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17
A nonergodic probabilistic cellular automaton with a unique invariant measure
 Stoch. Process. Appl
, 2011
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Stochastic Flips on Twoletter Words
"... This paper introduces a simple Markov process inspired by the problem of quasicrystal growth. It acts over twoletter words by randomly performing flips, a local transformation which exchanges two consecutive different letters. More precisely, only the flips which do not increase the number of pairs ..."
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This paper introduces a simple Markov process inspired by the problem of quasicrystal growth. It acts over twoletter words by randomly performing flips, a local transformation which exchanges two consecutive different letters. More precisely, only the flips which do not increase the number of pairs of consecutive identical letters are allowed. Fixedpoints of such a process thus perfectly alternate different letters. We show that the expected number of flips to converge towards a fixedpoint is bounded by O(n 3) in the worstcase and by O(n 5 2 ln n) in the averagecase, where n denotes the length of the initial word.
Probabilistic cellular automata, invariant measures, and perfect sampling
"... In a probabilistic cellular automaton (PCA), the cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. A PCA can be viewed as a Markov chain whose ergodicity is investigated. A classical cellular automaton (CA) is a particular case of PCA. ..."
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In a probabilistic cellular automaton (PCA), the cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. A PCA can be viewed as a Markov chain whose ergodicity is investigated. A classical cellular automaton (CA) is a particular case of PCA. For a 1dimensional CA, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA.
Coalescing Cellular Automata
, 2006
"... Abstract. We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We pr ..."
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Abstract. We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We prove coalescence for two elementary rules and show that there exists infinitely many coalescing CA. We then conduct an experimental study on all elementary CA and show that some rules exhibit a phase transition, which belongs to the universality class of directed percolation. 1
Three Emergent Phenomena in the MultiTurmite System and their Robustness to Asynchrony
, 2010
"... The multiturmite system is composed of concurrent Turing machines acting on a twodimensional grid. The machines obey simple local rules, namely Langton’s ant rules, their updating is considered under various simulation conditions: synchronous or asynchronous updating methods and different conflict ..."
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The multiturmite system is composed of concurrent Turing machines acting on a twodimensional grid. The machines obey simple local rules, namely Langton’s ant rules, their updating is considered under various simulation conditions: synchronous or asynchronous updating methods and different conflict resolution policies. We present three emergent phenomena: clocks, gliders and deadlocks. We study to which extent these phenomena are robust to changes in the updating and in the conflict resolution policy. Regularities of behaviour are observed from simulation results. We describe these regularities within the mathematical framework of discrete dynamical systems and show how their robustness can be analysed from a local “microscopic ” view. keywords concurrent Turing machines turmites; Langton’s ant; discrete dynamical systems; asynchronous updating; robustness analysis 1
Stochastic Flips on Dimer Tilings
"... This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called flips, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling ..."
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This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called flips, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixedpoints of such a process play the role of quasicrystals. We are here interested in the worstcase expected number of flips to converge towards a fixedpoint. Numerical experiments suggest a Θ(n 2) bound, where n is the number of tiles of the tiling. We prove a O(n 2.5) upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the averagecase.
E.: Stochastic minority on graphs
, 2008
"... Abstract. Cellular automata have been mainly studied on very regular graphs carrying the vertices (like lines or grids) and under synchronous dynamics (all vertices update simultaneously). In this paper, we study how the asynchronism and the graph act upon the dynamics of the classical Minority rul ..."
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Abstract. Cellular automata have been mainly studied on very regular graphs carrying the vertices (like lines or grids) and under synchronous dynamics (all vertices update simultaneously). In this paper, we study how the asynchronism and the graph act upon the dynamics of the classical Minority rule. Minority has been wellstudied for synchronous updates and is thus a reasonable choice to begin with. Yet, beyond its apparent simplicity, this rule yields complex behaviors when asynchronism is introduced. We investigate the transitory part as well as the asymptotic behavior of the dynamics under full asynchronism (also called sequential: only one random vertex updates at each time step) for several types of graphs. Such a comparative study is a first step in understanding how the asynchronous dynamics is linked to the topology (the graph). Previous analyses on the grid [1,2] have observed that Minority seems to induce fast stabilization. We investigate here this property on arbitrary graphs using tools such as energy, particles and random walks. We show that the worst case convergence time is, in fact, strongly dependent on the topology. In particular, we observe that the case of trees is non trivial. 1
Stochastic Cellular Automata: Correlations, Decidability and Simulations
, 2013
"... This paper introduces a simple formalism for dealing with deterministic, nondeterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature ..."
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This paper introduces a simple formalism for dealing with deterministic, nondeterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance, number conserving stochastic cellular automata require these local probabilistic correlations). We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the nondeterministic and stochastic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal nondeterministic cellular automaton.
Probabilistic cellular automata, invariant measures, and perfect sampling
, 2012
"... A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of ..."
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A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a 1dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA. Last, we focus on the PCA Majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect
On Convergence Properties of OneDimensional Cellular Automata with Majority Cell Update Rule
"... Abstract — We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus ..."
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Abstract — We are interested in simple cellular automata (CA) and their computational and dynamical properties. In our past and ongoing work, we have been investigating (i) asymptotic dynamics of various types of CA and (ii) different communication models for CA. In this paper, we specifically focus on the convergence properties of a very simple kind of totalistic CA, namely, those defined on onedimensional arrays where each cell or node updates according to the Boolean Majority function: the new state of a cell becomes 1 if and only if a simple majority of its inputs are currently in state 1, and it becomes 0 otherwise. We have observed in our prior work that such CA tend to have relatively simple asymptotic dynamics: a short transient chain followed by convergence to a “fixed point”. We now provide solid statistical evidence for these conjectures, based on our recent extensive computer simulations of Majority 1D CA. In particular, we study the convergence properties of such CA for two communication models: one is the classical, parallel CA model with perfectly synchronous cell updates, and the other are CA whose cells update sequentially, one at a time; we consider two variants of such sequential update regimes. We simulate CA whose sizes range up to 1,000 cells, and demonstrate very fast (in particular, sublinear), and very slowly decreasing with an increase in the total number of cells, speeds of convergence. Finally, we draw conclusions based on our extensive simulations and outline some interesting questions to be considered in the future work.