An efficiently computable characterization of stability and instability for linear cellular automata

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

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A characterization of important algorithms for quantum-dot cellular automata

Information Sciences ◽

10.1016/s0020-0255(98)10033-6 ◽

1999 ◽

Vol 113 (3-4) ◽

pp. 193-204 ◽

Cited By ~ 16

Author(s):

John C. Lusth ◽

Brandon Dixon

Keyword(s):

Cellular Automata ◽

Quantum Dot ◽

Quantum Dot Cellular Automata

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Recurrent Misconceptions in the Study of CA Reversibility on Triangular Grids

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421500140 ◽

2021 ◽

Vol 31 (01) ◽

pp. 2150014

Author(s):

Barbara Wolnik ◽

Maciej Dziemiańczuk ◽

Bernard De Baets

Keyword(s):

Cellular Automata ◽

Boundary Conditions ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

Triangular Grids

We present counterexamples illustrating that the characterization of the reversibility of linear cellular automata on finite triangular grids given by Uguz et al. [2017] and Uguz et al. [2019] is not valid, neither in the case of null boundary conditions nor in the case of periodic boundary conditions.

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Sequentializing cellular automata

Natural Computing ◽

10.1007/s11047-019-09745-7 ◽

2019 ◽

Vol 19 (4) ◽

pp. 759-772 ◽

Cited By ~ 1

Author(s):

Jarkko Kari ◽

Ville Salo ◽

Thomas Worsch

Keyword(s):

Cellular Automata ◽

Cellular Automaton ◽

Intermediate States ◽

The Right

Abstract We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single sweep of a bijective rule from left to right over an infinite tape. Such cellular automata are necessarily left-closing, and they move at least as much information to the left as they move information to the right.

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On characterization of state transition graph of additive cellular automata based on depth

Information Sciences ◽

10.1016/0020-0255(92)90120-w ◽

1992 ◽

Vol 65 (3) ◽

pp. 189-224 ◽

Cited By ~ 1

Author(s):

Aloke K. Das ◽

Tapas K. Nayak ◽

P. Pal Chaudhuri

Keyword(s):

Cellular Automata ◽

State Transition ◽

Transition Graph ◽

State Transition Graph

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Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/605098 ◽

2011 ◽

Vol 2011 ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

Pabitra Pal Choudhury ◽

Sudhakar Sahoo ◽

Mithun Chakraborty

Keyword(s):

Cellular Automata ◽

Cellular Automaton ◽

Boolean Functions ◽

State Transition ◽

The State ◽

Transition Diagram ◽

One Dimensional ◽

State Transition Diagram ◽

Linear Rule

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

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