1

Computability

CS 101ESpring 2007Michele Co

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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 toexpress many computations

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Can Computers be Programmed to Solve Any Problem?

Yes N

o

May

be

25%

14%

61%

1. Yes

2. No

3. Maybe

4

Can Humans Solve Any Problem?

Yes N

o

May

be

25%

5%

70%

1. Yes

2. No

3. Maybe

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Epimenides Paradox

This statement is false.Proof:1. Let A = This statement is false.2. Assume A is false

– Then, A must be true

3. Assume A is true– Then, A is false

No matter what truth value assigned, a contradiction is reached!

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The Halting Problem

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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!

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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

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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

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haltingTest1

public class haltingTest1 {public static void main(String [] args){

System.out.println("Alan Turing was a genius.");

}}

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haltingTest2

public class haltingTest2 {public static void main(String [] args){

for (int factor = 1; factor <= 10; factor ++){

System.out.println("Factor is: " + factor);}

}}

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haltingTest3

public class haltingTest3{public static void main(String [] args){

while(true) {

System.out.println("Hello world!");}

}}

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haltingTest4

public class haltingTest4{public static void main(String [] args){

int x = 10; while ( x > 0 ){ System.out.println(“hello world”); x++; }

}}

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Can Computers be Programmed to Solve Any Problem?

Yes N

o

May

be

16%9%

75%1. Yes

2. No

3. Maybe

15

Can Humans Solve Any Problem?

Yes N

o

May

be

21%

5%

74%1. Yes

2. No

3. Maybe

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Another Problem

Perfect Numbers

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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

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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 largerprecision type... BigInteger or larger

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Can Computers be Programmed to Solve Any Problem?

Yes N

o

May

be

14%5%

81%

1. Yes

2. No

3. Maybe

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Can Humans Solve Any Problem?

Yes N

o

May

be

27%

3%

70%1. Yes

2. No

3. Maybe

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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?

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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

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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!

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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