Efficient enumeration of three-state two-dimensional number-conserving cellular automata

A split-and-perturb decomposition of number-conserving cellular automata

“…Moreover, we can exploit eight symmetries of the von Neumann neighborhood in the two-dimensional space, so, there are only 16 significantly different split functions. As a result, we get that there are 1327 number-conserving local rules, which agrees with the findings of [10].…”

“…Since now we know all three-dimensional threestate number-conserving CAs, we will also be able to identify reversible ones -in particular, to see whether there are some nontrivial ones. In [10] we found that if the state set is t0, 1u or t0, 1, 2u, i.e., when we deal with binary or three-state CAs, there are only trivial number-conserving CAs that are reversible as well, namely, the identity and the shifts (in each of four possible directions). We conjecture that the same will hold for d " 3.…”

“…These conditions apply for any state set Q Ă R, whether it is finite or not. The main result presented in [27] allows to find all two-dimensional three-state number-conserving CAs [10] and all two-dimensional six-state rotation-symmetric number-conserving CAs [11]. Moreover, it allows to describe all affine continuous density-conserving CAs with state set Q " r0, 1s, which is infinite [8].…”

All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional

“…In Ref. [21], all 1327 ternary two-dimensional number-conserving CAs are enumerated and it is proven that there is no nontrivial reversible one. We conjecture that the same holds true for higher dimensions, i.e., that three states are too few to allow for the existence of a reversible number-conserving cellular automaton and we believe that it is possible to prove this fact by using the decomposition theorem.…”

Digital phagograms: predicting phage infectivity through a multilayer machine learning approach

“…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]).…”