# Questions tagged [cellular-automata]

The cellular-automata tag has no usage guidance.

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### A sensitive 2-dimensional cellular automaton with a blocking word

I'am a Ph.D student in the domain of discrete dynamical systems. My thesis is about spectral properties of cellular automata in higher dimension.
Kurka gives a classification for one dimensional ...

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### Growing gliders under rule 110

I found a glider in the evolution space of rule 110 that grows constantly in size. Normal gliders live in the so-called ether, e.g. the so-called E-glider:
Other – often complex – gliders exist in an ...

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### Relation between symbolic substitution and cellular automata

I recently asked this on Math Stackexchange recently in this thread. I was told that there is a relation between symbolic substitutions and cellular automata. I'm vaguely familiar with Cobham's ...

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### Elementary cellular automata in stochastic modes

There are several ways to run a given elementary cellular automaton in a stochastic way:
by giving for each of the eight local configurations 000,100,010 and so on a probability by which the rule is ...

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### Asymptotic densities of rules of elementary cellular automata

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes ...

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### Cellular automata with different generating-cells configuration

I recently faced the problem of generating "interesting" binary matrices for testing a heuristics for the $3DCC$-problem, i.e. to check if the graph represented by the adjacency matrix ...

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### A special kind of pseudo-garden eden states in cellular automata

I'm currently investigating Wolfram's elementary cellular automata on finite grids with periodic boundary conditions, i.e. on $\mathbb{Z}/k$ for different $k$.
It is clear that for each rule $R$ and ...

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### How slowly can one be absorbed in the absorbing phase of the directed percolation universality class?

In absorbing state transitions on a lattice, one often considers "active" and "inactive" sites, with update rules describing how activity spreads or decays. When the lattice sites ...

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### Lengths of paths through Conway’s Game of Life

This question is inspired by the following challenge from CodeGolf.SE: https://codegolf.stackexchange.com/q/251510/88765.
Given positive integer $N$, we can consider a version of Conway’s game of life ...

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### Mysteries of Wolfram's rule 18

[Unfortunately, I made some mistakes in my original question. I tacitly corrected them wherever I found them.]
Wolfram's rule 18 gives rise to fractal patterns, but when started with two black cells ...

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### Robust (=error-correcting) configurations in Conway's Game of Life

Motivation / discussion: Conway's Game of Life automaton is often suggested as a model for either real (biological) life and/or computation, but one important trend in both real life and computer ...

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### Possible shifts in finite elementary cellular automata

I investigated the long term behaviour of a pair of black cells ■■ on a circle of $N$ cells under the action of each of Wolfram's rules $R$. For each combination $(R,N)$ I determined the first ...

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### "Rule 30" in the infinite setting

This question tries to get right what went wrong in an earlier question.
Let $\{0,1\}^\mathbb{Z}$ denote the set of all functions $x:\mathbb{Z}\to \{0,1\}$. Let $+$ denote addition modulo $2$ on $\{0,...

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### Possible finite periodicities of "Rule 150" in the infinite setting

"Rule 150" is a fascinating one-dimensional and simple cellular automaton giving raise to some quite chaotic behaviour. This is the starting point of this question.
Let $\{0,1\}^\mathbb{Z}$ ...

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### Astonishing affinity of Wolfram's rule 110 to the numbers 2 and 7

I investigated the evolution of a single black cell on 1-dimensional grids with periodic boundary conditions of variable sizes $N$ under Wolfram's rule 110 which is the only one for which Turing ...

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### Local rule for the product of two cellular automata

Consider two one-dimensional cellular automata $(A^{\mathbb Z},F)$ and $(B^{\mathbb Z},G)$ with alphabets $A$ and $B$ and global rules $F: A^{\mathbb Z} \to A^{\mathbb Z}$ and $G: B^{\mathbb Z} \to B^{...

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### Does Conway's game of life admit a notion of energy?

(I am not sure if this is a mathematics or physics question so I am not sure where to post it. I am posting it here because the chief subject is an unreal universe that is purely a subject of ...

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### Reparameterizing Conway's Game of Life as a scalar reaction-diffusion equation

Consider the set of functions $\mathbb Z^2\to\{0,1\}$, the state space of Conway's Game of Life. One common way to define the update rule is to use the following 3x3 neighbor-counting convolution ...

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### Is there a description of cellular automata in form of sheaves?

Cellular automata are defined through rules in a local neighborhood and sheaves, as far as I understand, can be used to glue local data to global data. Has there been any effort to bring those two ...

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### Binary cellular automata: How slowly can an eroder remove $1$'s?

Consider some deterministic, monotonic, eroding binary cellular automata on some lattice $\mathbb{Z}^d$, and consider the set of initial states $I(L)$ in which all of the vertices are $0$ except for ...

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### Is Toom's rule robust under local but non-on-site noise?

Toom's rule is a 2-dimensional cellular automaton which is known to have two distinct stationary measures in the thermodynamic limit, even after small perturbations to a probabilistic cellular ...

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### Are there (probablistic) uniform 1D cellular automata which can fault-tolerantly store one bit?

In two dimensions, "Toom's rule" is known to be a cellular automaton which can fault-tolerantly store one bit of information. This means that, if we start with the all-0 configuration on an $...

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### Oscillator in Langton's ant

First of all, see Langton's ant Wikipedia page.
If we place a pair of ants looking north (using Golly or any another prog) on the coordinates $(x_1,y_1)$ and $(x_2,y_2)$ under the conditions:
$p=|x_1-...

