
Wolfram Rule 30 Prizes - soofy
https://www.rule30prize.org
======
lacker
It is pretty hard in general to prove that calculating things takes at least
O(n) time, so I wouldn't expect that one to be solved any time soon.

Personally, I am curious if there is any cellular automaton that has something
like 3-dimensional rotational symmetry. If our universe can be described by a
cellular automaton, it isn't obvious to me how such a symmetry could arise,
but I wouldn't be surprised if someone figured it out.

~~~
nswanberg
For what it's worth, Stephen decided that network was a better model for the
universe than a cellular automaton, and thought that space might emerge as a
property of that network rather than be "defined in", as in a cellular
automaton.

[https://www.wolframscience.com/nks/p475--space-as-a-
network/](https://www.wolframscience.com/nks/p475--space-as-a-network/)

(In the preceding section he discussed some constraints that a cellular
automaton would put on the universe model, but didn't dismiss the idea for
that reason)

~~~
jkp56
I can't open this link for some reason (uBO?), but the title reminds me one
very clever Wolfram's article where he brags about (as usual) about how he
derived GTR equations from his graph model. That article had a bunch of
comments with and one of them stating that in such a graph model there is
always a reference frame. Wolfram didn't respond to that comment.

~~~
mikhailfranco
I haven't followed this closely, but did SW provide a paper showing how he
derived GTR from the graph?

~~~
sokrates85
He made the claims but not sure if that paper ever appeared.

------
wwarner
An aside: 3D cellular automata
[https://www.youtube.com/watch?v=_W-n510Pca0](https://www.youtube.com/watch?v=_W-n510Pca0)

~~~
DonHopkins
Very beautiful and artistically rendered! Those would make great fireworks and
weapons in Minecraft!

From a different engineering perspective, Dave Ackley had some interesting
things to say about the difficulties of going from 2D to 3D, which I quoted in
an earlier discussion about visual programming:

[https://news.ycombinator.com/item?id=18497585](https://news.ycombinator.com/item?id=18497585)

David Ackley, who developed the two-dimensional CA-like "Moveable Feast
Machine" architecture for "Robust First Computing", touched on moving from 2D
to 3D in his retirement talk:

[https://youtu.be/YtzKgTxtVH8?t=3780](https://youtu.be/YtzKgTxtVH8?t=3780)

"Well 3D is the number one question. And my answer is, depending on what mood
I'm in, we need to crawl before we fly."

"Or I say, I need to actually preserve one dimension to build the thing and
fix it. Imagine if you had a three-dimensional computer, how you can actually
fix something in the middle of it? It's going to be a bit of a challenge."

"So fundamentally, I'm just keeping the third dimension in my back pocket, to
do other engineering. I think it would be relatively easy to imaging taking a
2D model like this, and having a finite number of layers of it, sort of a 2.1D
model, where there would be a little local communication up and down, and then
it was indefinitely scalable in two dimensions."

"And I think that might in fact be quite powerful. Beyond that you think about
things like what about wrap-around torus connectivity rooowaaah, non-euclidian
dwooraaah, aaah uuh, they say you can do that if you want, but you have to
respect indefinite scalability. Our world is 3D, and you can make little
tricks to make toruses embedded in a thing, but it has other consequences."

Here's more stuff about the Moveable Feast Machine:

[https://news.ycombinator.com/item?id=15560845](https://news.ycombinator.com/item?id=15560845)

[https://news.ycombinator.com/item?id=14236973](https://news.ycombinator.com/item?id=14236973)

The most amazing mind blowing demo is Robust-first Computing: Distributed City
Generation:

[https://www.youtube.com/watch?v=XkSXERxucPc](https://www.youtube.com/watch?v=XkSXERxucPc)

And a paper about how that works:

[https://www.cs.unm.edu/~ackley/papers/paper_tsmall1_11_24.pd...](https://www.cs.unm.edu/~ackley/papers/paper_tsmall1_11_24.pdf)

Plus there's a lot more here:

[https://movablefeastmachine.org/](https://movablefeastmachine.org/)

Now he's working on a hardware implementation of indefinitely scalable robust
first computing:

[https://www.youtube.com/channel/UC1M91QuLZfCzHjBMEKvIc-A](https://www.youtube.com/channel/UC1M91QuLZfCzHjBMEKvIc-A)

~~~
Iv
> Very beautiful and artistically rendered! Those would make great fireworks
> and weapons in Minecraft!

