1. Computable Problems

2. Computability
What does it mean to be computable?

3. Computability What does it mean to be computable?
Can decide an answer using
Finite number of steps
Finite list of instructions

4. Background: Proof By Contradiction

5. Proof By Contradiction
Assume something is true
Show it leads to a contradiction
Conclude the opposite

6. Proof By Contradiction
Assume something is true
Show it leads to a contradiction
Conclude the opposite

7. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle

8. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B

9. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B
Find a contradiction:
The measures of A + B + C = 180 (given)

10. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B
Find a contradiction:
The measures of A + B + C = 180 (given)
C = 180 (step 1 and our assumption)

11. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B
Find a contradiction:
The measures of A + B + C = 180 (given)
C = 180 (step 1 and our assumption)
Measure of C = 0 (step 2 and additive property of equality)

12. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B
Find a contradiction:
The measures of A + B + C = 180 (given)
C = 180 (step 1 and our assumption)
Measure of C = 0 (step 2 and additive property of equality)
Measure of C > 0 (given)

13. Proof By Contradiction
Given:
The 3 angles in a triangle are each > 0 degrees
The 3 angles in a triangle add to 180
Prove:
No triangle can have more than one 90 degree angle
Assume the opposite:
Triangle ABC has right angles A and B
Find a contradiction:
The measures of A + B + C = 180 (given)
C = 180 (step 1 and our assumption)
Measure of C = 0 (step 2 and additive property of equality)
Measure of C > 0 (given)
Assumption must be false – we can not have a triangle with right angles A and B

14. Background: Programs are Data

15. Programs Running Programs
Programs run on data
Programs are stored as data
Programs can run on other programs

16. Background: Yes/No Questions

17. Decidability
Decidability problem: Any problem that requires a yes/no answer
Does this data contain " "?
Will this program ever stop?
Will this program crash?
Is there a virus in this program?

18. Sample Program 1:
Always Yes Program
Anything
Rule:
Always say Yes
Yes

19. Sample Program 1: Always Yes Program Input Result hello.txt (2kB) YES
addressList.doc (86kB)
notepad.exe

20. Sample Program 2:
Always No Program
Anything
Rule:
Always say No No

21. Sample Program 2: Always No Program Input Result hello.txt (2kB) NO
addressList.doc (86kB)
notepad.exe

22. Sample Program 3: Size Checker Program File Rule: Is size > 10 Kb?
Yes/No

23. Sample Program 3: Size Checker Program Input Result hello.txt (2kB) NO
File
Input
Result
hello.txt (2kB) NO
addressList.doc (86kB)
YES
notepad.exe
Rule:
Is size > 10 Kb?
Yes/No

24. Does this program always say yes?
Always Yes?
Can I write a program that determines if other programs say Yes all the time?
Program
Rule:
Does this program always say yes?
Yes/No

25. Does this program always say yes?
Always Yes?
Can I write a program that determines if other programs say Yes all the time? Assume I can write AlwaysYes(Program) …
Program
Rule:
Does this program always say yes?
Yes/No

26. Does this program always say yes?
Always Yes?
Program
Input
Result
Yes No
Size Checker
Always Yes
Rule:
Does this program always say yes?
Yes/No

27. Does this program always say yes?
Always Yes?
Yes Program
Program
Input
Result
Yes No
Size Checker
Always Yes
Rule:
Does this program always say yes?
Yes/No
Yes Program:
Input
Result
hello.txt (2kB)
YES
addressList.doc (86kB)
notepad.exe

28. Does this program always say yes?
Always Yes?
No Program
Program
Input
Result
Yes No
Size Checker
Always Yes
Rule:
Does this program always say yes?
Yes/No
No Program:
Input
Result
hello.txt (2kB) NO
addressList.doc (86kB)
notepad.exe

29. Does this program always say yes?
Always Yes?
Size Program
Program
Input
Result
Yes No
Size Checker
Always Yes
Rule:
Does this program always say yes?
Yes/No
Size Check Program:
Input
Result
hello.txt (2kB) NO
addressList.doc (86kB)
YES
notepad.exe

30. Does this program always say yes?
Always Yes?
Always Yes ( Always Yes )  ???
Always Yes Program
File
Input
Result
Yes No
Size Checker
Always Yes ??
Rule:
Does this program always say yes?
Yes/No

31. Does this program always say yes?
Always Yes?
Always Yes ( Always Yes )  No
Always Yes Program
File
Input
Result
Yes No
Size Checker
Always Yes NO
Rule:
Does this program always say yes?
Yes/No
Everything seems fine…

