Hm, that poster seems like an interesting fellow. I’d like to meet him.
Seriously, I’m seeing two possible interpretations of the OP’s question, one of which is trivial (the program is either “yes” or “no”), and one of which is the question which was answered by Turing’s theorem. Is there still a nontrivial question which remains to be answered, here?
I’m thinking no. I’m trying to come up with “Does there exist a P-specific algorithm that determines halt(P) by actual analysis rather than just stating an answer?” But I realize that that’s not well defined, and I’m thinking at this point that it can’t be well defined without driving it into one or the other of our existing cases. So I’m happy to give up at this point.
As, as I have already intimated though without having actually stated it, am I, the grand high OP. And that, as I have already intimated though without having actually stated it, for the very reasons you two are discussing.
-FrL-
I knew I had seen this piece, was I was reluctant to bring it up until I could dig up the link:
Microsoft claims it has found a way around the halting problem for most cases.
Interesting only as an example in the study of corporate hype. Most programs actually written for a practical purpose, it’s very easy to determine whether they’ll terminate, because most programmers will put in escape clauses. So the program won’t be “Try this over and over again forever until it works”, it’ll be something like “Try this up to a million times until it works, and if after a million tries it hasn’t, spit out an error message and give up”. A tool which would automatically check a program for such escape clauses isn’t particularly interesting, nor is it useful except perhaps for evaluating rookie programmers. And it would be no help at all for programs which halt only on a nontrivial Collatz loop, or an even number without prime summands, or the like, which are the very cases which make Turing’s theorem interesting.