Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
There are $3^{3^2} = 19683$ 3-state OCAs, compared to $2^{2^3} = 256$ 2-state ECAs, which suggests that the behavior of the former class of CAs should be richer and likely to support all the same kinds of complex behavior, if not more, than the latter. But of course that's not a proof, just a vague heuristic argument. In particular, even though $3^2 > 2^3$, I don't see any obvious way to emulate arbitrary 2-state ECAs with 3-state OCAs.
Minor note: it seems to me you have some notational (and possibly logical) gaps in your definition of CT. For example, you use the symbols $≤$ and $≯$ without defining them, although it's fairly obvious how they should be defined. Also, I assume you (probably) intend $<$ to be a total order on numbers, but it's not clear to me that your axioms imply that. Your order axiom implies transitivity and irreflexivity of $<$ (at least assuming that there's no largest number!), but I'm not sure it implies asymmetry and it definitely doesn't imply totality: an empty relation trivially satisfies it!
…and since $\sum a_n < \infty$, for each $a_n > 0$ there can only be a finite number of elements $a_k$ such that $a_k ≥ a_n$. (If a finite sum wasn't required, $a_n = 1-1/n$ would be an example of a non-negative sequence that cannot be rearranged into descending order.)
As a minor detail, it might be worth pointing out that the rearrangement in the first paragraph is indeed (AFAICT) always possible, but only because the sequence elements are specified to be positive and to have a finite sum.
@MattF: That's a nice observation! AFAICT it only shows that a polygon maximizing your metric must have all sides of equal length (since any such polygon is invariant under the operation you describe), but the fact that a polygon with equal side lengths has maximal area when it's regular seems intuitively obvious enough anyway.
@MattF: I think the "geometric intuition" I was looking for was really more along the lines of "some way to intuitively see that a regular polygon really maximizes your metric". I think I did manage to finally convince myself of that, though. Basically, by Jensen's inequality, $\sum d_i^2≥\frac1n\left(\sum d_i\right)^2$, where $n$ is the number of sides, with equality only when all $d_i$ are equal. Thus, for fixed $A$, minimizing the total perimeter and dividing it into $n$ equal sides maximizes your metric. And for fixed and equal $d_i$ maximizing $A$ clearly gives a regular polygon.
Huh… I first misread your metric as $A \mathbin/ \left(\sum d_i\right)^2$ and thought "yeah, that makes perfect sense, it's measuring the area-to-perimeter ratio," but of course that's not what you actually wrote. Now I'm trying to figure out some geometric intuition for your actual metric.
It can't, obviously. :) I was just trying to highlight the fact that the process you get by having a non-zero update probability tend towards zero (while scaling time so that the average number of updates per cell per scaled time unit stays constant) is obviously not the trivial process shown in your last image, where the update probability is identically zero and thus nothing ever changes.
Comment: If you scale time by the inverse of the update probability in your "mode 2", so that each cell is on average updated once per scaled time step, then your "mode 3" is effectively the limit of "mode 2" as the update probability (per unscaled time step) tends to zero from above (and as the lattice size $k$ tends to infinity). Also, this scaled process is essentially a continuous-time Markov chain.
FWIW, the reason why you get fractals when starting from a single live cell in rule 18 is that this is yet another rule that behaves identically to rule 90 when all live cells are in odd-numbered (or even-numbered) positions. (There are 8 such ECA rules in total, with Wolfram codes $18 + 8a + 64b + 128c$ for $a,b,c \in \{0,1\}$.)
