Related papers: Cellular automata between sofic tree shifts
We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be…
Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective…
We study the density of periodic configurations for shift spaces defined on (the Cayley graph of) a finitely generated group. We prove that in the case of a full shift on a residually finite group and in that of a group shift space on an…
This paper presents an application of the Infinite Unit Axiom, introduced by Yaroslav Sergeyev, (see [11] - [14]) to the development of one-dimensional cellular automata. This application allows the establishment of a new and more precise…
We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated…
We reveal an algorithm for determining the complete prefix code irreducibility (CPC-irreducibility) of dyadic trees labeled by a finite alphabet. By introducing an extended directed graph representation of tree shift of finite type (TSFT),…
A one-dimensional cellular automaton $\tau : A^\mathbb{Z} \to A^\mathbb{Z}$ is a transformation of the full shift defined via a finite neighborhood $S \subset \mathbb{Z}$ and a local function $\mu : A^S \to A$. We study the family of…
Cellular automata (CA) are dynamical systems defined by a finite local rule but they are studied for their global dynamics. They can exhibit a wide range of complex behaviours and a celebrated result is the existence of (intrinsically)…
Cellular automata are a fundamental computational model with applications in mathematics, computer science, and physics. In this work, we explore the study of cellular automata to cases where the universe is a group, introducing the concept…
Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=\sigma(t)\sum_{v \preceq t}\alpha(v)\xi_v $$ for given weight functions $\alpha$ and $\sigma$ on $T$…
In this paper, we study linear cellular automata (CAs) on Cayley tree of order 2 over the field $\mathbb F_p$ (the set of prime numbers modulo $p$). We construct the rule matrix corresponding to finite cellular automata on Cayley tree.…
We consider continuous, translation-commuting transformations of compact, translation-invariant families of mappingsfrom finitely generated groups into finite alphabets. It is well-known that such transformations and spaces can be described…
We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in…
This paper concerns $\mu$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $\mu$-random configuration. More precisely, we investigate the…
The determinisation problem for min-plus (tropical) weighted automata was recently shown to be decidable. However, the proof is purely existential, relying on several non-constructive arguments. Our contribution in this work is twofold:…
We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting. In this article, we relax the notion of accepting run by allowing a certain quantity…
Exactly ergodicity in boundary-driven semi-infinite cellular automata (CA) are investigated. We establish all the ergodic rules in CA with 3, 4, and 5 states. We analytically prove the ergodicity for 12 rules in 3-state CA and 118320 rules…
Criteria for subnormality of unbounded injective weighted shifts on leafless directed trees with one branching vertex are proposed. The case of classical weighted shifts is discussed. The relevance of an inductive limit approach to…
In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with algebraic local rules and such that $\mathfrak{G}$ is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular…
Many dynamical systems can be naturally represented as `Bratteli-Vershik' (or `adic') systems, which provide an appealing combinatorial description of their dynamics. If an adic system X satisfies two technical conditions (`focus' and…