Mike's Notes
An excellent article from Quanta Magazine. Pipi 9 has some similarities to this.
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14/09/2025
Self-Assembly Gets Automated in Reverse of ‘Game of Life’
By: George Musser
Quanta Magazine 10/09/2025
George Musser is a contributing editor at Scientific American magazine
and the author of two books, Spooky Action at a Distance and The Complete
Idiot’s Guide to String Theory. He is the recipient of the 2011 American
Institute of Physics Science Writing Award and the 2010 American
Astronomical Society’s Jonathan Eberhart Planetary Sciences Journalism
Award. He was a Knight Science Journalism Fellow at MIT from 2014 to 2015.
He can be found on Mastodon and Bluesky.
In cellular automata, simple rules create elaborate structures. Now
researchers can start with the structures and reverse-engineer the
rules.
Alexander Mordvintsev showed me two clumps of pixels on his screen. They
pulsed, grew and blossomed into monarch butterflies. As the two butterflies
grew, they smashed into each other, and one got the worst of it; its wing
withered away. But just as it seemed like a goner, the mutilated butterfly
did a kind of backflip and grew a new wing like a salamander regrowing a
lost leg.
Mordvintsev, a research scientist at Google Research in Zurich, had not
deliberately bred his virtual butterflies to regenerate lost body parts; it
happened spontaneously. That was his first inkling, he said, that he was
onto something. His project built on a decades-old tradition of creating
cellular automata: miniature, chessboard-like computational worlds governed
by bare-bones rules. The most famous, the Game of Life, first popularized in
1970, has captivated generations of computer scientists, biologists and
physicists, who see it as a metaphor for how a few basic laws of physics can
give rise to the vast diversity of the natural world.
In 2020, Mordvintsev brought this into the era of deep learning by creating
neural cellular automata, or NCAs. Instead of starting with rules and
applying them to see what happened, his approach started with a desired
pattern and figured out what simple rules would produce it. “I wanted to
reverse this process: to say that here is my objective,” he said. With this
inversion, he has made it possible to do “complexity engineering,” as the
physicist and cellular-automata researcher Stephen Wolfram proposed in
1986(opens a new tab) — namely, to program the building blocks of a system
so that they will self-assemble into whatever form you want. “Imagine you
want to build a cathedral, but you don’t design a cathedral,” Mordvintsev
said. “You design a brick. What shape should your brick be that, if you take
a lot of them and shake them long enough, they build a cathedral for
you?”
Such a brick sounds almost magical, but biology is replete with examples of
basically that. A starling murmuration or ant colony acts as a coherent
whole, and scientists have postulated simple rules that, if each bird or ant
follows them, explain the collective behavior. Similarly, the cells of
your body play off one another to shape themselves into a single organism.
NCAs are a model for that process, except that they start with the
collective behavior and automatically arrive at the rules.
Alexander Mordvintsev created complex cell-based digital systems that use
only neighbor-to-neighbor communication.
Courtesy of Alexander Mordvintsev
The possibilities this presents are potentially boundless. If biologists
can figure out how Mordvintsev’s butterfly can so ingeniously regenerate a
wing, maybe doctors can coax our bodies to regrow a lost limb. For
engineers, who often find inspiration in biology, these NCAs are a potential
new model for creating fully distributed computers that perform a task
without central coordination. In some ways, NCAs may be innately better at
problem-solving than neural networks.
Life’s Dreams
Mordvintsev was born in 1985 and grew up in the Russian city of Miass, on
the eastern flanks of the Ural Mountains. He taught himself to code on a
Soviet-era IBM PC clone by writing simulations of planetary dynamics, gas
diffusion and ant colonies. “The idea that you can create a tiny universe
inside your computer and then let it run, and have this simulated reality
where you have full control, always fascinated me,” he said.
He landed a job at Google’s lab in Zurich in 2014, just as a new
image-recognition technology based on multilayer, or “deep,” neural networks
was sweeping the tech industry. For all their power, these systems were (and
arguably still are) troublingly inscrutable. “I realized that, OK, I need to
figure out how it works,” he said.
