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The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most of the cases. In this article, we introduce…
In a recent paper Sutner proved that the first-order theory of the phase-space $\mathcal{S}_\mathcal{A}=(Q^\mathbb{Z}, \longrightarrow)$ of a one-dimensional cellular automaton $\mathcal{A}$ whose configurations are elements of…
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…
Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a…
In this paper, under certain conditions we consider two-dimensional cellular automata with the Moore neighborhood. Namely, the characterization of 2D linear cellular automata defined by the Moore neighborhood with some mixed boundary…
One-to-one reversible automata are introduced. Their applicability to a modelling of the quantum mechanical measurement process is discussed.
Gauge-invariance is a fundamental concept in Physics -- known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of…
We present a preliminary study of a new class of two-input cellular automata called eventually number-conserving cellular automata characterized by the property of evolving after a finite number of time steps to states whose number of…
When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau…
Complexity has been a recurrent research topic in cellular automata because they represent systems where complex behaviors emerge from simple local interactions. A significant amount of previous research has been conducted proposing…
Computing the configuration of any one-dimensional cellular automaton at generation $n$ can be accelerated by constructing and running a composite rule with a radius proportional to $\log n$. The new automaton is the original one, but with…
A necessary and sufficient condition for a one-dimensional q-state n-input cellular automaton rule to be number-conserving is established. Two different forms of simpler and more visual representations of these rules are given, and their…
This paper investigates reversibility properties of 1-dimensional 3-neighborhood d-state finite cellular automata (CAs) of length n under periodic boundary condition. A tool named reachability tree has been developed from de Bruijn graph…
This article introduces new tools to study self-organisation in a family of simple cellular automata which contain some particle-like objects with good collision properties (coalescence) in their time evolution. We draw an initial…
A cellular automaton named Rule 184++C is proposed as a meta-model to investigate the flow of various complex particles. In this model, unlike the granular pipe flow and the traffic flow, not only the free-jam phase transition but also the…
Strongly correlated electron systems require the development of new theoretical schemes in order to describe their unusual and unexpected properties. The usual perturbation schemes are inadequate and new concepts must be introduced. In our…
Cellular automata, CA for short are continuous maps defined on the set of configurations over a finite alphabet A that commutes with the shift. They are characterized by the existence of local function which determine by local behavior the…
We provide algebraic criteria for the unitarity of linear quantum cellular automata, i.e. one dimensional quantum cellular automata. We derive these both by direct combinatorial arguments, and by adding constraints into the model which do…
An elementary cellular automaton with memory is a chain of finite state machines (cells) updating their state simultaneously and by the same rule. Each cell updates its current state depending on current states of its immediate neighbours…
In this work, the one-dimensional Cellular Automaton is extended to one that involves two sets of symbols and two global rules. As a main result, the Extended Curtis-Hedlund-Lyndon Theorem is demonstrated. Such constructions can be useful…