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From: nnymous109@gmail.com (nnymous109)
Newsgroups: comp.theory
Subject: Re: Yet another contribution to the P-NP question
Date: Sun, 29 Sep 2024 16:14:19 +0000
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> I tried to make one major suggestion to the author: explain (in English)
> in what way the core of the argument differs from the usual "it must
> examine all the cases" non-proofs that keep cropping up.
>

And there's what I most unsure of. I've heard of these "examine all
cases" non-proofs, but I don't know what exactly makes them fail (is it
just that they don't give any reason why we must examine all the cases
or is it something deeper?)

I would call the proof strategy I have come up with an "examine all the
cases" type proof except the underlying observation as to why we must do
that is that if x1 and x2 are different strings, unless there is some
extra information we have been given beforehand (about x1 and x2) that
we can take advantage of, there is in general no correspondence between
S.(M(x1)) and S.(M(x2)).

In the preceding paragraph, I am carrying over notation I used in my
first post today. Throughout this post, if there's any undefined
notation, it's because it's carried over from the same post.


> But there are some worrying signs.  If someone knows little mathematics,
> why describe a mapping as a homomorphism when there is no topology in
> play?  Does he or she just mean a bjection?  What has continuity to do
> with it?  There's a whiff of "that's a nice sounding word, I'll use it"
> here.
>


This is because it looked like something I saw in an algebra textbook
once. If M and N are recursions, and f : (U_M)* -> (U_N)*, so that
f(M(x)) = f(y) = N^b(f(x)) for some integer b. I'm thinking of f as
relabeling the computation*, and I'm using homomorphism to suggest that
analogy. Or it could just be my impostor syndrome at work :)

But again, if these words already conjure up very specific things and
conflating them would be troublesome, I'm perfectly happy to rename them
as is necessary.

> I'm prepared to take it seriously for a while.

Well, thank you. I think we're at the heart of it, so that at this
stage, we can make a really good estimation of whether there's something
here what considering further.

* - and to be exact, the domain of f is not (U_M)*, but only those
strings that may be elements of a computation by M