32. Does this program say Yes if run on itself?
Yes On Self
If I can write "Does this program always say yes"… I can write "Does this program say yes when it runs on itself"
Program
Rule:
Does this program say Yes if run on itself?
Yes/No

33. Yes On Self Input Result Yes No Size Checker (19kb) Always Yes

34. Yes On Self Input Result Yes YES No NO Size Checker (19kb) Always Yes

35. Yes On Self Input Result Yes YES No NO Size Checker (19kb) Always Yes
???

36. Yes On Self If Says Yes on Self says Yes Then the program should
Answer "Yes"

37. Yes On Self If Says Yes on Self says No Then the program should
Answer "No"

38. Yes On Self Says Yes on Self could decide No or Yes!?!?! Input Result
Size Checker (19kb)
Name Size Checker
Always Yes
Yes On Self
Yes or No
Says Yes on Self could decide No or Yes!?!?!

39. Yes On Self
If I can write "Does this program say yes when it runs on itself "… I can write "Say No if this program says Yes if run on itself"
Program

40. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
File
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.
Yes/No
Input
YesSelf
Anti
Yes
YES NO No
Size Checker (19kb)
Always Yes

41. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
What should AntiYesSelf day about itself?
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.

42. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
Assume: AntiYesSelf( AntiYesSelf( ) )  YES
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.
Yes

43. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
Assume: AntiYesSelf( AntiYesSelf( ) )  YES If it says YES when run on itself, it was supposed to say NO!
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.
Yes

44. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
Assume: AntiYesSelf( AntiYesSelf( ) )  NO
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes. No

45. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
Assume: AntiYesSelf( AntiYesSelf( ) )  NO If it says NO when run on itself, it was supposed to say YES!
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.
Yes

46. Say No if a program says Yes when run on itself. Otherwise say Yes.
Anti Yes On Self
Anti Yes On Self Can't Answer YES or NO!
Rule:
Say No if a program says Yes when run on itself. Otherwise say Yes.
Assume result is YES:
Assume result is No:

47. Chain:
If I can write Always Yes

48. Chain:
If I can write Always Yes
Then I can write Yes On Self

49. Chain: If I can write Always Yes Then I can write Yes On Self
Then I can write Anti Yes On Self

50. Chain: If I can write Always Yes Then I can write Yes On Self
Then I can write Anti Yes On Self
But Anti Yes On Self Can't Exist

51. Chain: If I can write Always Yes Then I can write Yes On Self
Then I can write Anti Yes On Self
But Anti Yes On Self Can't Exist
I can NOT write Always Yes

52. Crash Detection
What about detecting possible bugs?

53. Crash Detection
What about detecting possible bugs?

54. Crash Detection
What about detecting possible bugs?

55. Crash Detection
What about detecting possible bugs?

56. Crash Detection
What about detecting possible bugs?

57. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  YES

58. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  YES
Then AntiCrashSelf( AntiCrashSelf( ) ) Must Crash!

59. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  YES
Then AntiCrashSelf( AntiCrashSelf( ) ) Must Crash!

60. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  Crashes

61. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  Crashes
Then AntiCrashSelf( AntiCrashSelf( ) ) Must Output Yes

62. Crash Detection AntiCrashOnSelf runs on itself:
Assume: AntiCrashSelf( AntiCrashSelf( ) )  Crashes
Then AntiCrashSelf( AntiCrashSelf( ) ) Must Output Yes

63. The Dilemma

64. Crash Detection What about detecting possible bugs?
AntiCrashOnSelf runs on itself
YES: If it outputs Yes then it must have crashed
No: If it outputs No then it must not have crashed

65. Crash Detection
What about detecting possible bugs?

66. Crash Detection
What about detecting possible bugs?

67. Crash Detection What about detecting possible bugs?
Impossible to do perfectly
Must settle for pretty good most of the time

68. Alan Turing
Father of Computer Science

69. Alan Turing
Demonstrated the "Halting Problem" is uncomputable
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70. The Turing Machine Hypothetical device
Reads and write symbols on a tape
Decides what to do based on list of rules

71. Real Turing Machines

72. Unsolvable Problems
Turing machines can not solve some problems Ex: Look at a program and decide if it halts

73. Computers
Turing machine can compute anything computable by any system consisting of:
Finite number of steps
Finite list of instructions =

74. Unsolvable Problems Turing machines can not solve some problems
A Turing machine can solve any problem solvable by any computer …

75. Unsolvable Problems Turing machines can not solve some problems
A Turing machine can solve any problem solvable by any computer …
There are some problems unsolvable by computation