134 problems

Odd-index period formula conjecture. The period function satisfies

Even-index period formula conjecture. The period function satisfies

Positive rates conjecture. Every one-dimensional PCA with strictly positive transition rates is ergodic.

Let $G$ be a group and let $A$ be a finite set. Let $\tau\colon A^G\to A^G$ be a cellular automaton. It is post-surjective if, for every $x,y\in A^G$ such that $y$ is asymptotic to…

A cellular automaton is a shift-commuting endomorphism of a one-dimensional full shift. Two cellular automata are conjugate if there is a shift-commuting homeomorphism intertwining…

Let $S$ be a finite set, let $d>1$, and let $X=S^{\mathbb{Z}^d}$. A $d$-dimensional cellular automaton is a continuous map $T:X\to X$ commuting with the shift action of…

Cesàro-limit conjecture. The limit in Cesàro mean of the sequence

Low-growth existence conjecture. There exists $N>1$ and a surjective cellular automaton $f$ on $N$ symbols such that $\nu(f,S_N)<N$. The surrounding discussion poses related quanti…

Let $\Sigma_N$ be the space of doubly infinite sequences over an alphabet of $N$ symbols, and let $f:\Sigma_N\to\Sigma_N$ be a surjective one-dimensional cellular automaton. Let…

Let $m=(m^{(a)}_j)$ be a content such that … where … Let $\Omega(m)$ be the Bethe-ansatz multiplicity, let $W$ be the Weyl group of $A_n$, and let $\lambda(m)$ be the associated do…

Let $p\in P(m)$ be a state with soliton content $m$, and let $(T^{(r)}_l)^t(p)\neq0$ for every $t\geq0$. Denote by ${\mathcal P}^{(r)}_l$ the least common multiple defined earlier…

Let $B=B^{r_1,l_1}\otimes\cdots\otimes B^{r_L,l_L}$ be the tensor product of affine crystals for the $A^{(1)}_n$ cellular automaton, let $p\in B$ be a state, and let $T^{(a)}_k$ an…

Twin-period recurrence conjecture. Assume that $p$ and $p+3$ are twin periods. Then the next twin pair is $p'$ and $p'+3$, where

A reversible cellular automaton model has translationally invariant conserved charges, and one charge is obtainable from another through the boost operator. Boost-charge criterion.…

Let $G$ be a group universe, let $A$ be a finite alphabet, and let $\tau\colon A^G\to A^G$ be a cellular automaton. The cellular automaton $\tau$ is post-surjective if, whenever…

A nested glider is a positive nested glider or a negative nested glider: a positive nested $n$-stranded glider has positive speed and satisfies that $g_k^l$ is a glider for every…

Surjectivity characterization conjecture. $F$ is surjective if and only if either

Let $F$ be a surjective binary cellular automaton with radius $r$, and let $N^1_1(F,r,k)$ denote the corresponding symbol-frequency quantity; write $\mathrm{id}$ for the identity c…

A monotone interacting particle system applies a shrinker with rate one and the zero map with a small positive rate. Stability conjecture. Such a system should be stable against th…

Let CGL denote the paper’s coincidence-graph learning approach, applied to larger and more randomized graphs, and let the system’s recursive range measure how far graph interaction…

The model uses de-inforcement, meaning that plasticity reduces rather than amplifies signals in its internal graph representation. De-inforcement conjecture. The de-inforcement fun…

Unique-active-transition idempotency conjecture. The automaton $\tau$ is not idempotent if and only if there exists $t\in S$ such that

Completeness conjecture. There are exactly 120 elementary equivalence classes of proper liftings of diameter 6 and degree at least 2.

Let $\mathbb{G}$ be a finitely generated group, let $S oindent i mathbb{G}$ be a finite generating set, and let $operatorname{Cay}(mathbb{G},S)$ be its Cayley graph. A configuratio…

Let $\mathbb{G}$ be a group with non-trivial percolation threshold, and let $S$ be a finite generating set. The site-percolated additive cellular automaton has alphabet…