
P vs. NP - ikeboy
http://www.scottaaronson.com/blog/?p=3095
======
bmh100
If you are interested in the implications of P vs. NP, as opposed to a survey
of the situation around the problem, check out this excellent essay [1] by
Scott Aaronson. He goes into interesting consequences about the nature of
quantum computing, simulation of a human mind, and time travel, among other
topics.

There is also another interesting article [2] by Arkady Bolotin at Ben-Gurion
University in Israel, who argues that the lack of macroscopic quantum behavior
implies P!=NP. Basically, macroscopic quantum behavior would imply that the
universe is computing the solution to a NP-hard problem in violation with
currently accepted laws of physics. Also, see this debate between Bolotin and
an educated commenter [3].

[1]:
[http://www.scottaaronson.com/papers/philos.pdf](http://www.scottaaronson.com/papers/philos.pdf)
[pdf]

[2]: [https://medium.com/the-physics-arxiv-blog/the-astounding-
lin...](https://medium.com/the-physics-arxiv-blog/the-astounding-link-between-
the-p-np-problem-and-the-quantum-nature-of-universe-7ef5eea6fd7a#.hybznxnd5)

[3]:
[https://www.researchgate.net/post/P_NP_and_quantum_nature_of...](https://www.researchgate.net/post/P_NP_and_quantum_nature_of_universe-
any_thoughts)

~~~
ikeboy
>The problem is that the equation says nothing about how large an object needs
to be before it obeys Newtonian mechanics rather than the quantum variety.

I'm pretty sure this is BS.

The linked medium post does nothing to defend the implicit premise that the
universe can be computed in a reasonable amount of time by classical
computers.

~~~
frozenport
Indeed it is, interference phenomena have so far scaled up to fairly large
many atom structures in rather predictable ways. The real problem for Quantum
mechanics starts happening when needing to explain gravity.

~~~
dlubarov
I think you're talking past the parent, whose point is that our universe's
physics may not be polynomial time computable (or computable at all, for that
matter).

~~~
ikeboy
I made two different points. The first is that there's nothing that "obeys
Newtonian mechanics rather than the quantum variety.", it's quantum all the
way up. The second is what you said.

Guess it wasn't clear enough that the two points aren't related.

------
lisper
Do yourself a favor and take the time to read this paper not just to learn
about P=?NP but also to see a model of how technical writing ought to be done.
Scott Aaronson is truly the Richard Feynman of our day. Not only is he
freaking brilliant, but his writing style is accessible, not too mathy, even
humorous at times. This made me LOL:

"We see a similar “invisible fence” phenomenon in Leslie Valiant’s program of
“accidental algorithms” [246]. The latter are polynomial-time algorithms,
often for planar graph problems, that exist for certain parameter values but
not for others, for reasons that are utterly opaque if one doesn’t understand
the strange cancellations that the algorithms exploit. A prototypical result
is the following:

[Abstruse theorem elided, but it boils down to: there's this weird problem
with a parameter k that can be proven to have polynomial-time solutions for
some values of k and proven to be NP-hard for other values of k.]

Needless to say (because otherwise you would’ve heard!), in not one of these
examples have the “P region” and the “NP-complete region” of parameter space
been discovered to overlap."

~~~
pron
> to see a model of how technical writing ought to be done

Well, this is semi-popular technical writing directed at people who are _not_
theory-of-computation researchers. If all scientific writing would be written
like this, then all papers would be at least hundreds of pages long and skip
important details.

It's absolutely wonderful -- and I think vital -- that a researcher takes the
time to do popular writing, but most scientific writing can't be popular. This
kind of writing is usually reserved for _surveys_ , like this one. Popular
renderings of groundbreaking research also exist in publications like
Scientific American, and blog posts by researchers like Aaronson who like (and
are good at) communicating their findings in this way, but the important
details are often very much research-level and quite impenetrable to all but
very specialized researchers.

~~~
coldtea
> _Well, this is semi-popular technical writing directed at people who are not
> theory-of-computation researchers._

Nothing wrong about directing a technical writing to people other than
"theory-of-computation" researchers. Most people are not theory-of-computation
researchers, not just laymen, but also also statistics researchers, topology
researchers, geometricians, logicians, etc.

> _If all scientific writing would be written like this, then all papers would
> be at least hundreds of pages long and skip important details._

GP obviously means all scientific writing aimed at the same kind of audience.

Theory-of-computation researchers would obviously not need such a volume at
all in the first place: they already know this stuff, and they can just
directly read pure research on P vs NP, not some summary of what P vs NP
means.

That said, all scientific writing (including those aimed/produced by experts
in a field) could take from the quality of the wording, and the attention to
clarity. It doesn't mean it has to also follow the lengthy introduction to its
subject.

In other words, it seems to me like we're splitting hairs, while the intended
meaning was pretty clear from the GP's comment context.

~~~
pron
> GP obviously means all scientific writing aimed at the same kind of
> audience.

In that case I agree. :)

------
heliophobicdude
Abstract: In 1950, John Nash sent a remarkable letter to the National Security
Agency, in which—seeking to build theoretical foundations for cryptography—he
all but formulated what today we call the P=?NP problem, considered one of the
great open problems of science. Here I survey the status of this problem in
2016, for a broad audience of mathematicians, scientists, and engineers. I
offer a personal perspective on what it’s about, why it’s important, why it’s
reasonable to conjecture that P≠NP is both true and provable, why proving it
is so hard, the landscape of related problems, and crucially, what progress
has been made in the last half-century toward solving those problems. The
discussion of progress includes diagonalization and circuit lower bounds; the
relativization, algebrization, and natural proofs barriers; and the recent
works of Ryan Williams and Ketan Mulmuley, which (in different ways) hint at a
duality between impossibility proofs and algorithms.