Someone coded similar things in minetest, the open source clone of minecraft:

[https://forum.minetest.net/viewtopic.php?f=9&t=17608](https://forum.minetest.net/viewtopic.php?f=9&t=17608)

------
aarestad
What are the implications of positive/negative answers to these questions,
and/or what else aside from "because it's there" motivates answering these
questions?

~~~
uoaei
Who knows? Answers to these questions might spur discussion on connections
between cellular automata and other computational constructions. I think
that's Wolfram's angle -- justify the cellular-automata-is-everything tack
that he's been on the past 20-30 years.

~~~
DonHopkins
Maybe it's a scheme to drum up attention for his new upcoming line of "Rule 30
Wearable Cellular Automata Clothing and Fashion Accessories".

[https://www.kickstarter.com/projects/fbz/knityak-custom-
math...](https://www.kickstarter.com/projects/fbz/knityak-custom-mathematical-
knit-scarves)

They make great tattoos too:

[http://i.imgur.com/mct1AFX.jpg](http://i.imgur.com/mct1AFX.jpg)

[https://geekytattoos.wordpress.com/2011/04/14/wolfram-2-stat...](https://geekytattoos.wordpress.com/2011/04/14/wolfram-2-state-3-color-
turing-machine/)

That one is actually quite controversial:

[https://en.wikipedia.org/wiki/Wolfram%27s_2-state_3-symbol_T...](https://en.wikipedia.org/wiki/Wolfram%27s_2-state_3-symbol_Turing_machine)

------
DonHopkins
Oh My Gosh, It’s Covered in Rule 30s

[http://blog.stephenwolfram.com/2017/06/oh-my-gosh-its-
covere...](http://blog.stephenwolfram.com/2017/06/oh-my-gosh-its-covered-in-
rule-30s/)

HN discussion:

[https://news.ycombinator.com/item?id=14458955](https://news.ycombinator.com/item?id=14458955)

------
DonHopkins
I gave some links to visualizations of CA basins of attraction including Rule
30 in my previous post about "Garden of Eden" configurations:

[https://news.ycombinator.com/item?id=14468707](https://news.ycombinator.com/item?id=14468707)

There's a thing called a "Garden of Eden" configuration that has no
predecessors, which is impossible to get to from any other possible state.

For a rule like Life, there are many possible configurations that must have
been created by God or somebody with a bitmap editor (or somebody who thinks
he's God and uses Mathematica as a bitmap editor, like Stephen Wolfram ;),
because it would have been impossible for the Life rule to evolve into those
states. For example, with the "Life" rule, no possible configuration of cells
could ever evolve into all cells with the value "1".

[https://en.wikipedia.org/wiki/Garden_of_Eden_(cellular_autom...](https://en.wikipedia.org/wiki/Garden_of_Eden_\(cellular_automaton\))

For a rule that simply sets the cell value to zero, all configurations other
than pure zeros are garden of eden states, and they all lead directly into a
one step attractor of all zeros which always evolves back into itself, all
zeros again and again (the shortest possible attractor loop that leads
directly to itself).

There is a way of graphically visualizing that global rule state space, which
gives insight into the behavior of the rule and the texture and complexity of
its state space!

Andrew Wuensche and Mike Lesser published a gorgeous coffee table book
entitled "The Global Dynamics of Cellular Automata" that plots out the
possible "Garden of Eden" states and the "Basins of Attraction" they lead into
of many different one-dimensional cellular automata like rule 30.

[http://users.sussex.ac.uk/~andywu/gdca.html](http://users.sussex.ac.uk/~andywu/gdca.html)

The beautiful color plates begin on page 79 in the free pdf file:

[http://users.sussex.ac.uk/~andywu/downloads/papers/global_dy...](http://users.sussex.ac.uk/~andywu/downloads/papers/global_dynamics_of_CA.pdf)

I've uploaded the money shots to imgur:

[http://imgur.com/gallery/s3dhz](http://imgur.com/gallery/s3dhz)

Those are not pictures of 1-d cellular automata rule cell states on a grid
themselves, but they are actually graphs of the abstract global state space,
showing merging and looping trajectories (but not branching since the rules
are deterministic -- time flows from the garden of eden leaf tips around the
perimeter into (then around) the basin of attractor loops in the center,
merging like springs (GOE) into tributaries into rivers into the ocean (BOA)).

The rest of the book is an atlas of all possible 1-d rules of a particular
rule numbering system (like rule 30, etc), and the last image is the legend.

He developed a technique of computing and plotting the topology network of all
possible states a CA can get into -- tips are "garden of eden" states that no
other states can lead to, and loops are "basins of attraction".

Here is the illustration of "rule 30" from page 144 (the legend explaining it
is the last photo in the above album). [I am presuming it's using the same
rule numbering system as Wolfram but I'm not sure -- EDIT: I visually checked
the "space time pattern from a singleton seed" thumbnail against the
illustration in the article, and yes it matches rule 30!]

