![]() ![]() Here’s the graph for the GKL rule we discussed above, for the case n = 5. There are a total of 2 n possible configurations in this case, and we can represent all possible paths of evolution of the cellular automaton using a state transition graph. But given a particular rule, will it always reach consensus, or are there exceptions?Īs a first way to get a well-defined version of that question, we can consider finite cellular automata, say with a total of n cells, and cyclic boundary conditions. We’ve seen that there are 1D cellular automata that-at least in the examples we’ve looked at-achieve “majority consensus”. We could also consider rules that involve other “helper” colors that either disappear before the final state is reached, or define additional consensus states. The inner product in this Hilbert space is what induces the norm. Not every vector in Hilbert space is state vector (wave function) but any such vector can be normalized to get possible state vector. ![]() Here we’ve only discussed cellular automata with two possible colors for each cell. begingroup Wave function is a vector on Hilbert space, which is, technically speaking, a complete inner product space. ![]()
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