Commutative Cellular Automata (original) (raw)
Related papers
1997
The algebraic conditions under which two one-dimensional cellular automata can commute is studied. It is shown that if either rule is permutive, that is, one-to-one in its leftmost and rightmost inputs, then the other rule can be written in terms of it; if either rule is a group, then the other is linear in it; and if either is permutive and affine, that is, linear up to a constant, then the other must also be affine. We also prove some simple results regarding the existence of identities, idempotents (quiescent states), and zeroes (absorbing states).
Algebraic Properties of Cellular Automata
Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.
Universality and cellular automata
Machines, Computations, and Universality, 2005
The classification of discrete dynamical systems that are computationally complete has recently drawn attention in light of Wolfram's Principle of Computational Equivalence. We discuss a classification for cellular automata that is based on computably enumerable ...
Conservation of some dynamical properties for operations on cellular automata
Theoretical Computer Science, 2009
We consider the family of all the Cellular Automata (CA) sharing the same local rule but having different memory. This family contains also all the CA with memory m ≤ 0 (one-sided CA) which can act both on A Z and on A N . We study several set theoretical and topological properties for these classes. In particular, we investigate if the properties of a given CA are preserved when considering the CA obtained by changing the memory of the original one (shifting operation). Furthermore, we focus our attention to the one-sided CA acting on A Z starting from the one-sided CA acting on A N and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity ⇒ Dense Periodic Orbits (DPO)] can be restated in several different (but equivalent) ways. Furthermore, we give some results on properties conserved under iteration of the CA global map.
Linear cellular automata and finite automata
Cellular Automata: A Parallel Model, Kluwer, http://www …, 1999
Moreover, automata theory motivates many algorithms that are of use in the study of cellular automata. Below we will present a simple algorithm that tests whether a cellular automata is reversible, m-to-1 or surjective. For reversible cellular automata, the algorithm also ...
Some Algebraic Properties Of Linear Synchronous Cellular Automata
arXiv: Cellular Automata and Lattice Gases, 2017
Relation between global transition function and local transition function of a homogeneous one dimensional cellular automaton (CA) is investigated for some standard transition functions. It could be shown that left shift and right shift CA are invertible. The final result of this paper states that the set of all left and right shift CA together with the identity CA on the same set forms an abelian group.
Una A note on conjugacy classes for multistate cellular automata
2007
espanolEl objetivo de estas notas es clasificar por clases de conjugacion la coleccion de r-automatas celulares generada por las permutaciones de r reglas locales fijadas. EnglishThe main goal of this note is to class i fy by conjugacy classes the collect i on of r-cellular automata generated by the permutations of r fixed local rules.
Cellular automata and intermediate degrees
Theoretical Computer Science, 2003
We study a classification of cellular automata based on the Turing degree of the orbits of the automaton. The difficulty of determining the membership of a cellular automaton in any one of these classes is characterized in the arithmetical hierarchy.
A Formulation of Composition for Cellular Automata on Groups
IEICE Transactions on Information and Systems, 2014
We introduce the notion of 'Composition', 'Union' and 'Division' of cellular automata on groups. A kind of notions of compositions was investigated by Sato [10] and Manzini [6] for linear cellular automata, we extend the notion to general cellular automata on groups and investigated their properties. We observe the all unions and compositions generated by one-dimensional 2-neighborhood cellular automata over Z 2 including non-linear cellular automata. Next we prove that the composition is right-distributive over union, but is not left-distributive. Finally, we conclude by showing reformulation of our definition of cellular automata on group which admit more than three states. We also show our formulation contains the representation using formal power series for linear cellular automata in Manzini [6].
Global properties of cellular automata
Journal of Statistical Physics, 1986
Cellular automata are discrete mathematical systems that generate diverse, often complicated, behavior using simple deterministic rules. Analysis of the local structure of these rules makes possible a description of the global properties of the associated automata. A class of cellular automata that generate infinitely many aperiodic temporal sequences is defined, as is the set of rules for which inverses exist. Necessary and sufficient conditions are derived characterizing the classes of "nearest-neighbor" rules for which arbitrary finite initial conditions (i) evolve to a homogeneous state; (ii) generate at least one constant temporal sequence.