Section: New Results

Cellular automata as a model of computation

Participants : Nazim Fatès, Irène Marcovici.

The reversibility of classical cellular automata (CA) was examined for the case where the updates of the system are random. In this context, with B. Sethi and S. Das (India), we studied a particular form of reversibility: the possibility of returning infinitely often to the initial condition after a random number of time step,s this is the recurrence property of the system. We analysed this property for the simple rules and described the communication graph of the system [33].

We studied how to coordinate a team of agents to locate a hidden source on a two-dimensional discrete grid. The challenge is to find the position of the source with only sporadic detections. This problem arises in various situations, for instance when insects emit pheromones to attract their partners. A search mechanism named infotaxis was proposed to explain how agents may progressively approach the source by using only intermittent detections. With Q. Ladeveze, an intern, we re-examined in detail the properties of our bio-inspired algorithm that relies on the Reaction–Diffusion–Chemotaxis aggregation scheme to group agents that have limited abilities [38].

To study the robustness of asynchronous CA, we examined the coalescence phenomenon, which consists in observing the cases where two different initial conditions with the same sequence of updates quickly evolve to the same non-trivial configuration. With J. Francès de Mas, an intern, we studied the rules which always coalesce and those which exhibit a phase transition between a coalescing and non-coalescing behaviour. We proposed some formal explanations of non-trivial rapid coalescence giving lower bounds for the coalescence time of ECA 154 and ECA 62, and some first steps towards finding their upper bounds in order to prove that they have, respectively, quadratic and linear coalescence time [34].

We studied random mixtures of two deterministic Elementary Cellular Automata. There are 8088 such rules, called, diploid cellular automata. We used numerical simulations to perform some steps in the exploration of this space. As the mathematical analysis of such systems is a difficult task, we used numerical simulations to get insights into the dynamics of this class of stochastic cellular automata. We examined phase transitions and various types of symmetry breaking [17].