
Decades-Old Computer Science Conjecture Solved in Two Pages - parsimo2010
https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/
======
floatingatoll
On boolean sensitivies:
[https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/07/Boolea...](https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/07/Boolean_Sensitivity_FINAL560-1068x1720.jpg)

The closing quote:

> [Huang] was able to prove that in any collection of more than half the
> points in an n-dimensional cube, there will be some point that is connected
> to at least √n of the other points — and the sensitivity conjecture
> instantly followed from this result

The actual paper:
[https://arxiv.org/abs/1907.00847](https://arxiv.org/abs/1907.00847)

Another blog post:
[https://www.scottaaronson.com/blog/?p=4229](https://www.scottaaronson.com/blog/?p=4229)

An older HN post:
[https://news.ycombinator.com/item?id=20338281](https://news.ycombinator.com/item?id=20338281)

~~~
ssalka
This is pure intuition, but from reading this I get the sense that this
conjecture is some generalization of a fixed point theorem?

~~~
vecter
Can you explain your intuition for how this relates to fixed points?

~~~
zeckalpha
A Boolean array is isomorphic to a fixed point number.

The article discusses non-fixed equivalents, though, both a contextual
variable length and a quantum super position length bit array.

~~~
vecter
I don't understand what you said, but I and (I assume) GP were talking about
[https://en.wikipedia.org/wiki/Fixed_point_(mathematics)](https://en.wikipedia.org/wiki/Fixed_point_\(mathematics\))

~~~
zeckalpha
From the article:

> Other measures involve looking for the simplest way to write the Boolean
> function as a mathematical expression, or calculating how many answers the
> banker would have to show a boss to prove they had made the right loan
> decision. There’s even a quantum physics version of query complexity in
> which the banker can ask a “superposition” of several questions at the same
> time. Figuring out how this measure relates to other complexity measures has
> helped researchers understand the limitations of quantum algorithms.

I assumed you were discussing [https://en.wikipedia.org/wiki/Fixed-
point_arithmetic](https://en.wikipedia.org/wiki/Fixed-point_arithmetic) which
is a potential application domain for this research.

~~~
vecter
I'm not seeing the connection between this research and fixed-point
arithmetic.

~~~
eob123
I don’t exactly follow the above either, but I believe fixed point is meant in
a different sense here, as in a function f(x) such there a point x0 exists
where x0=f(x0).

[https://en.m.wikipedia.org/wiki/Fixed-
point_theorem](https://en.m.wikipedia.org/wiki/Fixed-point_theorem)

------
chmaynard
A fine article. A great science writer like Dr. Erica Kalrreich can make
anything interesting, even theoretical CS. Other articles by this author:

[https://www.quantamagazine.org/authors/erica-
klarreich/](https://www.quantamagazine.org/authors/erica-klarreich/)

~~~
dlkmp
It's interesting to read and extremely accessible to the point that nothing
can be learned about the actual problem and its solution.

------
dang
Recently discussed at
[https://news.ycombinator.com/item?id=20338281](https://news.ycombinator.com/item?id=20338281).

------
iamcreasy
Here is an excellent ELI5 on reddit:
[https://www.reddit.com/r/explainlikeimfive/comments/ci0q00/e...](https://www.reddit.com/r/explainlikeimfive/comments/ci0q00/eli5_the_sensitivity_conjecture_has_been_solved/)

------
gigatexal
What a fascinating article and kudos to Huang for solving it. I’m excited to
see what benefits this has in the future for computing. Not many articles do I
read all the way through (I’ve the attention span of a fly these days) but
this one I did.

------
Tepix
Good read! Does this new proof have an effect on cryptography given that it's
often desirable to have highly sensitive cryptographic functions (you flip one
bit in the input and get a very different output)?

~~~
rocqua
Sensitivity here is defined as

Take all possible input strings

Given any input string, the sensitivity of that string is how many bits you
could flip that cause the output to flip.

Now take the /maximum/ of all input string sensitivities.

For e.g. a hash function, you'd want either a minimum or something like a 1st
percentile.

~~~
userbinator
_For e.g. a hash function, you 'd want either a minimum or something like a
1st percentile._

As far as I know, all cryptographic hash functions are sensitive to single
bit-flips by design.

~~~
rocqua
It is definitely a design goal, but I would be very surprised if sha2 did not
have 2 inputs that only differ by a single bit, and have the same output.

Finding those inputs is essentially impossible, but for a true 'random oracle
' they are likely to exist.

~~~
SilasX
Note: correct me if I’m wrong, but to be relevant to this theorem and
definition of sensitivity, the two strings would have to be for the same
circuit and therefore be the same length, which is much harder than finding
hash collisions in general.

And I thought they had ways to construct hash functions so that all the inputs
of the same length have a different output?

~~~
Leszek
> And I thought they had ways to construct hash functions so that all the
> inputs of the same length have a different output?

That's trivially impossible for fixed-size hashes, by the pigeon hole
principle.

~~~
lonelappde
Not if the output is longer than the input, such as for passwords

~~~
gitrebase
I doubt any study of hash functions restricts itself to studying a very tiny
subset of all possible (infinite) inputs. The security of hash functions is
always studied under the assumption of arbitrary inputs.

------
garfieldnate
>the new proof is so simple that one researcher summed it up in a single
tweet.

Didn't look that simple to me! Reminds me of Andrew Ng showing his students
the simple one liner to solve the cocktail party problem in Octave. There's a
lot represented in that one line of code!

