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Simulations intrinsèques et complexités dans les réseaux d’automates

Abstract : The objects at the center of this thesis are discrete dynamical systems, understood through finite automata networks, defined as vectors of local transition functions (so that one function is associated with one automaton). These automata networks can be tools for modeling natural interaction systems (biological networks, particle networks in physics...) and the phenomena they induce. They can also be seen as a model of computation that can be studied per se. These objects are approached through the prism of complexity and (theoretical) simulation, each of these concepts forming the core of a part of the document. The first part is devoted to the relations between the interaction graph of a network, that represents the influences between its automata, and the underlying dynamics. First, the focus is put on the fixed points problem that aims to understand the information that gives an interaction graph on the number of fixed points of a network, and more precisely on the algorithmic complexity of this counting. Then, our interest is focused on a somewhat opposite property, namely expansiveness. A network is expansive if one can predict its initial configuration by observing only one of its components for a long enough time. In other words, a network is expansive if its dynamics is unstable and if any initial local perturbation has visible repercussions on each automaton. We characterize the interaction graphs that admit an expansive network and study the possibility of a network to be expansive according to several parameters like the size of the alphabet, the "expansiveness time" or the type of the computed functions (linear, abelian...). The second part is devoted to intrinsic simulation, namely the capacity of a network to contain the behavioral complexity and computational richness of another network. Specifically, we are interested in a specific simulation based on the ability of one network to simulate step by step the dynamics of another network. One of the parameters that is emphasized is the update mode, which represents the time steps in which automata update their state. It is well known that the dynamics of a network strongly depends on update modes. A natural question in this context is to understand what kind of dynamics can be simulated with a given update mode. First, we highlight that a sequentially updated network is less "powerful" than a network updated in parallel. While any dynamics can be simulated by a network evolving in parallel, we show that to be simulated sequentially, it may require larger networks for which we give bounds on the size. Next, we present the characteristics of "complete" networks in the sense that they can simulate all networks of a given size by varying their update modes. Finally, we emphasize several networks of minimum size, or of minimal "simulation time", and study the relations between these two parameters.
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Contributor : Florian Bridoux <>
Submitted on : Sunday, January 12, 2020 - 7:05:24 PM
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Florian Bridoux. Simulations intrinsèques et complexités dans les réseaux d’automates. Complexité [cs.CC]. Aix-Marseile Université, 2019. Français. ⟨tel-02436228⟩

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