Solutions to All-Colors Problem on Graph Cellular Automata

Complexity ◽

10.1155/2019/3164692 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Xiaoyan Zhang ◽

Chao Wang

Keyword(s):

Number Theory ◽

Dynamical Systems ◽

Cellular Automata ◽

Automata Theory ◽

Combinatorial Number Theory ◽

Linear Decoding

The All-Ones Problem comes from the theory of σ+-automata, which is related to graph dynamical systems as well as the Odd Set Problem in linear decoding. In this paper, we further study and compute the solutions to the “All-Colors Problem,” a generalization of “All-Ones Problem,” on some interesting classes of graphs which can be divided into two subproblems: Strong-All-Colors Problem and Weak-All-Colors Problem, respectively. We also introduce a new kind of All-Colors Problem, k-Random Weak-All-Colors Problem, which is relevant to both combinatorial number theory and cellular automata theory.

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Allowed patterns of β -shifts

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2911 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Sergi Elizalde

Keyword(s):

Number Theory ◽

Dynamical Systems ◽

Real Number ◽

Fractional Part ◽

Automata Theory ◽

Relative Order ◽

One Dimensional ◽

Chaotic Dynamical Systems ◽

International Audience ◽

Nombre Réel

International audience For a real number $β >1$, we say that a permutation $π$ of length $n$ is allowed (or realized) by the $β$-shift if there is some $x∈[0,1]$ such that the relative order of the sequence $x,f(x),\ldots,f^n-1(x)$, where $f(x)$ is the fractional part of $βx$, is the same as that of the entries of $π$ . Widely studied from such diverse fields as number theory and automata theory, $β$-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When $β$ is an integer, permutations realized by shifts have been recently characterized. In this paper we generalize some of the results to arbitrary $β$-shifts. We describe a method to compute, for any given permutation $π$ , the smallest $β$ such that $π$ is realized by the $β$-shift. Pour un nombre réel $β >1$, on dit qu'une permutation $π$ de longueur $n$ est permise (ou réalisée) par $β$-shift s'il existe $x∈[0,1]$ tel que l'ordre relatif de la séquence $x,f(x),\ldots,f^n-1(x)$, où $f(x)$ est la partie fractionnaire de $βx$, soit le même que celui des entrées de $π$ . Largement étudiés dans des domaines aussi divers que la théorie des nombres et la théorie des automates, les $β$-shifts sont des prototypes de systèmes dynamiques chaotiques unidimensionnels. Quand $β$ est un nombre entier, les permutations réalisées par décalages ont été récemment caractérisées. Dans cet article, nous généralisons certains des résultats au cas de $β$-shifts arbitraires. Nous décrivons une méthode pour calculer, pour toute permutation donnée $π$ , le plus petit $β$ tel que $π$ soit réalisée par $β$-shift.

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Additive problems in combinatorial number theory

Number Theory - Lecture Notes in Mathematics ◽

10.1007/bfb0083574 ◽

1989 ◽

pp. 123-139

Author(s):

Melvyn B. Nathanson

Keyword(s):

Number Theory ◽

Combinatorial Number Theory

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A STUDY OF BIFURCATIONS IN A CIRCULAR REAL CELLULAR AUTOMATON

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000234 ◽

1993 ◽

Vol 03 (02) ◽

pp. 293-321 ◽

Cited By ~ 1

Author(s):

JÜRGEN WEITKÄMPER

Keyword(s):

Hopf Bifurcation ◽

Dynamical Systems ◽

Cellular Automata ◽

Cellular Automaton ◽

Parallel Computers ◽

Discrete Dynamical Systems ◽

Two Dimensional ◽

Bifurcation Structure ◽

Logistic Mapping ◽

Henon Mapping

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

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HIGHER ORDER CELLULAR AUTOMATA

Advances in Complex Systems ◽

10.1142/s0219525905000403 ◽

2005 ◽

Vol 08 (02n03) ◽

pp. 169-192 ◽

Cited By ~ 11

Author(s):

NILS A. BAAS ◽

TORBJØRN HELVIK

Keyword(s):

Dynamical Systems ◽

Cellular Automata ◽

Level Structure ◽

Higher Order ◽

New Concepts ◽

Multi Level

We introduce a class of dynamical systems called Higher Order Cellular Automata (HOCA). These are based on ordinary CA, but have a hierarchical, or multi-level, structure and/or dynamics. We present a detailed formalism for HOCA and illustrate the concepts through four examples. Throughout the article we emphasize the principles and ideas behind the construction of HOCA, such that these easily can be applied to other types of dynamical systems. The article also presents new concepts and ideas for describing and studying hierarchial dynamics in general.

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Almost all $S$-integer dynamical systems have many periodic points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798113378 ◽

1998 ◽

Vol 18 (2) ◽

pp. 471-486 ◽

Cited By ~ 4

Author(s):

T. B. WARD

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Cellular Automata ◽

Zeta Function ◽

Periodic Points ◽

Radius Of Convergence ◽

Dynamical Zeta Function ◽

Hyperbolic Dynamical Systems ◽

Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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