Rule Primality, Minimal Generating Sets and Turing-Universality in the Causal Decomposition of Elementary Cellular Automata
Jürgen Riedel and Hector Zenil
We introduce several concepts such as prime and composite rule, tools and methods for causal composition and decomposition. We discover and prove new universality results in ECA, namely, that the Boolean composition of ECA rules 51 and 118, and 170, 15 and 118 can emulate ECA rule 110 and are thus Turing-universal coupled systems. We construct the 4-colour Turing-universal cellular automaton that carries the Boolean composition of the 2 and 3 ECA rules emulating ECA rule 110 under multi-scale coarse-graining. We find that rules generating the ECA rulespace by Boolean composition are of low complexity and comprise prime rules implementing basic operations that when composed enable complex behaviour. We also found a candidate minimal set with only 38 ECA prime rules — and several other small sets — capable of generating all other (non-trivially symmetric) 88 ECA rules under Boolean composition.
Keywords: Causal composition; multi-scale coarse-graining; renormalization; boolean composition; elementary cellular automata; turing-universality; ECA algebraic and group-theoretic properties.