Abstract
In the setting of symbolic dynamics on discrete finitely generated infinite groups, we define a model of finite automata with multiple independent heads that walk on Cayley graphs, called group-walking automata, and use it to define subshifts. We characterize the torsion groups (also known as periodic groups) as those on which the group-walking automata are strictly weaker than Turing machines, and those on which the head hierarchy is infinite.
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Notes
In the earlier article (Delorme and Mazoyer 2002), essentially the same observation is made in a slightly different setting on the group \(\mathbb {Z}^2\).
This is a group because by the compactness of X, the inverse of a CA is continuous, and the inverse of a bijection commuting with a group action is easily seen to commute with the group action.
The term ‘recursively presented’ comes from the fact that one may assume \(\{ w_i \;|\; i \in \mathbb {N}\}\) to be a recursive set of words.
Turing machines can do this on any group, as they can modify a configuration and create a tape for themselves. Our finite-state automata fail in this task on torsion groups, by Theorem 6.
Note that here we use the fact that \(x_{1_G} \ne 0\). In general, we can only conclude \(f^{n+m}(x) = \sigma ^R_h(f^n(x))\) for some \(h \in B_G((q + 1)r)\).
The fact that G is torsion prevents us, in general, from defining a bijective version of \(\phi\). Also, the subshift \(Y'\) may be strictly smaller than Y.
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Acknowledgements
Research supported by Comisión Nacional de Investigación Científica y Tecnológica, FONDECYT Grant 3150552 (Ville Salo).
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Salo, V., Törmä, I. Independent finite automata on Cayley graphs. Nat Comput 16, 411–426 (2017). https://doi.org/10.1007/s11047-017-9613-6
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DOI: https://doi.org/10.1007/s11047-017-9613-6