Pattern formation of generalized fuzzy elementary cellular automaton
Seiryu Shimizu, Tetsuji Tokihiro
TL;DR
This work addresses the limitation of fuzzifying elementary cellular automata via fuzzy disjunctive normal form by introducing generalized fuzzy elementary cellular automata (GFECA) that use flexible fuzzification functions $u,v,w$ and a shaping function $g$ to produce update rules $\tilde{f}_k(x,y,z) = g(f_k(u(x), v(y), w(z)))$ while preserving endpoint behavior. By defining fuzzification functions in $\Omega$ and demonstrating that endpoint consistency holds, the authors generate a broad family of CA-like dynamics, including cases that recover standard FECA and others with markedly richer patterns. They further explore pattern formation through GFECAs, including parameter-driven transitions that resemble phase changes, and quantify complexity via an effective dimension $d_\alpha$ tied to the $\ell^p$-norm. Extending to mixtures of rules, they show global rule mixing induces sharp dynamical transitions, with three-cell GFECAs already exhibiting fixed points, cycles, higher-dimensional attractors, and chaos, suggesting a fertile ground for modeling complex systems under uncertainty. The results point to potential integrations with machine learning for automatic discovery of fuzzy transition rules and broaden the applicability of CA-based modeling to natural and social phenomena. $d_\alpha$, $N_\alpha(p)$, and related fuzzification constructs provide a rigorous lens for analyzing and predicting pattern transitions in GFECAs.
Abstract
We propose a general method for constructing a fuzzy cellular automaton from a given cellular automaton. Unlike previous approaches that use fuzzy distinctive normal form, whose update function is restricted to third-order polynomials, our method accommodates a wide range of fuzzification functions, enabling the generation of diverse and complex time-evolution patterns that are unattainable with simpler heuristic models. We demonstrate that phase transitions in pattern formation can be observed by changing the parameters of the fuzzification function or the mixing ratio between two distinct evolution rules of elementary cellular automata. Remarkably, the resulting generalized fuzzy elementary cellular automata exhibit rich dynamical properties, including stable manifolds and chaos, even in minimal systems composed of just three cells.