14 research outputs found
Response Curves and Preimage Sequences of Two-Dimensional Cellular Automata
We consider the problem of finding response curves for a class of binary
two-dimensional cellular automata with -shaped neighbourhood. We show that
the dependence of the density of ones after an arbitrary number of iterations,
on the initial density of ones, can be calculated for a fairly large number of
rules by considering preimage sets. We provide several examples and a summary
of all known results. We consider a special case of initial density equal to
0.5 for other rules and compute explicitly the density of ones after
iterations of the rule. This analysis includes surjective rules, which in the
case of -shaped neighbourhood are all found to be permutive. We conclude
with the observation that all rules for which preimage curves can be computed
explicitly are either finite or asymptotic emulators of identity or shift.Comment: 7 pages, 3 figure
Intrinsic Simulations between Stochastic Cellular Automata
The paper proposes a simple formalism for dealing with deterministic,
non-deterministic and stochastic cellular automata in a unifying and composable
manner. Armed with this formalism, we extend the notion of intrinsic simulation
between deterministic cellular automata, to the non-deterministic and
stochastic settings. We then provide explicit tools to prove or disprove the
existence of such a simulation between two stochastic cellular automata, even
though the intrinsic simulation relation is shown to be undecidable in
dimension two and higher. The key result behind this is the caracterization of
equality of stochastic global maps by the existence of a coupling between the
random sources. We then prove that there is a universal non-deterministic
cellular automaton, but no universal stochastic cellular automaton. Yet we
provide stochastic cellular automata achieving optimal partial universality.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Multidimensional cellular automata and generalization of Fekete's lemma
Automata, Logic and Semantic
A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications
Cellular automata (CAs) are dynamical systems which exhibit complex global
behavior from simple local interaction and computation. Since the inception of
cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention
of several researchers over various backgrounds and fields for modelling
different physical, natural as well as real-life phenomena. Classically, CAs
are uniform. However, non-uniformity has also been introduced in update
pattern, lattice structure, neighborhood dependency and local rule. In this
survey, we tour to the various types of CAs introduced till date, the different
characterization tools, the global behaviors of CAs, like universality,
reversibility, dynamics etc. Special attention is given to non-uniformity in
CAs and especially to non-uniform elementary CAs, which have been very useful
in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Valeurs propres des automates cellulaires
On s'intéresse dans ce travail aux automates cellulaires unidimensionnels qui ont été largement étudiés mais où il reste beaucoup à faire. La théorie spectrale des automates cellulaires a notamment été peu abordée à l'exception de quelques résultats indirects. On cherche a mieux comprendre les cadres topologiques et ergodiques en étudiant l'existence de valeurs propres en particulier celles irrationnelles c'est à dire de la forme e^{2Ipa} où a est un irrationnel et I la racine carrée de l'unité. Cette question ne semble pas avoir été abordée jusqu'à présent. Dans le cadre topologique les résultats sur l'équicontinuité de K urka et Blanchard et Tisseur permettent de déduire directement que tout automate cellulaire équicontinu possède des valeurs propres topologiques rationnelles. La densité des points périodiques pour le décalage empêche l'existence de valeurs propres topologiques irrationnelles. La densité des points périodiques pour l'automate cellulaire semble être liée à la question des valeurs propres. Dans le cadre topologique, si l'automate cellulaire possède des points d'équicontinuité sans être équicontinu, la densité des points périodiques a comme conséquence le fait que le spectre représente l'ensemble des racines rationnelles de l'unité c'est à dire tous les nombres de la forme e^{2Ipa} avec a Q .Dans le cadre mesuré, la question devient plus difficile, on s'intéresse à la dynamique des automates cellulaires surjectifs pour lesquels la mesure uniforme est invariante en vertu du théorème de Hedlund. La plupart des résultats obtenus demeurent valable dans un cadre plus large. Nous commençons par montrer que les automates cellulaires ayant des points d'équicontinuité ne possèdent pas de valeurs propres mesurables irrationnelles. Ce résultat se généralise aux automates cellulaires possédant des points -équicontinu selon la définition de Gilman. Nous démontrons finalement que les automates cellulaires possédant des points -équicontinu selon la définition de Gilman possèdent des valeurs propres rationnellesWe investigate properties of one-dimensional cellular automata. This category of cellular automata has been widely studied but many questions are still open. Among them the spectral theory of unidimensional cellular automata is an open field with few indirect results. We want a better understanding of both ergodic and topological aspect by investigating the existence of eigenvalues of cellular automata, in particular irrational ones, i.e., those of the form e^{2Ipa} where a is irrationnal and I the complex root of -1. The last question seems not to have been studied yet.In the topological field the results of K urka & Blanchard and Tisseur about equicontinuous cellular automata have as direct consequence that any equicontinuous CA has rational eigenvalues. Density of shift periodic points leads to the impossibility for CA to have topological irrational eigenvalues. The density of periodic points of cellular automata seems to be related with the question of eignevalues. If the CA has equicontinuity points without being equicontinuous, the density of periodic points implies the fact that the spectrum contains all rational roots of the unity, i.e., all numbers of the form e^{2Ipa} with a Q .In the measurable field the question becomes harder. We assume that the cellular automaton is surjective, which implies that the uniform measure is invariant. Most results are still available in more general conditions. We first prove that cellular automata with equicontinuity points never have irrational measurable eigenvalues. This result is then generalized to cellular automata with -equicontinuous points according to Gilman's classification. We also prove that cellular automata with -equicontinuous points have rational eigenvaluesPARIS-EST-Université (770839901) / SudocSudocFranceF
Response curves of deterministic and probabilistic cellular automata in one and two dimensions
One of the most important problems in the theory of cellular automata (CA) is
determining the proportion of cells in a specific state after a given number of time
iterations. We approach this problem using patterns in preimage sets - that is, the
set of blocks which iterate to the desired output. This allows us to construct a
response curve - a relationship between the proportion of cells in state 1 after niterations
as a function of the initial proportion. We derive response curve formulae
for many two-dimensional deterministic CA rules with L-neighbourhood. For all
remaining rules, we find experimental response curves. We also use preimage sets to
classify surjective rules. In the last part of the thesis, we consider a special class of
one-dimensional probabilistic CA rules. We find response surface formula for these
rules and experimental response surfaces for all remaining rules