One noisy cellular automaton with an uncomputable noise diagram
Year: 2026
Type: preprint
Abstract: Can varying only the noise rate of one cellular automaton encode the halting problem? We construct one computably specified cellular automaton on Z^6, with a finite alphabet and a finite neighbourhood, whose ergodicity under uniform replacement noise does so. There are pairwise disjoint open intervals I_n with rational endpoints computable from n such that, at every rate in I_n, the perturbed automaton is ergodic exactly when a fixed universal machine halts on input n. In the nonhalting case it has at least two translation-invariant stationary laws; in the halting case it has one stationary law attracting every initial law.Finite rulers in a hierarchical tiling turn the noise rate into a cutoff in a sequence of local flags. Computation boards read that cutoff and simulate the selected input. A threshold checker preserves a global bit when the computation does not halt. When it halts, interval-valued sensors erase every surviving chart, and a common-noise coupling removes dependence on every initial state. The guaranteed intervals accumulate at zero and lie at astronomically small rates. The space-time Gibbs correspondence gives a finite-energy interaction on Z^7 with undecidable translation-invariant Gibbs uniqueness. Thus one fixed local rule has an uncomputable noise diagram.
Authors: Douglas, Seth, Mghirbi, Nidhal
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225831064
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