

Polynomial time factoring algorithm using Bayesian arithmetic - danpalmer
http://arxiv.org/abs/1212.4969

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gizmo686
When I opened this page, I was going to be the third comment, then I got
caught up trying to explain why their linked paper [1] (proving P=NP) is non-
sense. Unfourtuantly, my habit of assuming that my not understanding something
just means that I need to re-read it or play with that segment of the proof
myself costed me more time than I thought (even after I wrote in my partially
completed reply that the introduction alone was sufficient to dismiss the
paper as nonsense). Anyway, here is my overkill analysis:

I would be highly skeptical of their claim. First of all, if they did prove
P=NP (and long enough ago to write another paper), then I cannot imagine why
this is the first I am hearing about it, as solving this problem would likely
get a prominent spot in mainstream news, and definitively its own article on
HN.

Beyond that, the paper proving P=NP, even to my amateur eye, looks like
garbage. From the P=NP paper: "What is the powerful ingredient which allows a
dramatic speed-up of quantum computation over classical computation ? We
propose that this ingredient is an implicit use of the Bayesian probability
theory. Furthermore, we argue that both classical and quantum computation are
special cases of probability reasoning. On these grounds, introducing Bayesian
probability theory in classical computation as well, we reduce a typical NP
problem, namely 3-SAT, to a linear programming problem. According to
algorithmic complexity theory, this proves that P=NP."

"Any logical algorithm can be formulated as a linear programming problem.
Specially, a basic question of logical satisﬁability with n variables, namely
3-SAT, is equivalent to a linear programming problem with O(n^3) unknowns and
even in general with O(n) unknowns. According to algorithmic complexity
theory, this proves that P=NP."

"Again, the main reasons of the supposedly quantum eﬃciency are basically
unknown, but the common wisdom is that entanglement should be the key
ingredient. Indeed, in the quantum community, it is ‘widely believed that
classical systems cannot simulate highly entangled quantum systems’ [13]. By
contrast, we have nevertheless argued [14] that the concept of entanglement is
actually a quite classical attribute of contextual systems. Furthermore, any
classical computer can be regarded as a highly entangled system as far as the
entanglement is measured between the binary digits of the processed data
during the computation. In addition, we shall argue that the crucial
ingredient is not in the least a Hilbert space with its full quantum machinery
but only a ﬂexible randomization, or to be exact, the implicit use of Bayes
probabilities. This is like a grin without a cat in Alice and Bob adventures
in Quantum Land."

They devote two pages to history and what should be common knowledge to anyone
interested in this paper.

The actual proof is mathametical gibberish, however there are surprisingly
large contiguous sections of the proof that, by themselves, work.

[1] <http://arxiv.org/pdf/1205.6658v2.pdf>

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ColinWright
OK, I've really only skimmed it, but here's an impression ...

He spends a huge amount of time setting up the very very specific equations,
and then waves a magic wand saying "This is a Linear Programming Problem which
can be solved in polynomial time."

It looks to me like an integer linear programming problem, and that's known to
be NP-Complete. There doesn't seem to be anywhere that he takes a non-integer
solution and converts it in polynomial time to an integer solution (or
provably fails), nor is there anywhere that shows how to find an integer
solution in among all the possibly exponentially many solutions it might
produce.

~~~
danpalmer
As far as I understand from the two papers, and I haven't finished reading
them, nor do I expect to understand it all, but this appears to be very much
an LP problem, not ILP, from what I've read so far.

~~~
ColinWright
I don't have time now, and I'm unlikely to have time in the next six months,
but it looks awfully like he's setting up a standard LP problem, but I don't
see how he's extracting integer solutions from it.

I expect there to be errors, and I expect them to be subtle, and I expect that
if they're pointed out he'll push them around to some other place where
they're hard to find. Having said that, I haven't read it thoroughly (and
won't have the time to) and I'm disinclined to. Not that that's any great loss
- I'm not an expert.

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bzax
You might as well start with his proof of P=NP

<http://arxiv.org/pdf/1205.6658.pdf>

------
jellyksong
The abstract states, "When applied to NP-complete algorithms, this leads to
the fundamental conclusion that P = NP".

Isn't this a much more important conclusion?

~~~
jmurley
Agreed. Lots of papers like this are submitted, but none have been right so
far. Is there some reason to trust this one in particular? I'm not
sufficiently familiar with this material to evaluate it.

~~~
danpalmer
This happened a few years ago when someone posted some code on GitHub that
they claimed demonstrated P=NP, however it turned out not to.

This paper on the other hand is a more theoretical approach and appears to be
more concrete than previous attempts that I'm aware of.

While it may or may not be correct, (experience hinting at the latter) it is
interesting news, and an interesting paper nonetheless.

~~~
ColinWright
Actually, there appears to be nothing new, structurally or ideas-wise, and I
don't think it is very interesting. The concreteness is all about creating a
system of equations in terms of the bits in a multiplication. That's really
trivial. Large and messy, but nothing new.

There's no concrete indication that I've found in this paper about how to
solve those equations.

------
j2kun
Red flags: a single author, use of nonstandard terms like "algorithmic
complexity theory" (the standard term is simply "complexity theory"), appeals
to quantum computing theory (which is only tangentially related to the topic
of the paper), author only has two papers, no coauthors, and begins with
awkwardly irrelevant statements like, "Arithmetic is a part of abstract
mathematics."

