Four states are enough!

Ollinger, Nicolas; Richard, Gaëtan

doi:10.1016/j.tcs.2010.08.018

Cited by 24 publications

(12 citation statements)

References 17 publications

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“…Some of their first usages stand in the generation of prime numbers by Fischer [13] and Goto's solution to the Firing Squad Synchronization Problem (FSSP) [14]. Signals are commonly used: to solve the FSSP [20], to compute with rule 110 [4] or with only four states [18], to understand one CA [5] and so on. In [19] one important question is the automatic positioning of discrete signals.…”

Section: Exact Discretization Of 3-speed Rational Signal Machinesmentioning

confidence: 99%

Exact Discretization of 3-Speed Rational Signal Machines into Cellular Automata

Besson

Durand-Lose

2016

Cellular Automata and Discrete Complex Systems

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Section: Exact Discretization Of 3-speed Rational Signal Machinesmentioning

confidence: 99%

Exact Discretization of 3-Speed Rational Signal Machines into Cellular Automata

Besson

Durand-Lose

2016

Cellular Automata and Discrete Complex Systems

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“…Since then, these themes have received a considerable attention in the literature. Concerning universality, production of examples [1,18] was accompanied by progresses on the formalization and the theoretical analysis of the concept [17], in particular with the emergence of intrinsic simulations and universality [4,5]. Growing dynamics in cellular automata were also much studied, mostly through (classes of) examples with different points of view [11,12,7,2].…”

Section: Introductionmentioning

confidence: 99%

Universality in Freezing Cellular Automata

Becker¹,

Maldonado²,

Ollinger³

et al. 2018

Sailing Routes in the World of Computation

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Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.Context-sensitive simulation can get us over this hurdle as we show below; it is akin to the notion of conjugacy in symbolic dynamics [14].

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“…This research takes place in the quest for minimality in order to compute, and thus get unpredictable behavior. Let us cite [Rogozhin, 1996, Woods andNeary, 2007] for Turing machines, [Cook, 2004, Ollinger andRichard, 2011] for cellular automata, and [Margenstern, 2000] for a more general picture.…”

Section: Introductionmentioning

confidence: 99%