Questions tagged [cellular-automata]

For questions on cellular automata, a discrete model consisting of a regular grid of cells.

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Using the rule stated below, is there a way I can plant trees on a 2D plane such that there are no trees in the first k diagonals? [duplicate]

This question was part of the problem set for PROMYS Europe 2020, a maths (or math, just in case I made anyone unhappy) camp held at Oxford every year. It was the one which I couldn't do, and I am ...
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How to create a non-square 2D grid with spherical topology.

When programming Conway's Game of life on my computer. A problem arises; how to deal with the borders on the board? Do the cells at the border have to take into consideration less neighbours than the ...
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is it possible to embed a fractal inside another fractal

I have been looking at one-dimensional (1D) cellular automata (CAs) which generate two-dimensional (2D) fractal patterns. Among the 256 1D elementary CAs, I tried to list down the fractal generating ...
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A basic transition automaton of ECA 30

I am reading this thesis, which on page no 14 talks about modeling. It also says in page no 15 that: The automata we construct to model the basic transition scan words over $\Sigma^2$, where $\...
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Elementary cellular automaton showing eventually periodic behavior after large number of iterations.

In this video: Cellular Automata and Rule 30 Stephen Wolfram talks about such an elementary cellular automaton at 17:01. Does anybody have an idea which one exactly he could be talking about?
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What is a periodic pattern in 1D Cellular Automata

What is a periodic pattern in 1D cellular automata? I have this 3-color, code 1599 rule and I can see that a certain generation can be grouped together by their similar or foreseeable pattern, then ...
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Is there any 3D patterns, if we are to stack up each generation of “Conway's game of life” on top of each other?

1D Cellular Automata do not show any interesting patterns if we look at each state only one at a time in Succession, but if we put each state below each other, we can see patterns emerging. In the ...
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Fractal Pattern from Queen's Move Construction

This question relates to the OEIS sequence A279212. Fill an array by antidiagonals upwards; in the top left cell enter $a(0)=1$; thereafter, in the $n$-th cell, enter the sum of the entries of ...
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Sugarscape Axis'

How are the axis' of Growing Artificial Societies 51x51 grid interpreted? What do they denote? The configuration is unfamiliar to me being 5 sequential lots of [0,1,2 ... 9] each. It seems related to ...
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Name of these Sierpinski-like fractals?

A Sierpinski triangle can be created by Starting with a row consisting of a single 1 Each row below is horizontally offset by a half-cell Each cell is the sum-mod-2 of the two cells above it Now ...
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How does On/Off Reversal work in Life-Like Cellular Automata

I've recently been made aware of the concept of on-off rule reversal in life-like cellular automata. I understand the algorithm for calculating the rule reversal from a given rule. But I don't ...
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Are there any symmetries in rules of Life-Like Cellular Automata?

Life-like cellular automata have $2^{18}$ different possible rulesets, and I would like to test hypotheses/search through them all, however that's a lot of rulesets to test a simulation on, so I'm ...
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Diagonals in rule 30 - anything new?

I've had some fun with rule 30: https://brunni.de/findings30/ from the abstract: The usual pattern starting from a single cell with state 1 / black is examined. It is contructed from a richer ...
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Prove that a recurrence for Rule 30 is: B2=MOD(A1+B1+(1+B1)*C1,2)

Prove that Rule 30 satisfies the recurrence: $$T(1, k) = [k = N]$$ $$T(n,k)=(T(n-1,k-1)+T(n-1,k)+(T(n-1,k)+1) T(n-1,+1+k)) \bmod 2$$ where [ ] is the Iverson bracket. ...
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Finding Life-Like Cellular Automata Rules and Initial States with Guaranteed Still End States

My goal is to find all rulesets of life-like cellular automata and initial states that, when simulated in a toroidal grid (although non-toroidal would be interesting as well) of arbitrary finite ...
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Trouble understanding proof in “Aperiodicity in One-Dimensional Cellular Automata” paper by Erica Jen

From the proof of page 7 in https://digital.library.unt.edu/ark:/67531/metadc1443182/m2/1/high_res_d/7230855.pdf (page 10 of the PDF): The fact that $R$ is injective in its $(i+1)$-th component (...
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Noether's theorem in SmoothLife?

