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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

This paper proves the decidability of several important properties of additive cellular automata over finite abelian groups. First of all, we prove that equicontinuity and sensitivity to initial conditions are decidable for a nontrivial subclass of additive cellular automata, namely, the linear cellular automata over 𝕂ⁿ, where 𝕂 is the ring ℤ/mℤ. The proof of this last result has required to prove a general result on the powers of matrices over a commutative ring which is of interest in its own.
Then, we extend the decidability result concerning sensitivity and equicontinuity to the whole class of additive cellular automata over a finite abelian group and for such a class we also prove the decidability of topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to it. Finally, a decidable characterization of injectivity and surjectivity for additive cellular automata over a finite abelian group is provided in terms of injectivity and surjectivity of an associated linear cellular automata over 𝕂ⁿ.

Alberto Dennunzio, Enrico Formenti, Darij Grinberg, and Luciano Margara. From Linear to Additive Cellular Automata. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 125:1-125:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dennunzio_et_al:LIPIcs.ICALP.2020.125, author = {Dennunzio, Alberto and Formenti, Enrico and Grinberg, Darij and Margara, Luciano}, title = {{From Linear to Additive Cellular Automata}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {125:1--125:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.125}, URN = {urn:nbn:de:0030-drops-125321}, doi = {10.4230/LIPIcs.ICALP.2020.125}, annote = {Keywords: Cellular Automata, Decidability, Symbolic Dynamics} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e. , with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map.
As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case.

Alberto Dennunzio, Enrico Formenti, Darij Grinberg, and Luciano Margara. Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dennunzio_et_al:LIPIcs.MFCS.2019.68, author = {Dennunzio, Alberto and Formenti, Enrico and Grinberg, Darij and Margara, Luciano}, title = {{Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {68:1--68:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.68}, URN = {urn:nbn:de:0030-drops-110126}, doi = {10.4230/LIPIcs.MFCS.2019.68}, annote = {Keywords: Cellular Automata, Symbolic Dynamics, Complex Systems} }

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