Class 36: Halting Problem
Class 36: Slides [PPTX], Notes [PDF]
Here’s the movie from class today:
Mario – Determinism vs. Nondeterminism [video]
by Navid Hosseini, John Koelling, Trung Tran, and Ben Powell
The other movie we didn’t watch:
A Downfall Parody: P = NP [Video]
by Arthur Gordon, Allison Gurlitz, Stephen Lam, Eugene Moy
More conveying complexity submissions are here, including games, comic books, and stories.
PDF 
Sorry for commenting twice in one week, but I thought that this poem might help a bit with following the halting problem….
Scooping the Loop Snooper
an elementary proof of the undecidability of the halting problem
Geoffrey K. Pullum, University of Edinburgh
No program can say what another will do.
Now, I won’t just assert that, I’ll prove it to you:
I will prove that although you might work til you drop,
you can’t predict whether a program will stop.
Imagine we have a procedure called P
that will snoop in the source code of programs to see
there aren’t infinite loops that go round and around;
and P prints the word “Fine!” if no looping is found.
You feed in your code, and the input it needs,
and then P takes them both and it studies and reads
and computes whether things will all end as they should
(as opposed to going loopy the way that they could).
Well, the truth is that P cannot possibly be,
because if you wrote it and gave it to me,
I could use it to set up a logical bind
that would shatter your reason and scramble your mind.
Here’s the trick I would use – and it’s simple to do.
I’d define a procedure – we’ll name the thing Q -
that would take any program and call P (of course!)
to tell if it looped, by reading the source;
And if so, Q would simply print “Loop!” and then stop;
but if no, Q would go right back to the top,
and start off again, looping endlessly back,
til the universe dies and is frozen and black.
And this program called Q wouldn’t stay on the shelf;
I would run it, and (fiendishly) feed it itself.
What behaviour results when I do this with Q?
When it reads its own source, just what will it do?
If P warns of loops, Q will print “Loop!” and quit;
yet P is supposed to speak truly of it.
So if Q’s going to quit, then P should say, “Fine!” -
which will make Q go back to its very first line!
No matter what P would have done, Q will scoop it:
Q uses P’s output to make P look stupid.
If P gets things right then it lies in its tooth;
and if it speaks falsely, it’s telling the truth!
I’ve created a paradox, neat as can be -
and simply by using your putative P.
When you assumed P you stepped into a snare;
Your assumptions have led you right into my lair.
So, how to escape from this logical mess?
I don’t have to tell you; I’m sure you can guess.
By reductio, there cannot possibly be
a procedure that acts like the mythical P.
You can never discover mechanical means
for predicting the acts of computing machines.
It’s something that cannot be done. So we users
must find our own bugs; our computers are losers!
This is great!
There is one important issue that it takes a lot of “poetic license” with, but is commonly misunderstood. For example,
isn’t really correct. It should really be,
There are programs that can predict what other programs can do — the important point is the there is no program that can predict what every program can do. Indeed, it is easy to write a halts procedure that works correctly on many programs (for example, we know a program with no loops or recursive calls will always halt). Contrary to the last stanza, there are lots of programs that find bugs in programs (I worked on one for my Master’s project). The Terminator Project shows that it is possible to solve the HALTING problem for many complex programs also. But, as Turing proved, it is impossible to produce an algorithm that solves it correctly for all input programs. So, these tools can find some bugs in some programs, but there is no program that can find all bugs in all programs.