# Partially permutive cellular automata

@article{Eloranta1993PartiallyPC, title={Partially permutive cellular automata}, author={Kari Eloranta}, journal={Nonlinearity}, year={1993}, volume={6}, pages={1009-1023} }

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…

## 27 Citations

### Random walks in cellular automata

- Mathematics, Computer Science
- 1993

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 dynamics of defect ensembles in one-dimensional cellular automata

- Materials Science
- 1994

We investigate the dynamics of ensembles of diffusive defects in one-dimensional deterministic cellular automata. The work builds on earlier results on individual random walks in cellular automata.…

### Commuting Cellular Automata

- MathematicsComplex Syst.
- 1997

We study the algebraic conditions under which two cellular automata can commute. We show that if either rule is permutive, i.e., one-to-one on its leftmost and rightmost inputs, then the other rule…

### Enumeration of Maximal Cycles Generated by Orthogonal Cellular Automata

- Computer Science, MathematicsNatural Computing
- 2022

This paper considers an alternative approach to generate pseudorandom sequences through orthogonal CA (OCA), which guarantees a better amount of diffusion.

### Enumerating Orthogonal Latin Squares Generated by Bipermutive Cellular Automata

- Mathematics, Computer ScienceAUTOMATA
- 2017

The general case of nonlinear rules in bipermutive cellular automata, which could be interesting for cryptographic applications such as the design of cheater-immune secret sharing schemes, is addressed.

### Non-Abelian Cellular Automata

- Computer Science
- 1995

We show that a wide variety of nonlinear cellular automata can be written as a semidirect product of linear ones, and that these CAs can be predicted in parallel time [cal O](log[super 2] t). This…

### Algebraic Properties of the Block Transformation on Cellular Automata

- Computer ScienceComplex Syst.
- 1996

If the blocked rule satisfies an identity which holds for a broad class of algebras, then the underlying rule must have essentially the same structure, and must depend only on its leftmost and rightmost inputs; roughly speaking, that the block transformation cannot turn a nonlinear rule into a linear one.

### Defect particle kinematics in one-dimensional cellular automata

- MathematicsTheor. Comput. Sci.
- 2007

### Graph-theoretical characterization of invertible cellular automata

- Computer Science, Mathematics
- 2000

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- Mathematics, Computer Science
- 1993

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.

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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.…

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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.

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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).

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We show that some of the deterministic one-dimensional cellular automata studied recently by Wolfram exhibit a kind of spontaneous symmetry breaking. The associated kinks (Bloch walls) perform…