~~~
dilap
The handwritten letters from Nash are awesome. Highly recommend checking them
out.

------
alanh
Any survey that mentions Borges in its introductory section is A-OK in my
book.

 _Edit_ This reminds me of an assignment my Discrete Math professor, Dr.
Fishel, gave me, which shows how to create (in polynomial time) a Hamiltonian
circuit in any undirected graph in which every vertex is of the order
ceiling(n/2) where n is the number of vertices in the graph. A C++
implementation is provided. [2007] [https://alanhogan.com/asu/hamiltonian-
circuit/](https://alanhogan.com/asu/hamiltonian-circuit/)

~~~
zeroer
You must be forgetting some important prerequisite to the graph, because
there's no circuit on a graph of two vertices and a single edge between them.

~~~
alanh
n=3 is the minimum. I believe this is in the associated presentation.

------
mathgenius
His remarks on "geometric complexity theory" are very interesting (starting
p80):

 _I like to describe GCT as “the string theory of computer science.” Like
string theory, GCT has the aura of an intricate theoretical superstructure
from the far future, impatiently being worked on today. Both have attracted
interest partly because of “miraculous coincidences” (for string theory, these
include anomaly cancellations and the prediction of gravitons; for GCT,
exceptional properties of the permanent and determinant, and surprising
algorithms to compute the multiplicities of irreps). Both have been described
as deep, compelling, and even “the only game in town” (not surprisingly, a
claim disputed by the fans of rival ideas!). And like with string theory,
there are few parts of modern mathematics not known or believed to be relevant
to GCT._

------
gregschlom
>...and the recent works of Ryan Williams and Ketan Mulmuley, which (in
different ways) hint at a duality between impossibility proofs and algorithm

This is way over my head, but anyone has any insight on that? Sounds
intriguing.

~~~
j2kun
There is a large, emerging body of research in which one proves the
impossibility of solving problem A by finding an algorithm for solving a
different problem B.

For Mulmuley et al. (it's been a while since I was in the thick of this) it's
finding an algorithm for computing a particular decomposition of a
representation[1] having to do with determinants and permanents. This
essentially shows the matrix permanent formula can't be written as a
determinant of a small matrix (of smaller formulas of the variables), which is
the "algebraic" analogue of P vs NP, and which Mulmuley et al. believe will
pave a road toward P vs NP. Last I heard there was a serious blow to this
program. (edit [3])

For Ryan Williams, it's a more direct approach to separating complexity
classes such as ACC (a small circuit class) and NEXP (a large turing machine
class), where this[2] is the most amazing explanatory technical document I
have seen for almost anything. He also has this program whereby if you can
find faster algorithms for boolean satisfiability (SAT)—that is, faster
exponential-time algorithms—you get complexity class separations "for free."

[1]:
[https://en.wikipedia.org/wiki/Representation_theory](https://en.wikipedia.org/wiki/Representation_theory)
[2]: [https://arxiv.org/abs/1111.1261](https://arxiv.org/abs/1111.1261) [3]:
[https://arxiv.org/abs/1604.06431](https://arxiv.org/abs/1604.06431)

------
mroll
I've read both of his other survey articles linked in the post. Judging by
those this will certainly be interesting. Aaronson does a great job tying
formal theory to applications and getting across a good mental model for the
math he writes about. Looking forward to digging into this one!

------
d23
A guy at a company I used to work for made a great video on this. Very
understandable even without any background on the topic:
[https://www.youtube.com/watch?v=YX40hbAHx3s](https://www.youtube.com/watch?v=YX40hbAHx3s)

------
cyorir
I'll read this if I can afford the time! It would be nice if there were a
summary more detailed than the abstract but still short relative to the length
of the article (say, 10-pages long?).

~~~
powera
If I had to summarize it from one sentence from the article, I'd go with:

"The one prediction I feel confident in making is that the idea of “ironic
complexity theory”—i.e., of a profound duality between upper and lower bounds,
where the way to prove that there’s no fast algorithm for problem A is to
discover fast algorithms for problems B, C, and D—is here to stay."

------
DonHopkins
My eccentric CS professor had a t-shirt that said "P = NP? I don't know and I
don't care."

~~~
gspetr
I saw another amusing one: "P = NP" over a t-shirt depicting that X-Files
picture - "I want to believe".

------
nabla9
> 6.4 Ironic Complexity Theory

TIL

~~~
jonathanstrange
I suppose it has been invented in England (or Denmark).

~~~
Ar-Curunir
Nah, Stanford.

------
debt
Is there any math addressing how much more quickly NP-hard problems could be
computed when time is a flat circle?

~~~
gjm11
Yup, or at least when you have access to a "closed timelike curve" which I
think is close enough to what you're asking. See, e.g., this -- which is a
transcript of a lecture given by the author of the survey article in the OP.
[http://www.scottaaronson.com/democritus/lec19.html](http://www.scottaaronson.com/democritus/lec19.html)

With the help of a CTC you can solve any problem in class NP in polynomial
time; in fact you can do better and solve anything in PSPACE in polynomial
time; and the capabilities of quantum and classical computers are the same.

(The last of those is a result due to Scott Aaronson -- author of the survey
article in the OP -- and John Watrous.)

