<br><br><div class="gmail_quote">On Sat, Oct 9, 2010 at 9:11 AM, Stewart Gordon <span dir="ltr"><<a href="mailto:smjg_1998@yahoo.com">smjg_1998@yahoo.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
On 08/10/2010 00:39, Tomek Sowiński wrote:<br>
<snip><br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
What's the halting problem?<br>
</blockquote>
<br>
Basically, it's the challenge of determining algorithmically whether an arbitrary algorithm given arbitrary input will eventually halt or carry on running forever.<br>
<br>
The point is that the halting problem is known to be unsolvable. The standard proof of this is as follows. Suppose the halt analyser algorithm we seek exists. Call it WillHalt(Algorithm, Input). Then we can consider WillHalt(Algorithm, Algorithm).<br>
<br>
Then we can define a new algorithm, LoopIfHaltsOnItself(Algorithm), defined as<br>
<br>
if WillHalt(Algorithm, Algorithm) then<br>
loop forever<br>
else<br>
return<br>
<br>
Now try to analyse the outcome of LoopIfHaltsOnItself(LoopIfHaltsOnItself).<br>
<br>
<br>
Personally, I think it's a shame that the halting problem can't be solved. If it could, we could use it to solve many mathematical problems that have as it happens remained unsolved for centuries.<br><font color="#888888">
<br>
Stewart.</font></blockquote><div><br></div><div>Or more poetically, </div><div><br></div><div><meta http-equiv="content-type" content="text/html; charset=utf-8"><span class="Apple-style-span" style="font-family: Tahoma, arial, helvetica, sans-serif; font-size: 13px; font-style: italic; "><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
No general procedure for bug checks succeeds.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">Now, I won’t just assert that, I’ll show where it leads:<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
I will prove that although you might work till you drop,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
you cannot tell if computation will stop.</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
For imagine we have a procedure called P<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">that for specified input permits you to see<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
whether specified source code, with all of its faults,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">defines a routine that eventually halts.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">You feed in your program, with suitable data,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
and P gets to work, and a little while later<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">(in finite compute time) correctly infers<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
whether infinite looping behavior occurs.</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
If there will be no looping, then P prints out `Good.’<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">That means work on this input will halt, as it should.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
But if it detects an unstoppable loop,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">then P reports `Bad!’ — which means you’re in the soup.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">Well, the truth is that P cannot possibly be,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
because if you wrote it and gave it to me,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">I could use it to set up a logical bind<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
that would shatter your reason and scramble your mind.</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
Here’s the trick that I’ll use — and it’s simple to do.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">I’ll define a procedure, which I will call Q,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
that will use P’s predictions of halting success<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">to stir up a terrible logical mess.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">For a specified program, say A, one supplies,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
the first step of this program called Q I devise<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">is to find out from P what’s the right thing to say<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
of the looping behavior of A run on A.</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
If P’s answer is `Bad!’, Q will suddenly stop.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">But otherwise, Q will go back to the top,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
and start off again, looping endlessly back,<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">till the universe dies and turns frozen and black.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">And this program called Q wouldn’t stay on the shelf;<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
I would ask it to forecast its run on itself.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">When it reads its own source code, just what will it do?<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
What’s the looping behavior of Q run on Q?</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
If P warns of infinite loops, Q will quit;<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">yet P is supposed to speak truly of it!<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
And if Q’s going to quit, then P should say `Good’<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">— which makes Q start to loop! (P denied that it would.)</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">No matter how P might perform, Q will scoop it:<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
Q uses P’s output to make P look stupid.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">Whatever P says, it cannot predict Q:<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
P is right when it’s wrong, and is false when it’s true!</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
I’ve created a paradox, neat as can be —<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">and simply by using your putative P.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
When you posited P you stepped into a snare;<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">Your assumption has led you right into my lair.</p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">So where can this argument possibly go?<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
I don’t have to tell you; I’m sure you must know.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">By reductio, there cannot possibly be<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
a procedure that acts like the mythical P.</p><p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 10px; padding-right: 0px; padding-bottom: 10px; padding-left: 0px; font-size: 10pt; line-height: 1.4em; ">
You can never find general mechanical means<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">for predicting the acts of computing machines.<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">
It’s something that cannot be done. So we users<br style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; ">must find our own bugs. Our computers are losers!</p>
</span></div></div>- <meta http-equiv="content-type" content="text/html; charset=utf-8"><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: medium; font-style: italic; ">Geoffrey K. Pullum</span>