unsolvable)
Proof:
Basic idea:
We will assume that is decidable;
We will then prove that
every Turing-acceptable language
is also decidable
A contradiction!
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21. Suppose that is decidable
Input
string
Decider
for
accepts
YES NO
rejects
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22. Let be a Turing recognizable language
Let be the Turing Machine that accepts
We will prove that is also decidable:
we will build a decider for
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23. String description of Decider for Decider for YES accept accepts ? NO
This is hardwired and copied on the tape next to
input string s, and then the pair is input to H
Decider for
Decider for
YES
accept
(and halt)
accepts ? NO
reject
Input
string
(and halt)
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24. But there is a Turing-Acceptable language which is undecidable
Therefore,
is decidable
Since is chosen arbitrarily, every Turing-Acceptable language is decidable
But there is a Turing-Acceptable language
which is undecidable
Contradiction!!!!
END OF PROOF
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25. We have shown:
Undecidable
Decidable
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26. We can actually show:
Turing-Acceptable
Decidable
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27. Turing machine that accepts :
is Turing-Acceptable
Turing machine that accepts :
1. Run on input
2. If accepts
then accept
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28. processing input string ?
Halting Problem
Input:
Turing Machine
String
Question:
Does halt while
processing input string ?
Corresponding language:
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29. Suppose that is decidable; we will prove that
Theorem:
is undecidable
(The halting problem is unsolvable)
Proof:
Basic idea:
Suppose that is decidable;
we will prove that
every Turing-acceptable language
is also decidable
A contradiction!
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30. Suppose that is decidable
Input
string
halts on
input
YES
Decider for NO
doesn’t halt
on input
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31. Let be a Turing-Acceptable language
Let be the Turing Machine that accepts
We will prove that is also decidable:
we will build a decider for
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32. Decider for NO reject halts on ? YES accept Run with input reject
and halt
YES
Input
string
halts
and accepts
accept
and halt
Run
with input
halts
and rejects
reject
and halt
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33. But there is a Turing-Acceptable language which is undecidable
Therefore,
is decidable
Since is chosen arbitrarily, every Turing-Acceptable language is decidable
But there is a Turing-Acceptable language
which is undecidable
Contradiction!!!!
END OF PROOF
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34. An alternative proof Theorem: is undecidable Proof: Basic idea:
(The halting problem is unsolvable)
Proof:
Basic idea:
Assume for contradiction that
the halting problem is decidable;
we will obtain a contradiction
using a diagonilization technique
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35. Suppose that is decidable
Input
string
Decider
for
YES
halts on NO
doesn’t
halt on
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36. Looking inside Decider for YES Input string: halts on ? NO
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37. If halts on input Then Loop Forever Else Halt
Construct machine :
Loop forever
YES
halts on ? NO
If halts on input Then Loop Forever
Else Halt
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38. Construct machine : Copy on tape If halts on input Then loop forever
Else halt
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39. Then loops forever on input
Run with input itself
Copy
on tape If
halts on input
Then loops forever on input
Else halts on input
CONTRADICTION!!!
END OF PROOF
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40. We have shown:
Undecidable
Decidable
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41. We can actually show:
Turing-Acceptable
Decidable
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42. Turing machine that accepts :
is Turing-Acceptable
Turing machine that accepts :
1. Run on input
2. If halts on
then accept
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