A two-layer representation of four-state reversible number-conserving 2D cellular automata

Abstract

We present a novel representation of 1D reversible and numberconserving cellular automata with four states. Carrying this view over to two dimensions, we are able to construct 65 four-state reversible and numberconserving 2D cellular automata with the von Neumann neighborhood. A clever use of the split-and-perturb decomposition of number-conserving CAs allows to prove by elimination that this list is complete.

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“…However, for k = 3 and k = 4, we do not need to check the reversibility of the obtained local rules. Indeed, as we mentioned before, we found only three ternary ones (which are definitely reversible and we know their inverse) and for the quaternary NCCAs, descriptions of all 21 local rules presented in Table 3 were given by Dzedzej et al in [35] and from those descriptions it follows that all are reversible. We emphasize once again that both the results for k = 3 and k = 4 were already known -here we derived them only to illustrate our new method.…”

“…Moreover, we are convinced that this method can be used for a larger number of states (at least for nine states). We believe that the complete lists of reversible k-ary NCCAs will allow researchers to uncover the structure of these CAs, as was done in the case of quaternary reversible NCCAs by Dzedzej et al [35]. In this way, we hope it will be possible to devise a construction method for reversible NCCAs with an arbitrarily large number of states.…”

“…The decomposition theorem applies to any state set and any dimension. By means of this new tool, it has been proven possible to enumerate all NCCAs in the case of some state sets and dimensions, which, up to now, were beyond the capabilities of computers (see, for example, [34,35]), and also, what is even more valuable, it was possible to prove some general facts about NCCAs (see [36,37]).…”

“…It may be a bit surprising that the dynamics of reversible 5state NCCAs is definitely less diverse than for 4-state ones, but this is related to the fact that 5 is a prime number, so no two-layer representation of 5-state NCCAs is possible (see [35] for details).…”