On Monday, 29 July 2013 at 14:39:02 UTC, John Colvin wrote:
On Monday, 29 July 2013 at 13:05:10 UTC, JS wrote:
On Monday, 29 July 2013 at 12:36:36 UTC, John Colvin wrote:
On Monday, 29 July 2013 at 12:35:59 UTC, John Colvin wrote:
On Monday, 29 July 2013 at 12:17:22 UTC, JS wrote:
Even something like

if (random() == 3) break;

is decidable(it will halt after some time).

That program has a finite average runtime, but its maximum runtime is unbounded. You can't actually say it *will* halt. For any given input (in this case 0 inputs) one cannot tell whether the program will eventually halt, therefore it is undecidable.

I have formal background in CS so I might have got that totally wrong.

sorry, that should be "I have NO formal background in CS"

No, again, it isn't about infinite run time.

decidability != infinite run time.

to simplify, let's look at the program,

for(;;) if (random() == 0) break;

where random() returns a random number, not necessarily uniform, between 0 and 1.

Same problem just easier to see.

Since there must be a chance for 0 to occur, the program must halt, regardless of how long it takes, even if it takes an "infinite" amount of time.

That is, the run time of the program may approach infinity BUT it will halt at some point because by the definition of random, 0 must occur... else it's not random.

So, by the fact that random, must cover the entire range, even if it takes it an infinitely long time(so to speak), we know that the program must halt. We don't care how long it will take but just that we can decide that it will.

The only way you could be right is if random wasn't random and 0 was never returned... in that case the program would not halt... BUT then we could decide that it never would halt...

In both cases, we can decide the outcome... if random is known to produce 0 then it will halt, if it can't... then it won't.

But random must produce a 0 or not a 0 in an infinite amount of time. (either 0 is in the range of random or not).

That is, the halting state of the program is not random even though it looks like it. (again, it's not how long it takes but if we can decide the outcome... which, in this case, rests on the decidability of random)

Another way to see this, flipping a fair coin has 0 probability of producing an infinite series of tails.


After N flips, the probability of flipping exactly N tails is 1/N -> 0.

Ok, I think I get what you mean now. The 2 states of interest for the halting problem are, for a give input:

1) program *can* stop
2) program *will not* stop

is that correct?

A program will either halt or not halt, the question is, can we decide. Rather:

A program will either halt or not halt, or be impossible to tell.

We'd like to think we know if a program will stop or not but we can't always know that... there are just some strange programs that we can't figure out. The program is sort of a superposition of halting and not halting... sort of like schrodingers cat.

For example, it is impossible to know if schrodingers cat is alive or dead until we open the box(but suppose we never get to open the box).

Here is another example:

main() { readln(); }

Does the program halt or not?

Yes! Just because it depends on what the user does, does not mean the program change the fact that the program halts or not.

Basically the halting problem deals with the structural aspect of the program itself and not the inputs on it. (this does not mean that the input is not required)

Here is the only example that comes to mind.

main(S) { for(;;) if (S subset of S) halt; }

Ok? easy program right. Just checks if S is a subset of itself and halts, else doesn't.

But what happens when we call it with the set of all sets?

Such a program is indeterminate. Why? Because the set of all sets that do not contain themselves both contains itself and doesn't contain itself. (the if statement can't be computed)

i.e., Let S = Set of all sets that do not contain themselves.

If S doesn't contain itself in, by definition, S is a subset of itself... which is contradictory.

If S contains itself, then again, by definition, S is a set that does not contain itself, which is contradictory.

Hence, the program give above's halting state can't be known.. or, can't be determined, or is undecidable.

All you have to do is ask yourself if it halts(for the given input)? You can't even reason about it. Because if it does halt then it doesn't halt.

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