Questions tagged [cellular-automata]
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81 questions
1 vote
2 answers
86 views
Restriction via subconfigurations versus restriction as image of a cellular map
I ask this question in the context of cellular automata, where there are two different methods to define a subset of the set of configurations in a "local" manner. Consider a finite set $A$, ...
- 1,053
4 votes
0 answers
146 views
Are sums of surjective completely additive functions modulo all $n$ equally distributed?
Let us call a function $F \colon \mathbb N_{+} \to \mathbb Z$ balanced if for all $n > 1$, the function $\overline{F} = \tau \circ F:\mathbb N_{+} \to \mathbb Z/n$, where $\tau \colon \mathbb Z \to ...
- 12.3k
2 votes
0 answers
168 views
Why a discrete amenable group satisfy Følner condition?
A discrete group $G$ satisfying Følner condition means that there exists a net $(F_j)_{j\in J}$ of nonempty finite subsets of $G$ such that $\mathop{lim}\limits_{j}\frac{|F_j\triangle gF_j|}{|F_j|}=0$ ...
- 99
3 votes
0 answers
139 views
What is the smallest known number of states that a one-way cellular automaton needs to be universal?
We know there is an elementary cellular automaton (ECA) with 2 states (Rule 110) that is universal, i.e. Turing-complete. One-way cellular automata (OCA's) are a subcategory of ECA's where the next ...
- 121
1 vote
0 answers
151 views
Dynamical properties of cellular automata of small diameter
Let $f\colon\{0,1\}^k\to\{0,1\}$ be a function, $j$ an integer, and define the cellular automaton $F\colon\{0,1\}^\mathbb{Z}\to\{0,1\}^\mathbb{Z}$ by $F(x)_i=f(x_{i+j},\dotsc,x_{i+j+k-1})$. I wonder ...
- 29
1 vote
1 answer
143 views
Probabilistic 2D cellular automata with memory lifetime increasing like $e^{L^2}$
Consider 2-state probabilistic cellular automata on an $L\times L$ torus square lattice which has the all-$0$ and all-$1$ configurations as fixed points, thinking of something similar to Toom's rule ...
- 3,095
1 vote
1 answer
211 views
Properties of limit set for cellular automata
Is anyone familiar with results about properties of the limit set of the local rule for a cellular automaton? I haven't been able to find any good materials on the subject from an initial search, and ...
- 703
0 votes
0 answers
102 views
A cellular automaton with an image that is not closed
Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
- 891
2 votes
1 answer
237 views
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 ...
0 votes
0 answers
182 views
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 ...
- 9,934
0 votes
0 answers
120 views
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 ...
- 703
3 votes
2 answers
313 views
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 ...
- 9,934
1 vote
0 answers
129 views
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 ...
- 9,934
0 votes
0 answers
76 views
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 ...
- 13.9k
1 vote
1 answer
209 views
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 ...
- 9,934
1 vote
0 answers
50 views
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 ...
- 630
0 votes
0 answers
173 views
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 ...
- 3,509
2 votes
1 answer
497 views
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 ...
- 9,934
10 votes
0 answers
289 views
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 ...
- 38.1k
0 votes
0 answers
180 views
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 ...
- 9,934
2 votes
1 answer
286 views
"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,...
- 53.5k
1 vote
1 answer
562 views
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 $\{0,1\}^\mathbb{Z}$ ...
- 53.5k
1 vote
0 answers
461 views
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 ...
- 9,934
1 vote
1 answer
179 views
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^{...
- 51
47 votes
3 answers
8k views
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 ...
- 1,717
7 votes
1 answer
394 views
Rewrite Game of Life as a convolution and a nonlinearity
If you implement Conway's Game of Life on a computer, it's convenient to define the neighbor-counting kernel, $$ k = \left[ \begin{array}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \...
user130609
1 vote
1 answer
302 views
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 ...
- 3,367
3 votes
1 answer
218 views
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 ...
- 630
1 vote
1 answer
255 views
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 ...
- 3,095
3 votes
0 answers
184 views
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 $...
- 3,095
3 votes
0 answers
148 views
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-...
user231922
0 votes
0 answers
183 views
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 ...
- 1
6 votes
0 answers
240 views
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 ...
- 5,752
6 votes
1 answer
277 views
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 ...
- 6,934
4 votes
2 answers
972 views
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. ...
- 203
7 votes
0 answers
1k views
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 ...
- 303
0 votes
1 answer
238 views
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 ...
- 137
1 vote
1 answer
197 views
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 ...
- 413
1 vote
0 answers
151 views
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 ...
- 111
4 votes
0 answers
144 views
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 ...
- 41
7 votes
2 answers
534 views
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 $\...
- 27.5k
22 votes
4 answers
3k views
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, \...
- 27.5k
8 votes
1 answer
525 views
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 that ...
- 27.5k
31 votes
1 answer
2k views
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$. ...
- 27.5k
6 votes
0 answers
150 views
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^{\...
5 votes
4 answers
702 views
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 ...
4 votes
2 answers
665 views
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 ...
- 2,440
7 votes
2 answers
599 views
Vice-versa Erdős conjecture
Erdős conjectured that, except $1, 4$ and $256$, no power of $2$ is a sum of distinct powers of $3$. A vice-versa conjecture may be: except $1$, $9$ and $81$, all powers of $3$ contains two ...
- 79
2 votes
0 answers
326 views
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 ...
- 7,600
3 votes
0 answers
122 views
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 ...
- 395