
Insignificant Choice Polynomial Time - luu
https://arxiv.org/abs/2005.04598
======
lidHanteyk
Reading the abstract, this seems to be a claim on P vs NP. Specifically, that
P != NP. The claim goes through the front door, using descriptive complexity
theory, claiming a formal system for P and then claiming that SAT can't be
expressed in that system.

The main observations focus on broken symmetry. Polynomial-time algorithms
often have to start by choosing some unit of work, but without a requirement
that work units be ordered; therefore some choice is required in order to pick
a starting work unit. For example, to process a graph, one usually must start
by choosing an edge or vertex without limitation.

The claim is, effectively, that SAT requires _significant_ choices to be made
in which direction to compute in order to solve its instances in polynomial
time.

I confess that, although I've read the proof, I neither am convinced by it nor
can find problems with it. I need to study it more.

I'm not sure how the barriers are handled. The relativization barrier, I can
imagine, is handled by showing that abstract state machines with oracular
advice can turn seemingly-insignificant choices into significant choices, so
that the proof doesn't relativize. I don't know about algebrization, though.

In terms of Impagliazzo's five worlds, this result would indicate that we are
in Minicrypt or Cryptomania, as is commonly believed already.

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azdavis
The title seems innocent enough, but if I understand this correctly, this
purports to be a proof of P != NP.

I'm not familiar with a lot of the stuff talked about in the paper, but it's
still exciting to see another attempt at the P ?= NP problem. Of course,
purported proofs (for both P = NP and P != NP) have been published before[0].

[0]: [https://www.win.tue.nl/~gwoegi/P-versus-
NP.htm](https://www.win.tue.nl/~gwoegi/P-versus-NP.htm)

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marcelluspye
Big if true.

Clicking around arXiv and google scholar, I get the sense the author is
legitimate. But I wouldn't be surprised if some technical barrier was found by
another expert. I don't think the author has published anything in complexity
theory before, and the story of 'older academic attempts to enter new field
with old tools, immediately solves most important problem in field' is a tad
fishy to me, though not entirely without precedent. The idea sounds pretty
cool though; I'd definitely prefer to be mistaken.

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cevi
I am far from being an expert on descriptive complexity, and have only skimmed
the paper. One spot jumped out at me as a potential error: on page 21, the
sentence

"Next we exploit that a choice in state S will be insignificant iff all update
sets in Delta_r(S) are isomorphic, so..."

seems suspicious to me. One direction is likely obvious (that an isomorphism
between choices proves the insignificance of the choice), but the other
direction seems likely to be untrue (if I am interpreting correctly).

------
btilly
Given the history of this problem, I will wait until I see experts excited
about it before worrying about whether it might be a good argument.

~~~
potiuper
[https://zjui.intl.zju.edu.cn/en/content/874774](https://zjui.intl.zju.edu.cn/en/content/874774)
is not an "expert"? "Choiceless polynomial time" from Blass, Gurevich and
Shelah seems to change the common definition of polynomials using the axioms
of ZFC[hoice].

~~~
klyrs
That's the author. Wait til you see _independent_ experts showing excitement.

Andrew Wiles was plenty well respected but the review process turned up some
nontrivial errors in his proof of Fermat's Last [Conjecture]. The errors were
eventually fixed, but one should imagine that this isn't always the case.

~~~
contravariant
Famously Fermat himself gave a proof that is _assumed_ to contain at least
some nontrivial errors that couldn't be fixed.

Unfortunately this miraculous proof is lost to history as Fermat couldn't
spare the paper.

