1 / 24

Computability

Computability. CS 101E Spring 2007 Michele Co. So Far In Class. We’ve seen the following programming constructs if, if-else, if-else-if switch for while do-while And break continue. These are powerful constructs that allow us to express many computations.

bradley-manning
Télécharger la présentation

Computability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Computability CS 101E Spring 2007 Michele Co

  2. So Far In Class • We’ve seen the following programming constructs • if, if-else, if-else-if • switch • for • while • do-while • And • break • continue These are powerful constructs that allow us to express many computations

  3. Can Computers be Programmed to Solve Any Problem? • Yes • No • Maybe

  4. Can Humans Solve Any Problem? • Yes • No • Maybe

  5. Epimenides Paradox This statement is false. Proof: • Let A = This statement is false. • Assume A is false • Then, A must be true • Assume A is true • Then, A is false No matter what truth value assigned, a contradiction is reached!

  6. The Halting Problem

  7. The Halting problem • Given a program P, and input I, will the program P ever terminate? • Meaning will P(I) loop forever or halt? • Can a computer program determine this? • Can a human? • First shown by Alan Turing in 1936 • Before digital computers existed!

  8. Solving the Halting Problem • To “solve” the halting problem means we create a method CheckHalt(P,I) • P is the program we are checking for halting • I is the input to that program • And it will return “loops forever” or “halts” • Note it must work for any program, not just some programs

  9. Can a human determine if a program halts? • Given a program of 10 lines or less, can a human determine if it halts? • Assuming no tricks – the program is completely understandable • And assuming the computer works properly, of course • And we ignore the fact that an int will max out at 4 billion

  10. haltingTest1 public class haltingTest1 { public static void main(String [] args){ System.out.println("Alan Turing was a genius."); } }

  11. haltingTest2 public class haltingTest2 { public static void main(String [] args){ for (int factor = 1; factor <= 10; factor ++) { System.out.println("Factor is: " + factor); } } }

  12. haltingTest3 public class haltingTest3{ public static void main(String [] args){ while(true) { System.out.println("Hello world!"); } } }

  13. haltingTest4 public class haltingTest4{ public static void main(String [] args){ int x = 10; while ( x > 0 ){ System.out.println(“hello world”); x++; } } }

  14. Can Computers be Programmed to Solve Any Problem? • Yes • No • Maybe

  15. Can Humans Solve Any Problem? • Yes • No • Maybe

  16. Another Problem Perfect Numbers

  17. Perfect numbers • Numbers whose divisors (not including the number) add up to the number • 6 = 1 + 2 + 3 • 28 = 1 + 2 + 4 + 7 + 14 • The list of the first 10 perfect numbers:6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216 • The last one was 54 digits! • All known perfect numbers are even; it’s an open (i.e. unsolved) problem if odd perfect numbers exist • Sequence A000396 in OEIS

  18. Odd perfect number search public class searchForOddPerfectNumber { public static void main(String [] args){ int n = 1; // arbitrary-precision integer while (true) { int sumOfFactors = 0; for (int factor = 1; factor < n; factor++) { if ((n % factor) == 0) { sumOfFactors = sumOfFactors + factor; } // if } // for loop // if it’s a perfect number if (sumOfFactors == n) break; n = n + 2; } } } Using int for code clarity. Really need a larger precision type... BigInteger or larger

  19. Can Computers be Programmed to Solve Any Problem? • Yes • No • Maybe

  20. Can Humans Solve Any Problem? • Yes • No • Maybe

  21. Where does that leave us? • If a human can’t figure out how to do the halting problem, we can’t make a computer do it for us • It turns out that it is impossible to write such a CheckHalt() method • But how to prove this?

  22. CheckHalt()’s non-existence • Consider P(I): a program P with input I • Suppose that CheckHalt(P,I) exists • Tests if P(I) will either “loop forever” or “halt” • A program is a series of bits • And thus can be considered data as well • Thus, we can call CheckHalt(P,P) • It’s using the bytes of program P as the input to program P

  23. CheckHalt()’s non-existence • Consider a new method: Test(P): loops forever if CheckHalt(P,P) prints “halts” halts if CheckHalt(P,P) prints “loops forever” • Do we agree that Test() is a valid method? • Now run Test(Test) • If Test(Test) halts… • Then CheckHalt(Test,Test) returns “loops forever”… • Which means that Test(Test) loops forever • Contradiction! • If Test(Test) loops forever… • Then CheckHalt(Test,Test) returns “halts”… • Which means that Test(Test) halts • Contradiction!

  24. Why do we care about the halting problem? • It was the first algorithm that was shown to not be able to exist • You can prove an existential by showing an example (a correct program) • But it’s much harder to prove that a program can never exist

More Related