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On the question of ontological states in simple (pre-)quantum models

The quantum-mechanical features of Hamiltonian cellular automata (CA) are described, i.e., of CA with integer-valued variables and couplings that follow linear updating rules. We then discuss whether, in this class of CA, there is room for systems that evolve by permutations of a set of ontological states, thus providing examples for G. ‘t Hooft’s recent CA Interpretation of Quantum Mechanics. Multi-partite systems consisting of interacting two-state components are promising in this respect and may lead to physically interesting models.