~~~
fastball
Where is the tweet?

~~~
username3
[https://twitter.com/BooleanAnalysis/status/11458375764876124...](https://twitter.com/BooleanAnalysis/status/1145837576487612416)

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Exuma
Can someone explain the 'answer' simply? I 70% understand the initial
problem... I think

~~~
fastball
[https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/07/Boolea...](https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/07/Boolean_Sensitivity_FINAL560-1068x1720.jpg)

~~~
Exuma
I read that, but I believe that's describing the problem. Anyone care to
explain the answer in a similar simple way?

Actually one thing weird wiht that explanation is they call sensitive bits the
ones that don't change the output? You'd think it would be the red ones

~~~
fastball
That is the solution! See? It's so simple it seems like a formulation of the
problem itself!

The solution is to think of the input ('001') in terms of an n-dimensional
cube, where _n_ is the length of the input.

So for example, a binary logic with 5 bits ('01010') would require a
5-dimensional cube. From there, you check whether moving from one input
('00001') to an adjacent input ('00011') causes a flip in the output. If it
does, you label it as "red", and if it doesn't, you label it as "blue". Then
you merely find the vertex with the highest number of opposite colors, and the
number of opposite colors is the sensitivity.

~~~
Exuma
Interesting... so what does a "5 dimensional" cube look like? I am having
difficulty visualizing that

~~~
fastball
An easier way to think of it is just as a matrix where each vertex is
connected to _n_ other vertices. The reason an n-cube is used specifically is
because the bits correspond to spatial coordinates, which obviously require a
uniform structure (like a cube if we're talking 3D) in order to make sense.

This[0] article might help though.

[0]
[https://en.wikipedia.org/wiki/5-cube](https://en.wikipedia.org/wiki/5-cube)

~~~
Exuma
Thank you!

------
plg
nice use of a single .tex file to produce the full .pdf

[https://arxiv.org/e-print/1907.00847](https://arxiv.org/e-print/1907.00847)

(rename to 1907.00847.tex and then latexmk -pdf 1907.00847.tex)

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IshKebab
Can someone explain this in a slightly less dumbed-down way. Having trouble
following all the wooly explanations.

So far I get that we have a function that maps from a string of bits to a
single bit. The sensitivity of each input bit is the likelihood that it
affects the output, summed across all possible inputs.

What is the conjecture?

------
ohduran
The history of computer science is dauting. Unlike physics, though, it's the
history of incremental innovations that culminated in what we have today.
Thus, if you go back in time long enough, things eventually are manageable.

A good primer on the history of CS is probably Code, The Hidden Language of
Computer Hardware and Software, by Charles Pretzold. I've put together some
notes on it here: [http://alvaroduran.com/code](http://alvaroduran.com/code).

Any feedback is much appreciated!

------
PascLeRasc
Is sensitivity weighted by which stage it is for multi-stage logic chains?
I.E. for the example graphic, we could argue that the output of the OR into
the AND is an input, or in the bank loan example it might ask "Are you
married?" and if you answer yes it asks how much your partner makes, so it's a
second-stage input to the final boolean calculation - is that taken into
consideration when calculating sensitivity?

~~~
aesthesia
No. To compute sensitivity we consider the entire boolean function abstractly,
without any reference to a particular implementation. There are other
equivalent implementations of the same function with different arrangements
(and numbers) of gates, so this wouldn't necessarily make sense to include.
There are measures of complexity that do (at least sort of) take into account
the way that you implement a function with gates, and they typically look at
the circuit with the fewest gates or the shortest paths that computes the
function.

------
Bootwizard
Can anyone explain the significance of this finding? Any technologies that can
benefit from the application of this?

~~~
daxfohl
Usually with things like this, the result is already fairly well established,
or close enough at least, within the scale any real-world application would
require.

You can think of it like the four color theorem. A beautiful theoretical
result (though a far less beautiful proof), but the only practical
significance is now cartographers _know_ they'll never need that extra
crayon....

~~~
tathougies
I think the most interesting point of the paper isn't simply the proof the
conjecture but also the sqrt(n) bound. I doubt many downstream results were
predicated on that particular bound.

~~~
carlmr
>Huang’s result is even stronger than necessary to prove the sensitivity
conjecture, and this power should yield new insights about complexity
measures.

If I interpret this correctly it's a tighter bound than the original
conjecture, so it should allow better optimizations.

------
ghostbrainalpha
Pretty good for an "Amateur Mathematician", I can't wait to see what he does
once he goes pro.

~~~
nradclif
Huang is a professional mathematician at Emory.

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kian
Is this in any way related to the quantum algorithm for searching a space in
√n time?

~~~
kwaugh
No

------
Simon_says
Surprisingly good for a Quanta Magazine article.

~~~
HelloNurse
It's an important and interesting but unusually simple mathematical topic that
can be explained quite well without dumbing it down, leading to a
"surprisingly good" article compared to scientific divulgation in general.

~~~
yeukhon
What is the importance of such problem in real life?

