Comments
This is pretty cool. I have several points to make. 1. We all know that Cellular automata or more generally any dynamical system of sufficient complexity (and maybe not too much complexity) will be Turing complete, will have complicated "uncomputable" behavior, will have perhaps pattern formation, or gliders, solitons etc. So what is a valuable addition to this these computational investigations? I think when studying emergent computational behavior we really care about dynamics complexity / rules complexity. It's not impressive to get complicated dynamics out of a complicated system but the simplicity of game of life made it really impressive. I think in that regard LACE is pretty nice: the rule still feels very simple/natural and you can get much more structured/complex behavior with fewer cells.
2. Nevertheless in the end this blog shows mostly pretty pictures of computational, complex, emergent, chaotic behavior, which we've all seen before. And the key features that make the difference go something I would call physics-like are still missing. And I guess that would be complex stable patterns that can have complex stable interactions. Who knows maybe there are 10^16-celled patterns that have this but we don't know.
3. If I were you I would cut the whole preamble. It will make people take you less seriously than they should. You don't want to look like a crank.
By sushisource 2025-10-1616:203 reply +1 to this copy being a little bit over-the-top. This is neat, but, as you pointed out at the end of the day this is still computationally equivalent to normal 2d cellular automata. I suspect (not taking the time to prove this) that it's equal in a fairly obvious way, which is that you could just replace "links" with 8*<num link states> additional sub-states per cell. The only real difference is just in how it's visualized.
So, neat, but not exactly mindblowing.
Seems like maybe not the only instance of self-aggrandizement in this blog... the sidebar shows posts about things like the quantum mechanics of consciousness other such quackery.
Wow. One of the links is titled "Clarifying the Hierarchy of Information-Derived Gravitational Tensors in Consciousness Field Theory". The References section has 4 entries, all written by the author of this blog.
By airesearcher 2025-10-181:14 Actually - on that particular topic - you might want to read the articles and the math before forming an opinion - start from information geometry and fisher information metrics of information processing - which is what that particular line of thought is all about...Everything on that particular line of exploration is built up from fisher information where "consciousness" is defined as a high complexity information processing geometry and so its measurable. Because its defined as a measurable physical process, not as a magical quality, it's a mathematical argument. But of course that would require reading, and sufficient knowledge to understand it.
You may really like the PPS system which gives more life-like behavior. Here i got "hearts" beating to cause "blood to flow".
I became obsessed with these and made a few videos!
By airesearcher 2025-10-181:15 Very cool!
By airesearcher 2025-10-1617:281 reply In theory, if using a computationally universal CA, you can simulate any other CA with it. However it might require a lot of sub-steps to do so.
No claim is being made that this is a new kind of computation.
The observation that other CA can be equivalent is a weak critique at best, this CA may be a nice compact way of describing types of CA that have interesting properties. It is not terribly interesting that it may be subsumed by some other CA. It may be some interesting unstudied subset.
For instance the Game of Life is a subset of 2-d binary state CA, the rule only takes the totals of neighboring cells, and so is a subset of those CAs with rules that care about specific patterns of neighbors.
By airesearcher 2025-10-1618:39 The question is really whether this class of rule is a subset of 2D binary state CA, or whether it is a superset in fact.
By airesearcher 2025-10-1616:231 reply Good feedback -
These rules use very different principles than traditional cell-based rules - for example neighbor degree, number of connections, and eligibility criteria based on connectivity. So the cells are not becoming alive or dead based on the states of their neighbors, but rather on the topology of their neighborhoods.
The details are beyond the scope of a short write up, but are easy to explore in the rule-editor in the GUI of the code.
And preamble pruned of the historical anecdote behind this.
By michelpp 2025-10-170:56 An interesting approach to characterize graph topology, both locally and globally is to use a graphlet transform, there some interesting research happening around these types of topology signals, here's one that takes a very algebraic approach
By cluckindan 2025-10-178:06 I think your thoughts are valid, but they essentially call for CAs which would computationally bridge the gap from quantum to classical physics.
By airesearcher 2025-10-1613:33 LACE - Link Automata Computing Engine
(written in python, with optional taichi GPU-powered mode for large-scale simulations)
LACE is a new kind of cellular automata where rules operate on cell states and their links to other cells.
