Yesterday in a technical conversation I used the phrase ‘HP-complete’.
I had intended it, by analogy with ‘NP-complete’, to mean that if the problem under discussion could be solved, the solution would necessarily include a solution to the Halting Problem, i.e. the problem was as hard as the Halting Problem, i.e. uncomputable.
There are several other well known ‘-complete’ phrases which analogise in the same way – ‘Turing-complete’ and ‘AI-complete’ – and it seems to me that ‘HP-complete’ fits right into this framework and has a technically precise and useful meaning. But for some reason it isn't in common usage in the way that those other ‘-complete’ phrases are, so the person I was talking to didn't get what I meant and I had to explain. Bah. I don't see why it isn't part of the standard lexicon, for all the same reasons!
I suppose we already have the word ‘uncomputable’, so you could argue that we don't need ‘HP-complete’ as a synonym. But I think it's not quite a synonym, in that it also conveys a hint about why it's uncomputable, or at least about the train of thought that made me conclude that it was.
(I suppose it's conceivable, in my original context, that I should have chosen ‘HP-hard’ rather than ‘HP-complete’ – I don't think I'd intended to rule out the possibility that the problem under discussion was harder than the Halting Problem :-)