ConwayLife.com

After studying differential calculus, a thought came to my mind: Could CA be simulated through a system of equations? The answer is no, but it got me thinking about non-discrete systems. The following are 2 potential alternative systems to discrete automata.

1. State-continuous CA:
Generally, there are a finite number of states any given CA. However, this isn't necessary. For a totalistic rule, a cell would compute the sum of the states of each of it's neighbors, and according to it's birth and death rules, compute each. Then, it would interpolate (using an interpolation curve) between the two values according to it's state. In addition to an infinite range of states, the birth and death rules can be computed for an infinite number of inputs. In the non-totalistic rulespace, a specialized higher dimensional function would define the weights of each birth rule, essentially assigning a strength to each rule. I have no clue how to implement such a higher dimensional function, but one would need to exist in order for non-totalistic rules.
2. Position-continuous CA:
In the case of a CA without cells, or just an "A", The grid would be replaced by a scalar field, and each point in the scalar field would use a predetermined convolutional mask as a replacement for a sum.

I do not see this being useful as a means of actual prediction, as by definition, this would require infinite accuracy. They would be unpredictable, inconsistent, and getting repeatable results from such systems would be impossible. However, it is an interesting idea.

Time continuous CA would be possible to implement, but incredibly computationally difficult.

Have you heard of SmoothLife? It's basically a position-continuous time-continuous rule (you could also argue that states are continuous, I guess, though the line between continuous and non-continuous states in a position-continuous system is pretty fuzzy.

semirelevant side note: I wonder what other spaces you could have interesting automata on? Most cellular automata are in $\mathbb{Z}^{n}$ (and that's how the wiki defines it) though in practice I'm pretty sure in usage among forumites it's more or less agreed that you can do it on arbitrary tesselations of space.

continuous spatial automata (eg smoothlife) are on $\mathbb{R}^{n}$. I guess $\mathbb{C}^{n}$ probably wouldn't be terribly interesting (though maybe it's a little early for me to write it off! I just make this conclusion on the basis that $\mathbb{C}^{n}$ is pretty similar to $\mathbb{R}^{2n}$), but I wonder what would happen if your space were some subset/subclass of the surreals.

even less relevant side note: also, I guess you can do this with the time dimension as well. On the discord there have been discussions about infinite-time versions of GoL, based on infinite-time turing machines. (iirc some work was done but overall it was a bit underwhelming). I guess you could do surreal time as well


Ok, so, actually on the topic of non-discrete automata: one question that popped into my mind is whether you can have a nontrivial reversible space- and time-continuous automaton. In particular, could you have a continuous replicator rule? My intuition says it sounds like it would be impossible.