

If P vs NP formally independent then NP has very close to poly-time upper bounds - amichail
http://www.cs.technion.ac.il/~shai/ph.ps.gz

======
gjm11
This one may merit a summary, since there are a few subtleties. So, here goes.

Background: Ever since Goedel, we know that in any given system for doing
mathematics some statements are _neither provable nor refutable_. So, if after
much effort mathematicians fail to find either a proof or a disproof for some
conjecture, it's natural to wonder whether perhaps that's because neither
exists.

In some cases, an undecidability result of this kind would be pretty much
equivalent to an actual decision. For instance: Goldbach's conjecture says
"every even number >= 4 is the sum of two prime numbers". If this is
undecidable then, in particular, I can't write down an explicit
counterexample, so it might as well be true.

OK, so what about P=NP? Well, what David and Halevi have done is to show
_something a bit like_ the following: "If 'P=NP' is not decidable using the
axioms of Peano arithmetic, then any family of decision problems that's in NP
is 'almost in P'". But there are some fiddly details that might matter.

Detail #1: it's not actually "is not decidable", it's "is _provably_ not
decidable". The distinction between these two is important to anyone who's
interested in this stuff in the first place.

Detail #2: it's not even "is provably not decidable", it's "is provably not
decidable, where the undecidability proof is of a particular kind". They claim
that "any known technique for proving independence from sufficiently strong
theories of statements that are neither self-referential nor inherently proof-
theoretic" is of this kind. Since the whole field of independence proofs got
started when Goedel worked out how to make things "self-referential" that on
the face of it look like _completely the wrong kind of thing to sustain self-
reference_ , I can't help but be a bit unimpressed by this.

Detail #3: What they mean by "almost in P" is this: there are arbitrarily long
intervals of arbitrarily large numbers such that any problem whose size lies
in one of these intervals can be solved in almost-polynomial time -- i.e.,
O(n^f(n)) where f grows very, very slowly, in particular more slowly than log
log ... log n with any number of "log"s.

------
wcarss
<http://users.socis.ca/~wcarss/ph.pdf>

a bit more usable (article in pdf format)

~~~
roundsquare
Is it just me, or is this backwards?

------
amichail
Also see: [http://blog.computationalcomplexity.org/2009/09/is-pnp-
ind-o...](http://blog.computationalcomplexity.org/2009/09/is-pnp-ind-of-zfc-
respectable-viewpoint.html)

From comment 2: _If P!=NP is independent of Peano Arithmetic, then "almost"
P=NP._

~~~
gjm11
That's referring to this very paper.

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Devilboy
Sorry to be off-topic but is this an audio or video link or something? I can't
work out how to open it...

~~~
coryrc
It is gzip-compressed postscript. The following commands could come in handy:

gunzip (or gzip -d)

gv

ps2pdf

~~~
Devilboy
Er so I have to download a utility to gunzip it, and then download another one
to convert it to a PDF? That's pretty annoying.

EDIT: I'm just saying, if you're linking something to HN where hundreds of
people will click on it, isn't it more efficient and courteous to convert and
re-host it in a format that most people can read?

~~~
Rantenki
You want to read a mathematical proof about computability, and you need
somebody else to gunzip it for you first? LOL!

