Coarse-graining of cellular automata, emergence, and the predictability of complex systems.
We explore the reprogramming capabilities of computer programs by means of cellular automata (CA). We show a series of crossing boundary results including Wolfram Class 1 Elementary Cellular Automata (ECA) emulating a Class 2 ECA, a Class 2 ECA emulating a Class 3 ECA, and a Class 3 ECA emulating a Class 2 ECA, among results of similar type for general CA (neighbourhood range r = 3/2) including a Class 1 emulating a Class 3, Classes 3 and 4 emulating Class 4, and Class 4 emulating Class 3. All these emulations occur with only a constant overhead as a result of the block emulation method and thereby considered computationally efficient. By constructing emulation networks from an exhaustive search in the compiler space we show that topological properties such as ingoing and outgoing hub degrees determining emulation directions suggests a novel classification of class 4 behaviour and is in agreement to Turing universality capabilities. We also found that no hacking strategy based on compiler complexity or compiler similarity is suggested. We introduce a definition of CA relative prime rule similar to that of prime numbers acting as basic constructors of other rules under composition proving that all the ECA rule space can be constructed from a small subset none of which is a Wolfram Class 4 rule. We finally show a Turing universality result of a composition of ECA rules emulating rule 110. The approach leads to a topological perspective of the computer reprogramming and controlling space and the wide computing capabilities of even the simplest computer programs with appropriate initial conditions thereby providing strong evidence that Turing universality is ubiquitous. PACS Numbers: 05.45.-a ∗JR and HZ conceived the project, wrote the code, analysed the data, and wrote the paper together. Both authors contributed equally. †firstname.lastname@example.org ‡email@example.com 1 ar X iv :1 51 0. 01 67 1v 3 [ cs .F L ] 8 O ct 2 01 5