Conway's Game of Life, being discretized in both space and time domains, has no locally conserved quantities. SmoothLife, however, is a generalization of the Game of Life to a continuous and spatially-...
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Simplest set of replacement rules which can include mathematical logic?

Things like the various Type Theories appear to be based on replacement rules of one kind or another. This got me thinking, what would be the simplest set of replacement rules (maybe even just one ...
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Are Cellular Automata models compatible with the Holographic Principle?

Cellular Automata are discrete models which have a regular finite dimensional grid of cells, each in one of a finite number of states, such as on and off. There are various scientists that have ...
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Cell state probabilities for a probabilistic version of Conway's Game of Life

I am given a version of Conway's Game of Life where each cell can be in any of 3 states: Dead (D), Alive (A) or Dying (X). The following transition rules exist: An Alive cell next to a Dying cell ...
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About Tegmark's mathematical universe

I'm not sure anyone else than Tegmark himself can answer this, but why not give it a try. Would Tegmark consider a cellular automata a mathematical structure? If nature is mathematical, isn't it also ...
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Is Langton's Ant really Turing Complete

I recently watched a Numberphile video about Langton's Ant (and the extra footage). They mention that the ant always ends up creating a highway at some point in all the initial board configurations ...
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Infinitesimal Cellular Automaton

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following: Let $(X,*)$...
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Open problems in Cellular Automata field

here there is a link on Wolfram about 20 open problems of CA theory. Has anyone of them been solved or tested? I'm searching for literature.
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What reversible cellular automaton rule emulates all 256 Wolfram rules?

On page 648 of A New Kind of Science, there's a definition of a "universal" cellular automaton, which can emulate Wolfram's 256 elementary cellular automata. Furthermore, it emulates them in a "cell-...
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Reflector in Conway's Game of Life on Triangular Tessellations

Back in 1994, Carter Bays reports in "Cellular Automata in the Triangular Tessellation" that there are at least 6 Games of Life (GoL) living on triangular tessellations. A GoL has at least one glider (...
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Project ideas on Chaos theory, Cellular Automata, Fractals, Games, IA [closed]

I'm a computer science student and I need to find a final year project. What interests me the most is Chaos, IA, Games, Fractals, CA.. Something I liked was the chaos theory within sudoku. The ...
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Find minimal number of elements in matrix.

Consider a $A \in Mat_{n}(\{+1,-1\})$ (square matrix consisted of +1,-1). Now we can make and operation majority , i.e. $a_{i,j} = $ median of his neighborhoods(closest elements around him, i.e. ...
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Clustering computation in pair approximation model

Let's consider a square lattice of cells. Each cell can be either occupied by a species (1 or 2) or be empty (0). Each cell can be either in state 1, 2 or 0. In the pair approximation model, I ...
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Similarities of CA Rule 150 and Odd Collatz-function outputs

I made a "discovery" a couple of weeks ago in regards to the first iterates of (odd) numbers of the form $2^n-1$ where $n\in\mathbb{N}$. First iterates is a bit loose term here; what I mean is all of ...
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Contributing to a unsolved problem and writing a paper about it [closed]

How do I write a paper on the Collatz Conjecture without fooling or making an ass out of myself? Im not affiliated to any University at the moment (my past Uni was like 20 years ago, but I didnt ...
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Cellular Automata on the Collatz Conjecture

I have a Cellular Automaton that of any initial integer (initial condition of the automaton) generates states of Collatz sequences. The neighbourhood of the automaton is shaped like an L-tetromino (...
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How to translate mathematical intuition into a rigorous proof?

