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arXiv · 2610.14026

Towards New Parity Rules with Asynchronous Updating Inspired by Fatès's Rule

Rolf Hoffmann
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Abstract

The parity problem in 1D Cellular Automata is a classifier based on a local rule. If the number of ones in a ring of binary cells is odd then the automaton shall converge to all-one (11...11) else to all-zero (00...00). The objective is to find rules that are related to Fatès's Balanced Rule F but use less active neighborhoods (actions). The method is to analyze the effect of each actions and to find a functioning subset of the 12 actions used in rule F. New rules called HA, HB, HC, HF are defined with 10, 6, 4, 8 actions. Rule HC and HF need only three inputs whereas all others need four. The classification time of the rules was determined by averaging over many simulation runs with random initial configurations. It is almost the same for the rules HA and F, and is longer for the rules HB, HC, HF. In addition, the rule HD is defined, which updates three cells in parallel and works faster. In future work, the functioning and performance of the proposed rules should be treated in a more formal way, and related rules (also with different update schemes) could be considered.

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Rolf Hoffmann. 2026-09-30. Towards New Parity Rules with Asynchronous Updating Inspired by Fatès's Rule. https://arxiv.org/abs/2610.14026

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