A Novel Algorithm for Coarse-Graining of Cellular Automata
The coarse-graining is an approximation procedure widely used for simplification of mathematical and numerical models of multiscale systems. It reduces superfluous – microscopic – degrees of freedom. Israeli and Goldenfeld demonstrated in [1,2] that the coarse-graining can be employed for elementary cellular automata (CA), producing interesting interdependences between them. However, extending their investigation on more complex CA rules appeared to be impossible due to the high computational complexity of the coarse-graining algorithm. We demonstrate here that this complexity can be substantially decreased. It allows for scrutinizing much broader class of cellular automata in terms of their coarse graining. By using our algorithm we found out that the ratio of the numbers of elementary CAs having coarse grained representation to “degenerate” – irreducible – cellular automata, strongly increases with increasing the “grain” size of the approximation procedure. This rises principal questions about the formal limits in modeling of realistic multiscale systems.
Keywordsmultiscale systems coarse graining cellular automata
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- 1.Israeli, N., Goldenfeld, N.: Coarse-graining of cellular automata, emergence, and the predictability of complex systems. Phys. Rev. E. 73(2), 026203 (2006)Google Scholar
- 2.Israeli, N., Goldenfeld, N.: Computational irreducibility and the predictability of complex physical systems. Physical Review Letters 92(7), 074105 (2004)Google Scholar