# Questions tagged [cellular-automata]

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### Periodicity of oscillators in Langton's Ant and powers of $2$

This question based on previous one by me. As Christopher Purcell noticed in his comment, there exist conjecture (which has a lot of counterexamples) that if you take a pair of ants $(n,n+1)$ apart (...

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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 ...