Nonexistence of synchronous-update solution for density classification with an intermediate alphabet

Establish the nonexistence of any synchronous-update cellular automaton that, even when permitted to use an intermediate alphabet, maps every finite cyclic configuration over the binary alphabet B = {0, 1} with a strict majority to the corresponding uniform fixed point (b^n) and leaves no residual intermediate symbols; equivalently, prove that no synchronous-update cellular automaton satisfies this clean-convergence density classification specification.

Background

The paper presents a sequential (asynchronous, left-to-right) cellular automaton of radius 1/2 that solves the density classification task by using an intermediate alphabet and guarantees convergence to a clean fixed point with no residual auxiliary information. It further generalizes this construction to arbitrary finite alphabets and higher dimensions.

Classical impossibility results show that pure, deterministic one-dimensional cellular automata with parallel (synchronous) updates cannot solve the density classification task. Here, the authors propose a conjecture that even when allowing an intermediate alphabet, a synchronous-update solution to the exact clean-convergence specification does not exist. Establishing this would delineate minimal requirements for solvability and underscore the necessity of sequential updates for this task.

References

We are of the opinion that a solution of this exact problem using the synchronous update schedule does not exist. Proving this conjecture would be a fitting continuation of the present work, and would confirm that the present solution is in fact one example of some minimal requirements that are sufficient to solve the density classification task.

— A sequential solution to the density classification task using an intermediate alphabet  (2409.06536 - Perrotin et al., 2024) in Section 5 (Closing words)