Differentiable Logic Cellular Automata

This is very interesting. I've been chasing novel universal Turing machine substrates. Collecting them like Pokémon for genetic programming experiments. I've played around with CAs before - rule 30/110/etc. - but this is a much more compelling take. I never thought to model the kernel like a digital logic circuit.

The constraints of boolean logic, gates and circuits seem to create an interesting grain to build the fitness landscape with. The resulting parameters can be directly transformed to hardware implementations or passed through additional phases of optimization and then compiled into trivial programs. This seems better than dealing with magic floating points in the billion parameter black boxes.

This is exciting.

Michael Levin best posited for me the question of how animal cells can act cooperatively without a hierarchy. He has some biological experiments showing, for example, eye cells in a frog embryo will move to where the eye should go even if you pull it away. The question I don't think he could really answer was 'how do the cells know when to stop?'

Understanding non-hierarchical organization is key to understanding how society works, too. And to solve the various prisioner's delimmas at various scales in our self-organizing world.

It's also about understanding bare complexity and modeling it.

This is the first time I've seen the ability to model this stuff.

So many directions to go from here. Just wow.

I've been thinking a lot about "intelligence" lately, and I feel like we're at a decisive point in figuring out (or at least greatly advance our understanding of) how it "works". It seems to me that intelligence is an emergent natural behavior, not much different than classical Newtonian mechanics or electricity. It all seems to boil down to simple rules in the end.

What if everything non-discrete about the brain is just "infrastructure"? Just supporting the fundamentally simple yet important core processes that do the actual work? What if it all boils down to logic gates and electrical signals, all the way down?

Interesting times ahead.

There’s something compelling about these, especially w.r.t. their ability to generalize. But what is the vision here? What might these be able to do in the future? Or even philosophically speaking, what do these teach us about the world? We know a 1D cellular automata is Turing equivalent, so, at least from one perspective, NCA/these aren’t terribly suprising.

I love playing around with cellular automata for doing art. It's amazing what kind of patterns can emerge (example: https://gods.art/math_videos/hex_func27l_21.html). I may have to try to play with these DLCA.

There are a lot of cool ideas here. Maybe a small observation but the computation is stateful. Each cell has a memory and perception of it's environment. Compare this to say your modern NN which are stateless. Has there been any work on statefull LLMs for instance?

Self-plug here, but very related => Robustness and the Halting Problem for Multicellular Artificial Ontogeny (2011)

Cellular automata where the update rule is a perceptron coupled with a isotropic diffusion. The weights of the neural network are optimized so that the cellular automata can draw a picture, with self-healing (ie. rebuild the picture when perturbed).

Back then, auto-differentiation was not as accessible as it is now, so the weights where optimized with an Evolution Strategy. Of course, using gradient descent is likely to be way better.

Hmm.. could this be used for the ARC-AGI challenge? Maybe even combine with this recent one: https://news.ycombinator.com/item?id=43259182

This is wild. Long time lurker here, avid modeling and simulation user-I feel like there’s some serious potential here to help provide more insight into “emergent behavior” in complex agent behavior models. I’d love to see this applied to models like a predator/prey model, and other “simple” models that generate complex “emergent” outcomes but on massive scales… I’m definitely keeping tabs on this work!

The result checkerboard pattern is the opposite (the NOT) of the target pattern. But this is not remarked upon. Is it too unimportant to mention or did I miss something?

If I understand the article correctly, this research shows that you can compress some 2d image into a circuit design, that if replicated exactly many times in a grid, it will spontaneously output the desired image.

I'm interested in a nearby, but dissimilar project, almost it's reciprocal, wherein you can generate a logic design that is NOT uniform, but where every cell is independent, to allow for general purpose computing. It seems we could take this work, and use it to evolve a design that could be put into an FPGA, and make far better utilization than existing programming methods allow, at the cost of huge amounts of compute to do the training.

This is a groundbreaking shit, and it's not about checkerboards or lizards. The Navier-Stokes differential equation governing fluid motion is an update rule that predicts the next state of any given point given its neighborhood and a few previous states. The key insight is that all the complexity of the fluid and air motion, the formation of clouds and the motion of flames, is governed by a simple law, by an equation that's uniformly and simultaneously applied to each point in space and all it needs is the immediate neighborhood of that point and its immediate history. Discovering this equation by looking at real-life samples is what's called science. It should be possible in principle to apply this DLCA model to a video recording of smoke, constrained to a thin 2D layer for simplicity, and let this model derive the Navier-Stokes equation. When you take this one step further and consider that the update rule itself may be changing according to another update rule, we'll get into some interesting territory. This might be why in our brain neurons are connected to a neighborhood of thousand neurons instead of just ten or so.

Following the tradition, Google execs are going to dismiss this discovery as irrelevant to the ads business, and a couple years later when DLCA will have turned the world upside down, they'll try to take credit saying it's their employees who have made the discovery.

Continuous relaxation of boolean algebra is an old idea with much literature. Circuit synthesis is a really well-researched field, with an annual conference and competition [1]. Google won the competition 2 years ago. I wonder if you have tried your learner against the IWLS competition data sets. That would calibrate the performance of your approach. If not, why not?

[1] https://www.iwls.org/iwls2025/

The Conway's game of life example isn't so impressive. The network isn't really reverse engineering rules, it's being trained on data that is equivalent to the rules. It's sort of like teaching + by giving it 400 data points triplets (a,b,c) with 1 <= a,b <= 20 and c = a + b.

This is very interesting! I think an exciting direction would be to arrive at minimal circuits that are to some extent comprehensible by humans. Now, this might not be possible for every system, but certainly the rules of Conway‘s GoL can be expressed in less than 350 logic gates per cell?

