1. Undecidable problems:
RICE’s Theorem
Undecidable problems:
is empty?
is regular?
has size 2?
This can be generalized to all non-trivial
properties of Turing-acceptable languages
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2. Non-trivial property:
A property possessed by some
Turing-acceptable languages
but not all
Example:
: is empty?
YES NO NO
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3. More examples of non-trivial properties:
: is regular?
YES
YES NO
: has size 2? NO NO
YES
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4. A property possessed by ALL Turing-acceptable languages
Trivial property:
A property possessed by ALL
Turing-acceptable languages
Examples:
: has size at least 0?
True for all languages
: is accepted by some
Turing machine?
True for all
Turing-acceptable languages
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5. We can describe a property as the set
of languages that possess the property
If language has property then
Example:
: is empty?
YES NO NO
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6. Example: Suppose alphabet is : has size 1? NO YES NO NO
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7. Non-trivial property problem
Input:
Turing Machine
Question:
Does have the non-trivial
property ?
Corresponding language:
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8. Rice’s Theorem: is undecidable Proof: Reduce (membership problem) to
(the non-trivial property problem is unsolvable)
Proof:
Reduce
(membership problem) to or
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9. We examine two cases: Case 1: Examples: : is empty? : is regular?
: has size 2?
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10. Case 1: Since is non-trivial, there is a Turing-acceptable language
such that:
Let be the Turing machine that
accepts
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11. Reduce
(membership problem) to
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12. Decider for Reduction Compute Decider Given the reduction,
membership problem decider
Decider for
Non-trivial property
problem decider
Reduction
YES
YES
Compute
Decider NO NO
Given the reduction,
if is decidable,
then is decidable
A contradiction!
since
is undecidable
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13. We only need to build the reduction:
Compute
So that:
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14. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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15. For this we can run machine , that accepts language ,
with input string
Turing Machine
Accept
yes
yes
Write on tape, and
accepts ?
Simulate on input
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16. yes yes accepts does not accept Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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17. Therefore:
accepts
Equivalently:
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18. Case 2: Since is non-trivial, there is a Turing-acceptable language
such that:
Let be the Turing machine that
accepts
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19. Reduce
(membership problem) to
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20. Decider for Reduction Compute Decider Given the reduction,
membership problem decider
Decider for
Non-trivial property
problem decider
Reduction
YES
YES
Compute
Decider NO NO
Given the reduction,
if is decidable,
then is decidable
A contradiction!
since
is undecidable
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21. We only need to build the reduction:
Compute
So that:
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22. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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23. yes yes accepts does not accept Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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24. Therefore:
accepts
Equivalently:
END OF PROOF
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