[http://imgur.com/a/lKAbP](http://imgur.com/a/lKAbP)

"The Global Dynamics of Cellular Automata introduces a powerful new
perspective for the study of discrete dynamical systems. After first looking
at the unique trajectory of a system's future, an algoritm is presented that
directly computes the multiple merging trajectories of the systems past. A
given cellular automaton will "crystallize" state space into a set of basins
of attraction that typically have a topology of trees rooted on attractor
cycles. Portraits of these objects are made accessible through computer
generated graphics. The "Atlas" presents a complete class of such objects, and
is inteded , with the accompanying software, as an aid to navigation into the
vast reaches of rule behaviour space. The book will appeal to students and
researchers interested in cellular automata, complex systems, computational
theory, artificial life, neural networks, and aspects of genetics."

[https://en.wikipedia.org/wiki/Attractor](https://en.wikipedia.org/wiki/Attractor)

"Basins of attraction in cellular automata", by Andrew Wuensche:

[http://onlinelibrary.wiley.com/doi/10.1002/1099-0526(200007/...](http://onlinelibrary.wiley.com/doi/10.1002/1099-0526\(200007/08\)5:6%3C19::AID-
CPLX5%3E3.0.CO;2-J/full)

"To achieve the global perspective. I devised a general method for running CA
backwards in time to compute a state's predecessors with a direct reverse
algorithm. So the predecessors of predecessors, and so on, can be computed,
revealing the complete subtree including the "leaves," states without
predecessors, the so-called “garden-of-Eden" states.

Trajectories must lead to attractors in a finite CA, so a basin of attraction
is composed of merging trajectories, trees, rooted on the states making up the
attractor cycle with a period of one or more. State-space is organized by the
"physics" underlying the dynamic behavior into a number of these basins of
attraction, making up the basin of attraction field."

Pencil drawing from the very early days before automatic computer drawing was
perfected:

[http://www.ddlab.com/r30Pencil.gif](http://www.ddlab.com/r30Pencil.gif)

If you like the book, you'll love the code!

[http://www.ddlab.com/](http://www.ddlab.com/)

[http://www.ddlab.com/screensave3.png](http://www.ddlab.com/screensave3.png)

[http://uncomp.uwe.ac.uk/wuensche/2006_ddlab_slides1.pdf](http://uncomp.uwe.ac.uk/wuensche/2006_ddlab_slides1.pdf)

[http://uncomp.uwe.ac.uk/wuensche/meta.html](http://uncomp.uwe.ac.uk/wuensche/meta.html)

[http://uncomp.uwe.ac.uk/wuensche/boa_idea.html](http://uncomp.uwe.ac.uk/wuensche/boa_idea.html)

[http://uncomp.uwe.ac.uk/wuensche/downloads/papers/2008_dd_ov...](http://uncomp.uwe.ac.uk/wuensche/downloads/papers/2008_dd_overview_preprint.pdf)

~~~
codezero
The third link is a dead link, but it looks like it's up here now:
[http://users.sussex.ac.uk/~andywu/gdca.html](http://users.sussex.ac.uk/~andywu/gdca.html)

The PDF is here:
[http://users.sussex.ac.uk/~andywu/downloads/papers/global_dy...](http://users.sussex.ac.uk/~andywu/downloads/papers/global_dynamics_of_CA.pdf)

~~~
DonHopkins
Thank you! I fixed the links, and added the link to the Doctor-Seuss-like
pencil drawing.

------
d-d
Haha, this is just P versus NP for $970,000 less in prizes.

------
mar77i
So I typed a couple of digits into OEIS, which returned
[http://oeis.org/A080847](http://oeis.org/A080847), "mu(n) + 2", where mu is
the
[https://en.wikipedia.org/wiki/M%C3%B6bius_function](https://en.wikipedia.org/wiki/M%C3%B6bius_function)

This doesn't appear to be true for arbitrarily many digits, and I'm struggling
to see how the two fields may even be connected. Some properties appear
similar, eg. the Möbius function, too, appears to sum up to 0.

------
mhroth
If I were still in grad school, I would get lost in this. Alas, days gone
by...

------
Havoc
I love the fact that this is #1 on hn with a 30k prize.

Meanwhile the major tech corps would kill for that kind of bang per buck.

~~~
wavefunction
Reading the prize guidelines Wolfram gets the right to publish the proofs but
that's it, which seems fair. Big tech companies perhaps have to pay more
because they own the work product and it's presumably for some commercial
reason?

------
anderskaseorg
Computing the nth cell clearly takes at least 1 time unit, and by the
definition of O(), we have 1 = O(n). I’ll be claiming my $10,000 now.

(More seriously, don’t confuse O() with Ω() and Θ(), folks:
[https://en.wikipedia.org/wiki/Big_O_notation.](https://en.wikipedia.org/wiki/Big_O_notation.))

