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 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 Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

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 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 Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

Linear Time Algorithm for Identifying the Invertibility of Null-Boundary Three Neighborhood Cellular Automata

2010 ◽  
Vol 19 (1) ◽  
pp. 89-113 ◽  
Author(s):  
Nirmalya S. Maiti ◽  
Soumyabrata Ghosh ◽  
Shiladitya Munshi ◽  
P. Pal Chaudhuri ◽  
Keyword(s):  
Cellular Automata ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

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