He came up with “deep dreaming,” a process that takes whatever patterns a
neural network discerns in an image, then exaggerates them for effect. For a
while, the phantasmagoria that resulted — ordinary photos turned into a
psychedelic trip of dog snouts, fish scales and parrot feathers — filled the
internet. Mordvintsev became an instant software celebrity.
Among the many scientists who reached out to him was Michael Levin(opens a
new tab) of Tufts University, a leading developmental biologist. If neural
networks are inscrutable, so are biological organisms, and Levin was curious
whether something like deep dreaming might help to make sense of them, too.
Levin’s email reawakened Mordvintsev’s fascination with simulating nature,
especially with cellular automata.
From a single cell, this neural cellular automata transforms into the shape
of a lizard.
The core innovation made by Mordvintsev, Levin and two other Google
researchers, Ettore Randazzo and Eyvind Niklasson, was to use a neural
network to define the physics of the cellular automaton. In the Game of Life
(or just “Life” as it’s commonly called), each cell in the grid is either
alive or dead and, at each tick of the simulation clock, either spawns, dies
or stays as is. The rules for how each cell behaves appear as a list of
conditions: “If a cell has more than three neighbors, it dies,” for example.
In Mordvintsev’s system, the neural network takes over that function. Based
on the current condition of a cell and its neighbors, the network tells you
what will happen to that cell. The same type of network is used to classify
an image as, say, a dog or cat, but here it classifies the state of cells.
Moreover, you don’t need to specify the rules yourself; the neural network
can learn them during the training process.
To start training, you seed the automaton with a single “live” cell. Then
you use the network to update the cells over and over again for dozens to
thousands of times. You compare the resulting pattern to the desired one.
The first time you do this, the result will look nothing like what you
intended. So you adjust the neural network’s parameters, rerun the network
to see whether it does any better now, make further adjustments, and repeat.
If rules exist that can generate the pattern, this procedure should
eventually find them.
The adjustments can be made using either backpropagation, the technique
that powers most modern deep learning, or a genetic algorithm, an older
technique that mimics Darwinian evolution. Backpropagation is much faster,
but it doesn’t work in every situation, and it required Mordvintsev to adapt
the traditional design of cellular automata. Cell states in Life are binary
— dead or alive — and transitions from one state to the other are abrupt
jumps, whereas backpropagation demands that all transitions be smooth. So he
adopted an approach developed by, among others, Bert Chan at Google’s Tokyo
lab in the mid-2010s. Mordvintsev made the cell states continuous values,
anything from 0 to 1, so they are never strictly dead or alive, but always
somewhere in between.
Mordvintsev also found that he had to endow each cell with “hidden”
variables, which do not indicate whether that cell is alive or dead, or what
type of cell it is, but nonetheless guide its development. “If you don’t do
that, it just doesn’t work,” he said. In addition, he noted that if all the
cells updated at the same time, as in Life, the resulting patterns lacked
the organic quality he was seeking. “It looked very unnatural,” he said. So
he began to update at random intervals.
Finally, he made his neural network fairly beefy — 8,000 parameters. On the
face of it, that seems perplexing. A direct translation of Life into a
neural network would require just 25 parameters, according to simulations
done in 2020 by Jacob Springer, who is now a doctoral student at Carnegie
Mellon University, and Garrett Kenyon of Los Alamos National Laboratory. But
deep learning practitioners often have to supersize their networks, because
learning to perform a task is harder than actually performing it.
Moreover, extra parameters mean extra capability. Although Life can
generate immensely rich behaviors, Mordvintsev’s monsters reached another
level entirely.
Fixer Upper
The paper that introduced NCAs to the world in 2020 included an
applet(opens a new tab) that generated the image of a green lizard. If you
swept your mouse through the lizard’s body, you left a trail of erased
pixels, but the animal pattern soon rebuilt itself. The power of NCAs not
just to create patterns, but to re-create them if they got damaged,
entranced biologists. “NCAs have an amazing potential for regeneration,”
said Ricard Solé of the Institute of Evolutionary Biology in Barcelona, who
was not directly involved in the work.