Check out the Gallery in https://www.novaspivack.com/science/introducing-lace-a-new-k... to see the familiar Game of Life rule, but with links.
* Quick Examples **
Game of Life, with links: https://videopress.com/v/lTZ8e4hD
Amazing Dragons (LACE rules): https://videopress.com/v/lQ5Bghsj
** MANY more examples in the Gallery (in the blog post cited above)
Rules can use topological properties of cells and neighborhoods, such as number of connections, neighbor degree, and other metrics.
The added topological dimension enables rules that can have more interesting behavior than traditional "cells-only" CA rules, opening up a fascinating new computational world of new species of stable patterns - oscillators - gliders, puffers, and more.
For details on how these rules work, get the repo and open various rules in the rule editor, where all their parameters are explained. There are many new classes of rules to experiment with.
** You can get the repo and learn more at: https://github.com/novaspivack/lace
So what actually is it? None of the rules in the videos look particularly striking compared to other Life-like cellular automata and 2d cellular automata in general. As you say, their behaviour includes oscillators, spaceships, patterns that grow endlessly... all things that are well-known from other cellular automata. So the videos didn't really show off why they're interesting.
I don't mind the rambling about "planets, galaxies, galaxy clusters, superclusters… and beyond …." but some technical detail would be nice too!
By airesearcher 2025-10-1616:242 reply These rules use very different principles than traditional cell-based rules - for example neighbor degree, number of connections, and eligibility criteria based on connectivity.
So in short, the cells are not becoming alive or dead based on the states of their neighbors, but rather on the topology of their neighborhoods.
The details are beyond the scope of a short write up, but are easy to explore in the rule-editor in the GUI of the code.
Here is an example of a rule that is markedly different from a typical "life-like" rule: https://videopress.com/v/lQ5Bghsj
The level of structure and self-organization is striking, to me at least.
Also in all the rules - the links are visible and can have binary or real-valued states as well as the cells. So this enables pretty rich topology which rules can utilize.
Could you try explaining it in a comment? Not the general principle, but just the rules for one particular automaton. Whichever one is your favourite. Or Amazing Dragons, if you don't have a favourite.
The amazing part of cellular automata is the emergence of complicated behaviour from simple rules. Life's rules can be written in three sentences, maybe less.
Forgive my quibbling, but I don't understand what this is doing that other projects in this space haven't done before. Adding states and transition rules to edges is new to me...
I did try running your project, but I had to tweak it to get it to work with the instructions in the repo. I seem to be missing a few packages -- mpmath, sympy, typing_extensions. Can you add those to the requirements.txt file?
Let's see if I understood this right. For the Betweenness Amazing Dragons rule:
* Compute the "betweenness" of each living cell, which is 1 divided by its degree. Cells which are not connected to anything have infinite/undefined betweenness, but it doesn't matter.
* Then, for each cell, sum up the betweenness of its connected neighbours.
* If the total betweenness of a dead cell is in the range [(1.3, 3.6)], it is born and becomes alive at the next generation.
* If the total betweenness of a living cell is in the range [(0.9, 2.6)], it survives and remains alive to the next generation.
* Exception: any cell with 0, 1, 7 or 8 neighbours (in total, ignoring betweenness) dies anyway after the rules above were applied.
... That's not quite right, there's some references to "eligibility" that I can't make sense of. What else am I missing?
By airesearcher 2025-10-1617:501 reply I've added a bit more explanation
https://github.com/novaspivack/lace/blob/master/Rule_Explana...
https://github.com/novaspivack/lace/blob/master/Betweenness_...
These cover only one metric and one rule, but give some more info
Here is a more detailed explanation: https://github.com/novaspivack/lace/blob/master/Realm_of_Lac...
It'd be better to have fewer videos on the page and select for more striking examples like this one, and put them early. I got fatgue from see so many examples.
Also, consider getting off the grid and maybe doing some topology-based automata in combination with a more traditional network presentation paradigm like a force-directed layout. That would give you a much more 'biological' look which would draw a lot of people's attention.
By mkl 2025-10-179:33 That's the very first LACE video on the page (was it updated?). It does come after a bunch of traditional cellular automata videos though.
