Synthesis of Scalable Single Length Cycle, Single Attractor Cellular Automata in Linear Time

This paper proposes the synthesis of single length cycle, single attractor cellular automata (SACAs) for arbitrary length. The n-cell single length cycle, single attractor cellular automaton (SACA), synthesized in linear time O(n), generates a pattern and finally settles to a point state called the single length cycle attractor state. An analytical framework is developed around the graph-based tool referred to as the next state transition diagram to explore the properties of SACA rules for three-neighborhood, one-dimensional cellular automata. This enables synthesis of an (n+1)-cell SACA from the available configuration of an n-cell SACA in constant time and an (n+m)-cell SACA from the available configuration of n-cell and m-cell SACAs also in constant time.

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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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Definition and Application of a Five-Parameter Characterization of One-Dimensional Cellular Automata Rule Space

Artificial Life ◽

10.1162/106454601753238645 ◽

2001 ◽

Vol 7 (3) ◽

pp. 277-301 ◽

Cited By ~ 33

Author(s):

Gina M. B. Oliveira ◽

Pedro P. B. de Oliveira ◽

Nizam Omar

Keyword(s):

Phase Transition ◽

Dynamical Systems ◽

Cellular Automata ◽

Dynamic Behavior ◽

Discrete Dynamical Systems ◽

Transition Diagram ◽

One Dimensional ◽

Rule Space ◽

Spatially Extended

Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

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Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121420053 ◽

2021 ◽

pp. 1-19

Author(s):

Augusto Modanese

Keyword(s):

Cellular Automata ◽

Parallel Computation ◽

Time Complexity ◽

Regular Languages ◽

Constant Time ◽

Language Recognition ◽

One Dimensional ◽

Acceptance Condition

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes [Formula: see text] and (uniform) [Formula: see text]. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

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A faster algorithm for the Birthday Song Singers Synchronization Problem (FSSP) in one-dimensional CA with multiple speeds

Acta Informatica ◽

10.1007/s00236-020-00383-6 ◽

2021 ◽

Vol 58 (4) ◽

pp. 451-462

Author(s):

Thomas Worsch

Keyword(s):

Cellular Automata ◽

Linear Time ◽

Constant Factor ◽

One Dimensional ◽

Synchronization Problem ◽

Firing Squad ◽

Problem Instances ◽

Firing Squad Synchronization Problem ◽

The One ◽

Firing Squad Synchronization

AbstractIn cellular automata with multiple speeds for each cell i there is a positive integer $$p_i$$ p i such that this cell updates its state still periodically but only at times which are a multiple of $$p_i$$ p i . Additionally there is a finite upper bound on all $$p_i$$ p i . Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.

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