1. Busch Complexity Lectures: Reductions
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2. Problem is reduced to problem
If we can solve problem
then we can solve problem
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3. function (reduction) such that:
Definition:
Language
is reduced to
language
There is a computable
function (reduction) such that:
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4. There is a deterministic Turing machine
Recall:
Computable function :
There is a deterministic Turing machine
which for any string computes
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5. If: a: Language is reduced to b: Language is decidable
Theorem:
If: a: Language is reduced to
b: Language is decidable
Then: is decidable
Proof:
Basic idea:
Build the decider for
using the decider for
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6. Decider for Reduction compute Decider for END OF PROOF Input string
YES
Input
string
YES
accept
accept
compute
Decider
for
(halt)
(halt) NO NO
reject
reject
(halt)
(halt)
END OF PROOF
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7. Example:
is reduced to:
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8. We only need to construct:
Turing Machine
for reduction
DFA
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9. Let be the language of DFA
Turing Machine
for reduction
DFA
construct DFA
by combining and so that:
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10. Prof. Busch - LSU

11. Decider for Reduction compute Input string Decider YES YES NO NO
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12. If: a: Language is reduced to b: Language is undecidable
Theorem (version 1):
If: a: Language is reduced to
b: Language is undecidable
Then: is undecidable
(this is the negation of the previous theorem)
Proof:
Suppose is decidable
Using the decider for
build the decider for
Contradiction!
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13. If is decidable then we can build:
Decider for
Reduction
YES
Input
string
YES
accept
accept
compute
Decider
for
(halt)
(halt) NO NO
reject
reject
(halt)
(halt)
CONTRADICTION!
END OF PROOF
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14. that some language is undecidable we only need to reduce a
Observation:
In order to prove
that some language is undecidable
we only need to reduce a
known undecidable language to
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15. while processing input string ?
State-entry problem
Input:
Turing Machine
State
String
Question:
Does
enter state
while processing input string ?
Corresponding language:
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16. (state-entry problem is unsolvable)
Theorem:
is undecidable
(state-entry problem is unsolvable)
Proof:
Reduce
(halting problem) to
(state-entry problem)
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17. Decider for Reduction Compute Decider Given the reduction,
Halting Problem Decider
Decider for
state-entry 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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18. We only need to build the reduction:
Compute
So that:
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19. A transition for every unused tape symbol of
Construct from :
special
halt state
halting
states
A transition for every unused
tape symbol of
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20. special halt state halting states halts halts on state
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21. Therefore: halts on input halts on state on input Equivalently:
END OF PROOF
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22. Blank-tape halting problem
Input:
Turing Machine
Question:
Does
halt when started with
a blank tape?
Corresponding language:
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23. Theorem: is undecidable Proof: Reduce (halting problem) to
(blank-tape halting problem is unsolvable)
Proof:
Reduce
(halting problem) to
(blank-tape problem)
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24. Decider for Reduction Compute Decider Given the reduction,
Halting Problem Decider
Decider for
blank-tape 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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25. We only need to build the reduction:
Compute
So that:
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26. Construct from : no yes Run with input If halts then halt
Accept and halt no
Tape is blank?
yes
Run
Write on tape
with input
If halts then halt
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27. halts when started on blank tape
Accept and halt no
Tape is blank?
yes
Run
Write on tape
with input
halts on input
halts when started on blank tape
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28. halts when started on blank tape
halts on input
halts when started on blank tape
Equivalently:
END OF PROOF
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29. If: a: Language is reduced to b: Language is undecidable
Theorem (version 2):
If: a: Language is reduced to
b: Language is undecidable
Then: is undecidable
Proof:
Suppose is decidable
Then is decidable
Using the decider for
build the decider for
Contradiction!
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30. Suppose is decidable Decider for reject accept (halt) (halt)
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31. Suppose is decidable Then is decidable Decider for Decider for
(we have proven this in previous class)
Decider for NO
YES
reject
accept
Decider
for
(halt)
(halt)
YES NO
accept
reject
(halt)
(halt)
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32. If is decidable then we can build:
Decider for
Reduction
YES
Input
string
YES
accept
accept
compute
Decider
for
(halt)
(halt) NO NO
reject
reject
(halt)
(halt)
CONTRADICTION!