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### Spread of a disease on a modular chessboard (torus) - lower bound

I learned about the following result from one of Peter Winkler's books:
It is impossible to infect the entire $n\times n$ chessboard (usual chessboard) starting from fewer than $n$ infected cells.
The ...

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### 2D quadrant sandpile: emergent highway structure

Consider the top-right quadrant of the plane divided into unit cells, each cell containing some number of chips. A cell containing at least two chips can fire two chips, one to the cell above it and ...

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### Topologically mixing cellular automata on groups

For which group-alphabet pairs $(G, A)$ does $(G, A^G)$ admit a topologically mixing cellular automaton?
Definitions:
Let $G$ be a (discrete) group. An alphabet is a finite set of cardinality at ...

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### Busy Beaver in cellular automata

The Busy Beaver function is usually studied for Turing machines. Of course, it's not a computable function, but for 2-state Turing machines, the first four values are known, and the fifth conjectured.
...

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### Absolute oscillator in Langton's Ant

We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...

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### Probabilistic approach for cellular automata

Few months ago my scientific adviser asked me to use probabilistic ideas in such problem :
Consider a matrix NxN. Each element of matrix is a number 1 or 0. We may change all elements of this matrix ...

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### Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?

Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...

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### Minimal period for a bounded Langton's ant moving on a tessellation

We consider Langton's ant on the 2D plane, but we replace the square lattice by a Voronoi tessellation obtained from a finite set of points (it could be another tessellation, however directions such ...

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### Percolation in torus under threshold rule

As part of my graduate research I am currently studying the last section in the paper "Random Majority Percolation" by Balister, Bollobas et al. The paper itself is very complicated but the last two ...

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### The von Neumann algebra generated by a non-closable operator

Let $H$ be a separable Hilbert space and let $M$ be a densely defined operator $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\...

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### The 1-step vanishing polyplets on Conway's game of life

A $n$-polyplet is a collection of $n$ cells on a grid which are orthogonally or diagonally connected.
The number of $n$-polyplets is given by the OEIS sequence A030222: $1, 2, 5, 22, 94, 524, 3031, \...

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### The graph of Rule 110 and vertices degree

Consider the elementary cellular automaton called Rule 110 (famous for being Turing complete):
It induces a map $R: \mathbb{N} \to \mathbb{N}$ such that the binary representation of $R(n)$ is ...

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### Vanishing line on Conway's game of life

If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$.
...

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### At what rates can creatures in a conservative cellular automata expand?

Let $A$ be a finite set, and suppose $0\in A$.
For each $a\in A\setminus\{0\}$, let $w_{a}$ be a positive integer called the weight of $a$, and let $w_{0}=0$. Give $A$ the discrete topology and $A^{\...

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### Why do some linear cellular automata over $Z_{2}$ on the torus have small order?

At https://dmishin.github.io/js-revca/index.html, you can play around with reversible cellular automata. I noticed that on that site, that for the reversible linear cellular automata (which I have ...

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### Intermediate results for Langton's ant highway conjecture

This paper states the following theorem about Langton's ant:
The set of cells that are visited infinitely often by the ant (for a given initial configuration) has no corners. A corner of a set is a ...

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### Vice-versa Erdos conjecture

Erdos conjectured that, except $1, 4$ and $256$, no power of $2$ is a sum of powers of $3$.
A vice-versa conjecture may be: except $1$, $9$ and $81$ all powers of $3$ contains two consecutive ...

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### Life. Intermediate stages

My question is pure mathematics when restricted to the cellular automata theory.
John von Neumann got the grasp of and defined life. Many years later biologists supported von Neumann's definition of ...

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### Lattice Boltzman derivation for vorticity eqn $\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega$

So as showed by Frisch et al. (a), the 2D Euler equation $$v_{t}+ v\cdot \nabla v=\mu \Delta v$$ can be derived by the Hexagonal-placed automaton (for low velocity).
I am curious about the existence ...

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### How to get $\omega$-regular expression from buchi automaton

Is there an algorithm or a trick on how to get $\omega$-regular expressions from Buchi automatons? If yes, is there also some way to do create minimal such regular expressions?
It is extremely ...

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### Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the ...

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### Is the variety of algebras $(A,*,+)$ that satisfy $(x*y)+(y*z)=(x+y)*(y+z)$ generated by its finite algebras?

The finite algebras $(A,*,+)$ that satisfy the identity $(x*y)+(y*z)=(x+y)*(y+z)$ are precisely the algebras such that the one-dimensional cellular automata produced by $*$ and $+$ are commutative ...

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### Does the 2-shift map have a root automorphism?

By the 2-shift map I mean the map $T:\{0,1\}^\mathbb{Z}\to \{0,1\}^\mathbb{Z}$ that shifts the sequence leftwise. By a root I mean an homeomorphism $\psi:\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ that ...

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### Does Langton's ant cover every n by 6 gridded torus?

This post follows this other post about times cover by Langton's ant of $n$ by $n$ gridded torus.
For $n$ by $n$ gridded torus, I've checked for $n \le 1000$ that the ant covers all. This fact needs ...

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### Time for Langton's ant to cover a "square" torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel in
...

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### Periodic configurations for elementary cellular automata

Let $L$ be an elementary cellular automaton. Then $L$ acts on $\{0,1\}^{\mathbb{Z}}$. We say that a configuration $w\in\{0,1\}^{\mathbb{Z}}$ is periodic if $L^{(n)}(w)=w$ for some $n\in\mathbb{N}$.
...

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### Math behind Conway's Game of Life [closed]

I'm an undergraduate student intending to double major in Computer Science and Pure Math, and I wish to do research on cellular automata in the future. My question is: what kind of mathematical basis ...