The butterfly and lizard images are not realistic animal simulations; they
do not have hearts, nerves or muscles. They are simply colorful patterns of
cells in the shape of an animal. But Levin and others said they do capture
key aspects of morphogenesis, the process whereby biological cells form
themselves into tissues and bodies. Each cell in a cellular automaton
responds only to its neighbors; it does not fall into place under the
direction of a master blueprint. Broadly, the same is true of living cells.
And if cells can self-organize, it stands to reason that they can
self-reorganize.
Cut off the tail of an NCA lizard and the form will regenerate
itself.
Sometimes, Mordvintsev found, regeneration came for free. If the rules
shaped single pixels into a lizard, they also shaped a lizard with a big
gash through it into an intact animal again. Other times, he expressly
trained his network to regenerate. He deliberately damaged a pattern and
tweaked the rules until the system was able to recover. Redundancy was one
way to achieve robustness. For example, if trained to guard against damage
to the animal’s eyes, a system might grow backup copies. “It couldn’t make
eyes stable enough, so they started proliferating — like, you had three
eyes,” he said.
"A kind of computer that looks like an NCA instead would be a vastly more
efficient kind of computer." - Blaise Agüera y Arcas
Sebastian Risi(opens a new tab), a computer scientist at the IT University
of Copenhagen, has sought to understand what exactly gives NCAs their
regenerative powers. One factor, he said, is the unpredictability that
Mordvintsev built into the automaton through features such as random update
intervals. This unpredictability forces the system to develop mechanisms to
cope with whatever life throws at it, so it will take the loss of a body
part in stride. A similar principle holds for natural species. “Biological
systems are so robust because the substrate they work on is so noisy,” Risi
said.
Last year, Risi, Levin and Ben Hartl, a physicist at Tufts and the Vienna
University of Technology, used NCAs to investigate how noise leads to
robustness. They added one feature to the usual NCA architecture: a memory.
This system could reproduce a desired pattern either by adjusting the
network parameters or by storing it pixel-by-pixel in its memory. The
researchers trained it under various conditions to see which method it
adopted.
If all the system had to do was reproduce a pattern, it opted for
memorization; fussing with the neural network would have been overkill. But
when the researchers added noise to the training process, the network came
into play, since it could develop ways to resist noise. And when the
researchers switched the target pattern, the network was able to learn it
much more rapidly because it had developed transferable skills such as
drawing lines, whereas the memorization approach had to start from scratch.
In short, systems that are resilient to noise are more flexible in
general.
Even if disturbed, the textures created by NCAs have the ability to heal
themselves.
The researchers argued that their setup is a model for natural evolution.
The genome does not prescribe the shape of an organism directly; instead, it
specifies a mechanism that generates the shape. That enables species to
adapt more quickly to new situations, since they can repurpose existing
capabilities. “This can tremendously speed up an evolutionary process,”
Hartl said.
Ken Stanley, an artificial intelligence researcher at Lila Sciences who has
studied computational and natural evolution, cautioned that NCAs, powerful
though they are, are still an imperfect model for biology. Unlike machine
learning, natural evolution does not work toward a specific goal. “It’s not
like there was an ideal form of a fish or something which was somehow shown
to evolution, and then it figured out how to encode a fish,” he noted. So
the lessons from NCAs may not carry over to nature.
Auto Code
In regenerating lost body parts, NCAs demonstrate a kind of problem-solving
capability, and Mordvintsev argues that they could be a new model for
computation in general. Automata may form visual patterns, but their cell
states are ultimately just numerical values processed according to an
algorithm. Under the right conditions, a cellular automaton is as fully
general as any other type of computer.
The standard model of a computer, developed by John von Neumann in the
1940s, is a central processing unit combined with memory; it executes a
series of instructions one after another. Neural networks are a second
architecture that distributes computation and memory storage over thousands
to billions of interconnected units operating in parallel. Cellular automata
are like that, but even more radically distributed. Each cell is linked only
to its neighbors, lacking the long-range connections that are found in both
the von Neumann and the neural network architectures. (Mordvintsev’s neural
cellular automata incorporate a smallish neural network into each cell, but
cells still communicate only with their neighbors.)