~~~
AlEinstein
Would you mind providing a little more detail of your reasoning?

For example, say that someone shows that the most efficient algorithm for
computing the value (0 or 1) of the central column of the nth row of rule 30
starting with a single 1 cell (i.e., the system under consideration for the
prize) takes time n^2. Wouldn’t one then say that the complexity of the
calculation is O(n^2)? One couldn’t say that it’s O(n), surely?

------
wwarner
love this phrase: "naturally constructed" numbers, suggesting that the numbers
we use every day are themselves computational conveniences, and possibly that
another number construction could be useful at other scales.

~~~
pohl
I thought maybe this was a reference to constructivist mathematics, to refer
to only this numbers that are algorithmic and exclude numbers we know exist in
ℝ but have to be proven to exist using the axiom of choice.

~~~
wwarner
maybe, probably, it was. But the idea that a measure of one of these rules can
be interpreted as a number sparked my imagination.

I have to say also that interpreting the grid as space itself, and color of
the square as the presence or absence of a particle is pretty compelling. When
I think of it that way, then the proportion of each color matters a lot, as it
would be the way equilibrium is expressed.

------
breck
Has anyone seen a version of Rule 30 for a 2D array, instead of 1D? I'm doing
some searching but can't find one. Somewhat tangential to my angle of attack,
but would be helpful to me.

~~~
penagwin
What? Rule 30 is on a 2D plane.

EDIT: Oh I understand now! Each generation is 1D. Thanks for explaining! :D

~~~
kmote00
Quote from previous link, in answer to a similar question comparing CAs (1d)
to Conway's Life (2d):

"The pictures you're looking at, one of the axes is time. One row "becomes"
the row below it. Each time-step iteration adds a row to the bottom, while the
rows above it are immutable (because they are in the past)."

------
AnimalMuppet
Anybody got a mirror? The link seems to be dead (or overwhelmed).

------
abetusk
For some context, Wolfram published a book of collected papers called
"Cellular Automata and Complexity" [1] where he classified Cellular Automata
(CA) into four broad categories:

    
    
        I. Convergent uniform final state
        II.  Convergent simple pattern final state
        III. "Random"
        IV. "Complexity"
    

Where "Complexity" essentially means Turing Machine Equivalent [2]. See also
[3].

Later, Wolfram self-published "A New Kind of Science" (ANKoS). ANKoS is
abysmal and barely readable but, from what I can understand from what little I
read of it, Wolfram updated his understanding and instead basically classified
CA into two categories, either 'simple' or 'complex', where, again, 'complex'
means Turing Machine Equivalent.

If I remember correctly, I think Wolfram even used Rule 30 as a basis for
random number generators somewhere (Mathematica?).

Considering the tenor of the problems, maybe Wolfram is still essentially
classifying CA into his four initial categories as it looks like he's pushing
for the idea that "Rule 30 == 'random'".

My personal take on this is that the idea that there are basically "two"
classifications of CA, either 'simple' (non-Turing Machine equivalent) and
'complex' (Turing Machine equivalent) is correct. Though it might be the case
where the more 'random' a CA looks, the harder it is to do the reduction to
TME, maybe even going so far as having a diverging reduction cost.

Regardless, this is why these questions might be interesting. Is Rule 30 TME?
If so, then that answers question 3 (O(n) simulation is essentially saying
there's no "shortcut" and you're basically doing full computation). Questions
1 and 2 are, in my opinion, shades of the same TME question, where non-
periodic is another way of saying there's no short-circuit computation and
calculating the average is getting at the unpredictability (read 'required
computability') of the cells.

If Rule 30 isn't TME but still requires O(n) cost to predict, that's also
pretty interesting as it requires as much computing power to predict it but
isn't as powerful as a Turing Machine. From a broader perspective, this gets
at the whole "randomness vs. determinism" idea, as this is a deterministic
system that, for many purposes, behaves randomly. This idea is probably old
news to this crowd but there's a close link to TME and randomness that is
still not completely understood and investigations into CA of this sort are
another way to tackle this idea.

[1] [https://www.amazon.com/Cellular-Automata-Complexity-
Collecte...](https://www.amazon.com/Cellular-Automata-Complexity-Collected-
Papers/dp/0201626640/)

[2]
[https://en.wikipedia.org/wiki/Turing_machine_equivalents](https://en.wikipedia.org/wiki/Turing_machine_equivalents)

[3] [https://www.wolframscience.com/nks/p231--four-classes-of-
beh...](https://www.wolframscience.com/nks/p231--four-classes-of-behavior/)

