Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata

Round 1

Reviewer 1 Report

This paper elucidates the refinement of mean-field theory as applied to cellular automata, employing stochastic perturbations predicated upon the system's evolutionary rules. The essence of novelty within this discourse resides in the application of venerable classical mean-field theory polynomials, complemented by the infusion of serendipitous perturbations. These perturbations are ascertained through a discerning scrutiny of the cellular automata's evolutionary rule, discerning whether it begets a heightened incidence of specific neighborhood typologies in comparison to their counterparts. From the results in this artical, this method is useful.

 

However, there are still some issues with the article:

1. Pay heed to the image formatting: it is advisable to center them all.

2. The descriptions of the images are overly succinct; it is suggested to augment them with essential information to enhance the article's readability.

3. In the sixth paragraph of the introduction, reference is made to previous methods predominantly employing a posteriori perspective. Kindly elucidate concisely the specific advantages of the method proposed in this article, which obviates the necessity for running system evolutions.

4. Regarding Figure 1, please provide explanations for each subfigure.

5. For Figure 2(a), please denote which is the Current pattern and which represents the New state for the center cell.

6. The representation in Figure 2(b) and (c) appears rather cryptic. It might be beneficial to segregate the initial configuration of evolution from the resultant configuration for the sake of clarity.

7. In the seventh paragraph of Section 2, please expound upon the process of renaming and reordering, thus augmenting the article's readability.

8. The matrix format in Figures 7 and 8 should align with the previous matrix forms in the article. The size of the numerals should remain consistent; some numerals are rendered too diminutive for clear comprehension.

9. Could you consider supplementing a comparative analysis between the detection outcomes of mean-field analysis with and without the utilization of Random perturbation?

10. The first paragraph in page 7 is used to explain related works of perturbations, it should be moved to the Introduction or Conclusion.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

A comparison section with similar works is missing.

Acceptable

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Mean-field approximation is a standard way to approximate long-term behavior of cellular automata (and variants of it are also used (even more) in other fields of science). While there is no real guarantee that this approximation has anything to do with the actual behavior of a CA, it is cheap to analyze, and there is no reason why it would not be useful, so it is worth trying.

This paper suggests applying a random perturbation to mean-field approximation, to estimate how stable the mean-field approximation is, with the intuition that if the random version has very different characteristics from the standard mean-field approximation, then perhaps something interesting is going on, and specifically the authors want to find gliders (pockets of information moving on a stable background).

For example, for elementary CA 110 (the standard example of a CA where gliders do something interesting, indeed universal computation), the approximations are very different, and the authors give concrete values that automatically detect a bunch of CA (including 110) that are revealed to indeed produce gliders. I don't know how much the authors needed to tweak the parameters to make the algorithm find a reasonable set (i.e. whether this was truly automatic), but as they note this is just a first attempt at such a study, so this is not a big issue.

(It seems the authors do not make any claim that the random mean-field approximation is better for approximating true behavior than the standard one. Indeed, for 110 it looks like the random variant of mean-field produces just as incorrect a result as the standard mean-field. This was not initially clear to me, and the authors could perhaps more explicitly state this, to avoid the readers concentrating on the wrong thing.)

There are a number of repeated paragraphs. I assume that at the last step of production, the authors decided to delete all comments, but only removed the comment (%) sign, leading to the comments becoming visible. Or something equivalent. 

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My two cents on the heuristics, which might be completely wrong as I did not do any experiment myself (NB. when I wrote this I was under the impression (that the authors are not to blame for) that you want to improve on the results of mean-field approximation, rather than the more subtle (though also less directly measurable) thing you actually attempted i.e. finding gliders):

The choices of g(x) and renormalization look a bit random, my intuition says that the additive scaling (subtracting |S| from l_x) and the additive correction at line l249 are not "correct", and somehow should be more multiplicative... or there should at least be a parameter that allows weighing the discrepancies differently. Perhaps the reality is that pretty much any formula will produce something interesting, though.

The choice of looking at one more step in the calculation of l_x also seems strange, now we don't know if it's the random approximation of mean field that is doing the work, or whether it's the fact you are essentially going one more step... With this definition, it seems like it would make more sense to compare your random mean field approximation to standard mean field applied to the square of the CA now (w.r.t. composition), no? (Or maybe yours can be seen as a more computationally efficient in-between idea?)

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Some detailed comments:

l157 "which can be found experimentally as"

I guess you assume that you have a way to sample the initial configurations, rather than fix an initial measure (I'm more used to the latter).

l165 "This assumption is repeated for subsequent values of k."

Of course then it is no longer _true_, but rather postulating it simplifies calculations, right? And is why we call it the mean-field *approximation*. For initial k, it is only true if you assume a Bernoulli distribution (i.e. i.i.d.) initially, an assumption that I missed. (You say "Assuming that initially c^0 is defined randomly, one can assume independence in the occurrence of each state in c^0", which I suppose clarifies you mean i.i.d., but this is not usually implied by  the word "random".)

l178  "In part (B), the highest estimated density belongs to the 3 state, followed by the density of the 1 state, while the other states have lower estimated densities. However, experimentally, the densities of the states 1 to 3 are similar"

I think state names are not consistent with figure.

The paragraph starting at l213 is more or less repeated from above.

The paragraph starting at l221 is yet again the same paragraph repeated with minor changes.

l242 this paragraph is more or less repeated from above

Author Response

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Author Response File: Author Response.pdf