Algorithmical and mathematical approaches of causal graph dynamics

Simon Martiel 1
1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe MC3
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : Cellular Automata constitute one of the most established model of discrete physical transformations that accounts for euclidean space. They implement three fundamental symmetries of physics: causality, homogeneity and finite density of information. Even though their origins lies in physics, they are widely used to model spatially distributed computation (self-replicating machines, synchronization problems,...), as well as a great variety of multi-agents phenomena (traffic jams, demographics,...). While being one of the most studied model of distributed computation, their rigidity forbids any trivial extension toward time-varying topology, which is a fundamental requirement when it comes to modelling phenomena in biology, sociology or physics: for instance when looking for a discrete formulation of general relativity. Causal graph dynamics generalize cellular automata to arbitrary, bounded degree, time-varying graphs. In this work, we generalize the fundamental structure results of cellular automata for this type of transformations. We endow our graphs with a compact metric space structure, and follow two approaches. An axiomatic approach based on the notions of continuity and shift-invariance, and a constructive approach, where a local rule is applied synchronously on every vertex of the graph. Compactness allows us to show the equivalence of these two definitions, extending the famous result of Curtis-Hedlund-Lyndon’s theorem. Another physics-inspired symmetry is then added to the model, namely reversibility.
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Simon Martiel. Algorithmical and mathematical approaches of causal graph dynamics. Other [cs.OH]. Université Nice Sophia Antipolis, 2015. English. ⟨NNT : 2015NICE4043⟩. ⟨tel-01188690⟩

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