# Partially permutive cellular automata

@article{Eloranta1993PartiallyPC,
title={Partially permutive cellular automata},
author={Kari Eloranta},
journal={Nonlinearity},
year={1993},
volume={6},
pages={1009-1023}
}
• K. Eloranta
• Published 1 November 1993
• Computer Science, Mathematics
• Nonlinearity
One-dimensional cellular automata are analysed via their generalized permutivity. Invariant subalphabets provide a systematic way of identifying periodic and aperiodic tilings as well as stationary distributions invariant under the cellular automaton iteration. In the case of several invariant subalphabets a hierarchy of interaction phenomena arise. In particular the interaction of subalphabets can generate random walks as well as their degenerate forms. A comprehensive scheme emerges that…

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## References

SHOWING 1-10 OF 27 REFERENCES

### Random walks in cellular automata

Topological defects or phase boundaries discerned in a number of one-dimensional cellular automata appear to perform random walks as well as simpler motions are analyzed to result in a complete classification in the partially permutive case.

### The kink of cellular automaton rule 18 performs a random walk

• Mathematics
• 1992
We give an exact characterization of the movement of a single kink in the elementary cellular automaton Rule 18. It is a random walk with independent increments as well as independent delay times.

### The Interaction Dynamics of the Kinks in the Cellular Automaton Rule 18

A series of computer simulations on the joint motion of two kinks in the elementary cellular automaton Rule 18 show a remarkably close approximation to the case of independent annihilating random walks except at close range interaction where combinatorial conditions impose an asymmetry on the dynamics.

### Particlelike structures and their interactions in spatiotemporal patterns generated by one-dimensional deterministic cellular-automaton rules.

• Physics
Physical review. A, Atomic, molecular, and optical physics
• 1991
A detailed description of such reactions'' sheds new light on the large-time behavior of range-1 Rule 54 with a very slow decrease of the particle number, as ${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensure Math{\gamma}}$ (Â£0.15).

### The attractor—basin portrait of a cellular automaton

• Computer Science
• 1992
The attractor-basin portrait of nonlinear elementary CA rule 18 is described, whose global dynamics is largely determined by a single regular attracting domain, and it is confirmed that in rule 18, isolated dislocation trajectories, as well as a dislocation gas, agree extremely well with the classical model of annihilating diffusive particles.