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Elementary, finite and linear vN-regular cellular automata.

Castillo-Ramirez, Alonso and Gadouleau, Maximilien (2020) 'Elementary, finite and linear vN-regular cellular automata.', Information and computation., 274 . p. 104533.


Let G be a group and A a set. A cellular automaton (CA) over AG is von Neumann regular (vN-regular) if there exists a CA over AG such that = , and in such case, is called a weak generalised inverse of . In this paper, we investigate vN-regularity of various kinds of CA. First, we establish that, over any nontrivial conguration space, there always exist CA that are not vN-regular. Then, we obtain a partial classication of elementary vN-regular CA over f0; 1gZ; in particular, we show that rules like 128 and 254 are vN-regular (and actually generalised inverses of each other), while others, like the well-known rules 90 and 110, are not vN-regular. Next, when A and G are both nite, we obtain a full characterisation of vN-regular CA over AG. Finally, we study vN-regular linear CA when A = V is a vector space over a eld F; we show that every vN-regular linear CA is invertible when V = F and G is torsion-free elementary amenable (e.g. when G = Zd; d 2 N), and that every linear CA is vN-regular when V is nite-dimensional and G is locally nite with char(F) - o(g) for all g 2 G.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2020 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:No date available
Date deposited:03 March 2020
Date of first online publication:02 March 2020
Date first made open access:02 March 2021

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