Generalized FDNF fuzzification of elementary cellular automata and its nonlinear pattern dynamics

Abstract

Fuzzy disjunctive normal form (FDNF) gives the canonical multi-affine extension of an elementary cellular automaton (ECA) rule to the unit cube. Although it preserves the Boolean rule on binary states, its multi-affine structure can smooth high-contrast CA patterns and restrict continuous-state dynamics. We introduce generalized FDNF rules \[ \widetilde f_k^{g,u,v,w}(x,y,z) = g\left(f_k(u(x),v(y),w(z))\right), \] where the transformations $g,u,v,w: [0,1] \to [0,1]$ fix the endpoints. The identity maps recover ordinary FDNF, while threshold-like, discontinuous, non-monotone, and expanding choices yield rule-preserving fuzzy ECAs. We demonstrate, in representative rules, that the transformation shape strongly affects pattern dynamics: threshold-like maps promote ECA-like pattern recovery, parameter deformations interpolate toward FDNF-like smoothing, and discontinuities induce gap-generated regimes. Pattern changes are summarized by contrast, fuzziness, and a finite-resolution participation-type support exponent. In three-cell systems, an expanding non-monotone transformation yields stable period-six cycles for rule 210, verified by interval arithmetic, coexisting with an expanding invariant line set; rules 51 and 85 inherit one-dimensional expanding dynamics. The framework provides a rule-preserving bridge from Boolean cellular automata to fuzzy and continuous-state nonlinear dynamics.