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33. Alternatively: Decider for Reduction compute Decider forNO
Input
string
YES
reject
accept
compute
Decider
for
(halt)
(halt)
YES NO
accept
reject
(halt)
(halt)
CONTRADICTION!
END OF PROOF
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34. that some language is undecidable we only need to reduce some
Observation:
In order to prove
that some language is undecidable
we only need to reduce some
known undecidable language to
or to
(theorem version 1)
(theorem version 2)
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35. Undecidable Problems for Turing Recognizable languages
Let be a Turing-acceptable language
is empty?
is regular?
has size 2?
All these are undecidable problems
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36. Let be a Turing-acceptable language
is empty?
is regular?
has size 2?
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37. Empty language problem
Input:
Turing Machine
Question: Is
empty?
Corresponding language:
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38. (empty language problem)
Theorem:
is undecidable
(empty-language problem is unsolvable)
Proof:
Reduce
(membership problem) to
(empty language problem)
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39. Decider for Reduction Compute Decider Given the reduction,
membership problem decider
Decider for
empty 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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40. We only need to build the reduction:
Compute
So that:
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41. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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42. Louisiana The only possible accepted string yes yes Turing Machine
Write on tape, and
accepts ?
Simulate on input
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43. yes yes accepts does not accept Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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44. Therefore:
accepts
Equivalently:
END OF PROOF
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45. Let be a Turing-acceptable language
is empty?
is regular?
has size 2?
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46. Regular language problem
Input:
Turing Machine
Question: Is
a regular language?
Corresponding language:
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47. (regular language problem)
Theorem:
is undecidable
(regular language problem is unsolvable)
Proof:
Reduce
(membership problem) to
(regular language problem)
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48. Decider for Reduction Compute Decider Given the reduction,
membership problem decider
Decider for
regular 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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49. We only need to build the reduction:
Compute
So that:
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50. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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51. yes yes not regular accepts does not accept regular Turing Machine
Write on tape, and
accepts ?
Simulate on input
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52. Therefore: accepts is not regular Equivalently: END OF PROOF
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53. Let be a Turing-acceptable language
is empty?
is regular?
has size 2?
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54. Does have size 2 (two strings)?
Size2 language problem
Input:
Turing Machine
Question:
Does have size 2 (two strings)?
Corresponding language:
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55. (size 2 language problem)
Theorem:
is undecidable
(size2 language problem is unsolvable)
Proof:
Reduce
(membership problem) to
(size 2 language problem)
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56. Decider for Reduction Compute Decider Given the reduction,
membership problem decider
Decider for
size2 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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57. We only need to build the reduction:
Compute
So that:
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58. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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59. yes yes 2 strings accepts does not accept 0 strings Turing Machine
Write on tape, and
accepts ?
Simulate on input
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60. Therefore: accepts has size 2 Equivalently: END OF PROOF
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61. 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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62. Non-trivial property:
A property possessed by some
Turing-acceptable languages
but not all
Example:
: is empty?
YES NO NO
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63. More examples of non-trivial properties:
: is regular?
YES
YES NO
: has size 2? NO NO
YES
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64. 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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65. 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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66. Example: Suppose alphabet is : has size 1? NO YES NO NO
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67. Non-trivial property problem
Input:
Turing Machine
Question:
Does have the non-trivial
property ?
Corresponding language:
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68. 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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69. We examine two cases: Case 1: Examples: : is empty? : is regular?
: has size 2?
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70. 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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71. Reduce
(membership problem) to
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72. 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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73. We only need to build the reduction:
Compute
So that:
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74. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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75. 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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76. yes yes accepts does not accept Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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77. Therefore:
accepts
Equivalently:
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78. 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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79. Reduce
(membership problem) to
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80. 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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81. We only need to build the reduction:
Compute
So that:
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82. Construct from : Tape of input string yes yes Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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83. yes yes accepts does not accept Turing Machine Accept
Write on tape, and
accepts ?
Simulate on input
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84. Therefore:
accepts
Equivalently:
END OF PROOF
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