I've seen a lot of questions about how to develop mathematical intuition, but often I have the opposite problem. Several times I have run into a situation where I want to solve a math problem, and I ...
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How to interpret wolfram rule 108 as a simplified populationmodel

I have made a Matlab-program as to iterate wolfram-rule 108 on any chosen begin situation. However my question is how this rule can be interpreted as a simplified version of a population model. (...
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Mandelbrot and Julia fractals for $z_{n+2} = z_{n+1}^2 + z_n^2 + c$

The Mandelbrot and Julia type fractals are very Well known. But such fractals follow from $$z_n = f(z_{n-1},c)$$ In other words a recursion that only depends on the previous value and a constant. (...
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Second-order Cellular Automaton definition

What does second-order term mean in a Cellular Automaton? I read on second-order cellular automaton (wikipedia) that a second-order has two time-states, but does the definition of an order mean the ...
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In the linked image, why does a pattern form, as opposed to random noise?

Image in question This image was generated line by line, with each pixel's value determined by a set of rules: If the pixel is on the edge, set to the same color as the previous pixel 1 away from ...
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Does Day and Night on a Klein bottle have a steady state?

Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement ...
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Population Preserving Cellular Automata

Let's consider the class $CA$ of cellular automata where a cell is thought to exist in the $2$-dimensional grid ${\bf Z}\times{\bf Z}$, has binary states and the usual neighbourship relation of of ...
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How does a cellular automaton “know” when to halt?

I came across the (proven) claim that the cellular atomaton "rule $110$" is universal, meaning that it can do every calculation in principle. But to calculate a result, the automaton must "halt" ...
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Simplest universal cellular automaton

Is there more simple one-dimensional universal CA than Rule 110? One where next state of cell is determined not by 3 but by 2 cells?
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Average time bootstrap percolation

Presentation of the model: we consider the regular lattice created from $\mathbb{Z}^d$. At $t=0$, each site is said "active" independently with a probability $p$, "inactive" otherwise. At $t$, if a ...
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Invertibility of Cellular Automata over $A^G$ for locally finite G.

This is an exercise from "Cellular Automata and Groups" by T. Ceccherini-Silberstein and M. Coornaert. Let $G$ be a locally finite group and let $A$ be an arbitrary set. Let $\tau: A^G \rightarrow A^...
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Any good reference for start in Cellular Automata?

The question is as simple as the title. I want to know more about Cellular Automata, and his applications in Chemistry and Biology. I have read that, maybe, there is a relation of the topic with Graph ...
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How to show that the probability that $\mathbb{Z}^d$ is internally spanned is equal to 1? (bootstrap percolation)

Presentation of the model: we consider the regular lattice created from $\mathbb{Z}^d$. At $t=0$, each site is said "active" independently with a probability $p$, "inactive" otherwise. At $t$, if a ...
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Are there injective maps from a higher- to a lower-dimensional cellular automaton?

Suppose we have a cellular automaton on $\mathbb Z^n$ with cell values in the finite set $V$, with update function $u : V^{\mathbb Z^n} \to V^{\mathbb Z^n}$, and similarly another cellular automaton ...
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Chaos in discrete space and time systems. Definition, tests, etc?

I have been working with a cellular automata model and have been struggling to characterize its dynamics. It seems to be displaying aperiodic behaviour in a certain part of the parameter space, but I ...
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Expanding Nebula: Problem in Cellular Automata or Coding Theory

First post to Math Stackexchange. Recently I was posed with a mathematically intensive computer programming challenge that I am completely at a loss for solving. To give you some info on my math ...
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Real life application of Conway's Game of life

Does Conway's Game of Life has any real life application? I mean applications that are used today. If so, please add references, because I couldn't find anything, except for some hypothetical ...
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How can I prove that Game of Life's evolution function is continuous?

Conway's Game of Life is a cellular automaton, but also a discrete dynamical system. In all the papers, books, notes I have read on it, it is never never never shown that its evolution function is ...