"You are forcing it not to memorize that answer, but to learn a process to
develop the solution." - Stefano Nichele
Long-range connections are a major power drain, so if a cellular automaton
could do the job of those other systems, it would save energy. “A kind of
computer that looks like an NCA instead would be a vastly more efficient
kind of computer,” said Blaise Agüera y Arcas, the chief technology officer
of the Technology and Society division at Google.
But how do you write code for such a system? “What you really need to do is
come up with [relevant] abstractions, which is what programming languages do
for von Neumann–style computation,” said Melanie Mitchell of the Santa Fe
Institute. “But we don’t really know how to do that for these massively
distributed parallel computations.”
A neural network is not programmed per se. The network acquires its
function through a training process. In the 1990s Mitchell, Jim Crutchfield
of the University of California, Davis, and Peter Hraber at the Santa Fe
Institute showed how cellular automata could do the same. Using a genetic
algorithm, they trained automata to perform a particular computational
operation, the majority operation: If a majority of the cells are dead, the
rest should die too, and if the majority are alive, all the dead cells
should come back to life. The cells had to do this without any way to see
the big picture. Each could tell how many of its neighbors were alive and
how many were dead, but it couldn’t see beyond that. During training, the
system spontaneously developed a new computational paradigm. Regions of dead
or living cells enlarged or contracted, so that whichever predominated
eventually took over the entire automaton. “They came up with a really
interesting algorithm, if you want to call it an algorithm,” Mitchell
said.
She and her co-authors didn’t develop these ideas further, but
Mordvintsev’s system has reinvigorated the programming of cellular automata.
In 2020 he and his colleagues created an NCA that read handwritten digits, a
classic machine learning test case. If you draw a digit within the
automaton, the cells gradually change in color until they all have the same
color, identifying the digit. This year, Gabriel Béna of Imperial College
London and his authors, building on unpublished work by the software
engineer Peter Whidden, created algorithms for matrix multiplication and
other mathematical operations. “You can see by eye that it’s learned to do
actual matrix multiplication,” Béna said.
Stefano Nichele, a professor at Østfold University College in Norway who
specializes in unconventional computer architectures, and his co-authors
recently adapted NCAs to solve problems from the Abstraction and Reasoning
Corpus, a machine learning benchmark aimed at measuring progress toward
general intelligence. These problems look like a classic IQ test. Many
consist of pairs of line drawings; you have to figure out how the first
drawing is transformed into the second and then apply that rule to a new
example. For instance, the first might be a short diagonal line and the
second a longer diagonal line, so the rule is to extend the line.
Neural networks typically do horribly, because they are apt to memorize the
arrangement of pixels rather than extract the rule. A cellular automaton
can’t memorize because, lacking long-range connections, it can’t take in the
whole image at once. In the above example, it can’t see that one line is
longer than the other. The only way it can relate them is to go through a
process of growing the first line to match the second. So it automatically
discerns a rule, and that enables it to handle new examples. “You are
forcing it not to memorize that answer, but to learn a process to develop
the solution,” Nichele said.
Other researchers are starting to use NCAs to program robot swarms. Robot
collectives were envisioned by science fiction writers such as Stanisłav Lem
in the 1960s and started to become reality in the ’90s. Josh Bongard, a
robotics researcher at the University of Vermont, said NCAs could design
robots that work so closely together that they cease to be a mere swarm and
become a unified organism. “You imagine, like, a writhing ball of insects or
bugs or cells,” he said. “They’re crawling over each other and remodeling
all the time. That’s what multicellularity is really like. And it seems — I
mean, it’s still early days — but it seems like that might be a good way to
go for robotics.”
To that end, Hartl, Levin and Andreas Zöttl, a physicist at the University
of Vienna, have trained virtual robots — a string of beads in a simulated
pond — to wriggle like a tadpole. “This is a super-robust architecture for
letting them swim,” Hartl said.
For Mordvintsev, the crossover between biology, computers and robots
continues a tradition dating to the early days of computing in the 1940s,
when von Neumann and other pioneers freely borrowed ideas from living
things. “To these people, the relation between self-organization, life and
computing was obvious,” he said. “Those things somehow diverged, and now
they are being reunified.”