This also reminds me of using Hopfield networks to store images. Seems like Hopfield networks are a special case of this where the activation function of each cell is a simple sum, but I’m not sure. Another difference is that Hopfield networks are fully connected, so the neighborhood is the entire world, i.e., they are local in time but not local in space. Maybe someone can clarify this further?

It’s interesting to see how differentiable logic/binary circuits can be made cheap at inference time.

But what about the theoretical expressiveness of logic circuits vs baselines like MLPs? (And then of course compared to CNNs and other kernels.) Are logic circuits roughly equivalent in terms of memory and compute being used? For my use case, I don’t care about making inference cheaper (eg the benefit logical circuits brings). But I do care about the recursion in space and time (the benefit from CAs). Would your experiments work if you still had a CA, but used dumb MLPs?

So this does not need large training data sets like traditional models?

The lizard and the Game of life example seem to illustate that you only need one data points to create or "reverse" engineer a an algorithm that "generates" something Equal to the data point.

How is this different from using a neural network and then over fitting it?

Maybe that instead learning trained weights, the Cellular Automata learns a combination of logic (a circuit).

So the underlying, problems with over fitting an neural network (a model being un able to generalise) still hold for this "logic cellular automata"?

> magine trying to reverse-engineer the complex, often unexpected patterns and behaviors that emerge from simple rules. This challenge has inspired researchers and enthusiasts that work with cellular automata for decades.

Can someone shed some light on what makes this a problem worth investigating for decades, if at all?

Could this be used to train an LLM? It seems the hidden states could be used to learn how to store history.

This is a very interesting paper. Question though: it seems the cells gates since they're updated using a "global" gradient descent that it isn't truly parallel.

Is there any promise towards a strictly local weight adjustment method ?

Can anybody point out what's special about this?

Late here, but a few comments: the main idea of the authors was to combine differential logic gates (an amazing invention I had not heard of) with cellular automata as they say in the paper, or more accurately I would say a grid topology of small neural networks (cells). The cells get and send information to their neighbors.

The idea would be you create some sort of outcome for fitness (say an image you want the cells to self organize into, or the rules of Conway’s game of life), set up the training data, and because it’s fully differentiable, Bob’s your uncle at the end.

Depending on what you think about computational complexity, this may or may not shock you.

But since they’ve been doing gradient descent on differentiable logic gates at the end of the day, when the training is done, they can just turn each cell into binary gates, think AND OR XOR, etc. You then have something that can be used for inference crazy fast. I presume it could also be laid out and sent to a fab, but that work is left for a later paper. :)

This architecture could do a LOTTT of things to be clear. But sort of as a warm up they use all the Conway life start and end rules to train cells to implement Conway. Shockingly this can be done in 5 gates(!). I note that they mention almost everywhere that they hand prune unused gates - I imagine this will eventually be automated.

They then go on to spec small 7k parameter or so neural networks that when laid out in cells can self organize into different black and white or color images, and can even do so on larger base grids than they were trained, and are resilient to noise being thrown at them. They then demonstrate that async networks (each cell updates randomly) can be trained, and are harder to train but more resilient to noise.

All this is quite a lot to take in, and spectacular in my opinion.

One thing they mention, a lot, is that a lot of hyperparameter tuning is required for “harder” problems. I can imagine like 50 lines of research out of this paper, but one of them would certainly be adding stability in to the training process. Arc-AGI is mentioned here, and is an awesome idea — could you get a “free lunch” with Arc? Or some of Arc? Different network topologies are yet another interesting question, hidden information, “backing layers” - e.g. why not give each cell 20 private cells that info goes out to and comes back in? Why not make some of those cells talk to some other cells? Why not send radio waves as signals across the custom topology and train an efficient novel analog radio? Why not give each cell access to a shared “super sized” 100k, 1mmk parameter “thinking node”? What would a good topology be for different tasks?

I’ll stop here. Amazing paper. Quite a number of PhD papers will be generated out of it, I expect.

I’d like to see Minecraft implemented though. Seems possible. Then we could have Bad Apple in Minecraft on raw circuits.

It seems to me this is a concept of how an AGI would store memories of things it has seen or sensed and later recall them?

Wouldn’t you need a custom non-von-Neuman architecture to leverage the full power of CA?

Can someone ELI5 for a Muggle?

Is there any code available?

I wonder what Stephen Wolfram has to say about this.

I wish we were all commenting about the ideas embedded in this paper. It intrigues me, but is out of my comfort zone. Love to read more content-related insights or criticisms rather than the long thread on the shamefully smooth, engaging, and occasionally rote style.

This writing feels so strongly LLM flavored. It's too bad, since I've really liked Alexander Mordvintsev's other work.

Loved this work! See some crazy cool implications.

Ive always wondered, if CA's are a canidate for making the smallest solution to computer programming problems.

Excitement: Id love to see a chatgpt running in hardware on a FPGA! that would be wild.

If a method was found for training these types of models in real time would be amazing for industrial applications. Click button and it learns the problem and can take updates as a assembly line goes along. Think QA problems

Questions: What are the training requirements for a scaled up version?

Can DLCA work with problems that require floating point?

Can the digital circuits generate float equivalents?

Could adding more advanced logical constructs like used in chip design benefit training?

How difficult would it be to convert a digital cicuit into a FPGA? What speedup gains could be achived?

Where are the ruff edges of this approch? Does it have some current scaling problems

Only criticism of this work is seeing some failures and what are its short comings.

Thought: Can this work be applied to a LLM? Is their and technical roadblocks to application of this to a llm, say lack of a ReLu or some sigmoid activation function . Does fpga's have ability for float like behavior? Idk

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