Rule Primality, Minimal Generating Sets and Turing-Universality in the Causal Decomposition of Elementary Cellular Automata

New Universality results in Elementary Cellular Automata by emulation of Rule 110 just published by the Journal of Cellular Automata: . Also available at:  includes a minimal set of only 38 ECAs generating all others under composition and 2 new Turing universal 4-colour cellular automata.


"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 two 4-colour Turing-universal cellular automata 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."