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In this paper, we prove that there is a weakly universal cellular automaton on the pentagrid with three states which is rotation invariant and which uses \`a la Moore neighbourhood. Moreover, at each step of the computation, the set of non…

Discrete Mathematics · Computer Science 2015-11-02 Maurice Margenstern

The objective is to find a Cellular Automata rule that can form a 2D point pattern with a maximum number of points (1-cells). Points are not allowed to touch each other, they have to be separated by 0-cells, and every 0-cell can find at…

Computational Geometry · Computer Science 2022-02-15 Rolf Hoffmann

A necessary and sufficient condition for a one-dimensional q-state n-input cellular automaton rule to be number-conserving is established. Two different forms of simpler and more visual representations of these rules are given, and their…

adap-org · Physics 2007-05-23 Nino Boccara , Henryk Fuks

A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e.…

Discrete Mathematics · Computer Science 2011-08-25 Pierre Guillon , Gaétan Richard

We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional (3D) toric…

Quantum Physics · Physics 2021-01-26 Michael Vasmer , Dan E. Browne , Aleksander Kubica

Cellular automata have been useful artificial models for exploring how relatively simple rules combined with spatial memory can give rise to complex emergent patterns. Moreover, studying the dynamics of how rules emerge under artificial…

Cellular Automata and Lattice Gases · Physics 2014-07-11 Theodore P. Pavlic , Alyssa M. Adams , Paul C. W. Davies , Sara Imari Walker

Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the…

Quantum Physics · Physics 2023-03-15 Jonathan F. San Miguel , Dominic J. Williamson , Benjamin J. Brown

We study the effect of mixing two rules on the dynamics of one-dimensional cellular automata by large scale numerical simulations. We calculate the decay of the magnetization for the Domany-Kinzel automaton (XOR/AND mixing) to its…

Condensed Matter · Physics 2007-05-23 Stam Nicolis

Cellular automata can show well known features of quantum mechanics, such as a linear updating rule that resembles a discretized form of the Schr\"odinger equation together with its conservation laws. Surprisingly, a whole class of…

Quantum Physics · Physics 2016-08-26 Hans-Thomas Elze

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the…

Probability · Mathematics 2012-07-25 Mark McCann , Nicholas Pippenger

In this paper we use the cellular automaton (CA) approach to model one-dimensional binary diffusion in solids. Employing a very simple state change rule we define an asynchronous CA model and take its continuum limit to obtain the governing…

Computational Physics · Physics 2019-11-20 Helena Ribera , Brian Wetton , Timothy Myers

We study a cellular automaton model, which allows diffusion of energy (or equivalently any other physical quantities such as mass of a particular compound) at every lattice site after each timestep. Unit amount of energy is randomly added…

Condensed Matter · Physics 2009-10-22 T. C. Chan , H. F. Chau , K. S. Cheng

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the…

Probability · Mathematics 2007-09-10 Mark McCann , Nicholas Pippenger

In this paper, under certain conditions we consider two-dimensional cellular automata with the Moore neighborhood. Namely, the characterization of 2D linear cellular automata defined by the Moore neighborhood with some mixed boundary…

Dynamical Systems · Mathematics 2024-06-11 B. A. Omirov , Sh. B. Redjepov , J. B. Usmonov

The objective is the design of a Cellular Automata rule that can form patterns with 'touching' loops. A loop is defined as a closed path of 1-cells in a 2D grid on a zero background and with a zero border. A path cell is connected with two…

Cellular Automata and Lattice Gases · Physics 2024-10-18 Rolf Hoffmann

This paper deals with the theory and application of 2-Dimensional, nine-neighborhood, null- boundary, uniform as well as hybrid Cellular Automata (2D CA) linear rules in image processing. These rules are classified into nine groups…

Discrete Mathematics · Computer Science 2008-04-16 Pabitra Pal Choudhury , Birendra Kumar Nayak , Sudhakar Sahoo , Sunil Pankaj Rath

An important question to be addressed regarding system control on a time interval $[0, T]$ is whether some particular target state in the configuration space is reachable from a given initial state. When the target of interest refers only…

Cellular Automata and Lattice Gases · Physics 2024-03-07 Franco Bagnoli , Samira El Yacoubi , Raul Rechtman

We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to $N$. The behaviour of the automaton is mainly determined by the support of the random variable that selects one…

Probability · Mathematics 2011-01-07 F. M. Dekking , L. van Driel , A. Fey

Using the concept of the Boolean derivative we study damage spreading for one dimensional elementary cellular automata and define their maximal Lyapunov exponent. A random matrix approximation describes quite well the behavior of…

Statistical Mechanics · Physics 2007-05-23 F. Bagnoli , R. Rechtman , S. Ruffo

We systematically study the boundaries of one-dimensional, 2-color cellular automata depending on 4 cells, begun from simple initial conditions. We determine the exact growth rates of the boundaries that appear to be reducible. Morphic…

Cellular Automata and Lattice Gases · Physics 2015-03-13 Charles D. Brummitt , Eric Rowland