scholarly journals 60/102 Null Boundary Cellular Automata based expander graphs

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Sung-Jin Cho ◽  
Un-Sook Choi ◽  
Han-Doo Kim ◽  
Yoon-Hee Hwang ◽  
Jin-Gyoung Kim
Keyword(s):  
Cellular Automata ◽  
Communication Networks ◽  
Spectral Gap ◽  
Expander Graphs ◽  
International Audience

International audience Expander graphs are useful in the design and analysis of communication networks. Mukhopadhyay et al. introduced a method to generate a family of expander graphs based on nongroup two predecessor single attractor Cellular Automata(CA). In this paper we propose a method to generate a family of expander graphs based on 60/102 Null Boundary CA(NBCA) which is a group CA. The spectral gap generated by our method is maximal. Moreover, the spectral gap is larger than that of Mukhopadhyay et al.

scholarly journals The fractal structure of cellular automata on abelian groups

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Johannes Gütschow ◽  
Vincent Nesme ◽  
Reinhard F. Werner
Keyword(s):  
Cellular Automata ◽  
Fractal Structure ◽  
Cell Structure ◽  
Ring Structure ◽  
Entangled States ◽  
Universal Model ◽  
Basic Building Block ◽  
Quantum Cellular Automata ◽  
Sierpinski Triangle ◽  
International Audience

International audience It is a well-known fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$-Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurement-based models of quantum computing.

scholarly journals Number conserving cellular automata: new results on decidability and dynamics

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Bruno Durand ◽  
Enrico Formenti ◽  
Aristide Grange ◽  
Zsuzsanna Róka
Keyword(s):  
Cellular Automata ◽  
Linear Time ◽  
Number Conservation ◽  
International Audience

International audience This paper is a survey on our recent results about number conserving cellular automata. First, we prove the linear time decidability of the property of number conservation. The sequel focuses on dynamical evolutions of number conserving cellular automata.

scholarly journals On the analysis of ''simple'' 2D stochastic cellular automata

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Damien Regnault ◽  
Nicolas Schabanel ◽  
Eric Thierry
Keyword(s):  
Cellular Automata ◽  
Time Step ◽  
Energy Functions ◽  
Stochastic Cellular Automata ◽  
Von Neumann ◽  
Rich Variety ◽  
Moore Neighborhood ◽  
International Audience ◽  
Definition Of ◽  
New Phenomena

International audience Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata

scholarly journals Orbits of the Bernoulli measure in single-transition asynchronous cellular automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Henryk Fukś ◽  
Andrew Skelton
Keyword(s):  
Cellular Automata ◽  
Initial Density ◽  
Response Curves ◽  
Asynchronous Cellular Automata ◽  
Final Density ◽  
Short Cylinder ◽  
Bernoulli Measure ◽  
Single Transition ◽  
International Audience ◽  
Transition Rules

International audience We study iterations of the Bernoulli measure under nearest-neighbour asynchronous binary cellular automata (CA) with a single transition. For these CA, we show that a coarse-level description of the orbit of the Bernoulli measure can be obtained, that is, one can explicitly compute measures of short cylinder sets after arbitrary number of iterations of the CA. In particular, we give expressions for probabilities of ones for all three minimal single-transition rules, as well as expressions for probabilities of blocks of length 3 for some of them. These expressions can be interpreted as "response curves'', that is, curves describing the dependence of the final density of ones on the initial density of ones.

scholarly journals Larger than Life: Digital Creatures in a Family of Two-Dimensional Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Kellie M. Evans
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Nearest Neighbor ◽  
Two Dimensional ◽  
Game Of Life ◽  
Large Neighborhood ◽  
International Audience ◽  
Parameter Values

International audience We introduce the Larger than Life family of two-dimensional two-state cellular automata that generalize certain nearest neighbor outer totalistic cellular automaton rules to large neighborhoods. We describe linear and quadratic rescalings of John Conway's celebrated Game of Life to these large neighborhood cellular automaton rules and present corresponding generalizations of Life's famous gliders and spaceships. We show that, as is becoming well known for nearest neighbor cellular automaton rules, these ``digital creatures'' are ubiquitous for certain parameter values.

scholarly journals Representing Reversible Cellular Automata with Reversible Block Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Jérôme Durand-Lose
Keyword(s):  
System Theory ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Linear Time ◽  
Fixed Size ◽  
Mathematical System ◽  
Reversible Cellular Automaton ◽  
International Audience ◽  
Infinite Lattices ◽  
Reversible Cellular Automata

International audience Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus <i>(Physica D 45)</i> improved by Kari in 1996 <i>(Mathematical System Theory 29)</i>.

scholarly journals The Size of One-Way Cellular Automata

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Martin Kutrib ◽  
Jonas Lefèvre ◽  
Andreas Malcher
Keyword(s):  
Cellular Automata ◽  
Real Time ◽  
Finite Automata ◽  
Fixed Number ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Order Of Magnitude ◽  
International Audience ◽  
Number Of Cells

International audience We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.

scholarly journals Asynchronous Cellular Automata and Brownian Motion

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Philippe Chassaing ◽  
Lucas Gerin
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Cellular Automata ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Time Step ◽  
Interacting Particle ◽  
Asynchronous Cellular Automata ◽  
International Audience ◽  
A Cell

International audience This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.

scholarly journals Multidimensional cellular automata and generalization of Fekete's lemma

2008 ◽  
Vol Vol. 10 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Silvio Capobianco
Keyword(s):  
Growth Rate ◽  
Cellular Automata ◽  
New Variant ◽  
Number Sequences ◽  
Combinatorial Result ◽  
International Audience ◽  
Multidimensional Cellular Automata

Automata, Logic and Semantics International audience Fekete's lemma is a well-known combinatorial result on number sequences: we extend it to functions defined on d-tuples of integers. As an application of the new variant, we show that nonsurjective d-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index.

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