5-State Rotation-Symmetric Number-Conserving Cellular Automata are not Strongly Universal

TL;DR

This paper characterizes 5-state rotation-symmetric number-conserving cellular automata on the von Neumann neighborhood and demonstrates they cannot be strongly Turing universal, though they can embed some boolean circuit elements.

Contribution

The paper provides a complete classification of 5-state RNCA and proves their limitations in computational universality, while showing how to embed certain circuit components.

Findings

5-state RNCA are not strongly Turing universal

Complete characterization of 5-state RNCA rules

Embedding of boolean circuit elements in 5-state RNCA

Abstract

We study two-dimensional rotation-symmetric number-conserving cellular automata working on the von Neumann neighborhood (RNCA). It is known that such automata with 4 states or less are trivial, so we investigate the possible rules with 5 states. We give a full characterization of these automata and show that they cannot be strongly Turing universal. However, we give example of constructions that allow to embed some boolean circuit elements in a 5-states RNCA.