A1 Refereed original research article in a scientific journal
On von Neumann regularity of cellular automata
Authors: Salo Ville
Publisher: SPRINGER
Publication year: 2023
Journal: Natural Computing
Journal name in source: NATURAL COMPUTING
Journal acronym: NAT COMPUT
Number of pages: 12
ISSN: 1567-7818
eISSN: 1572-9796
DOI: https://doi.org/10.1007/s11047-022-09935-w
Web address : https://doi.org/10.1007/s11047-022-09935-w
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/178723717
Abstract
We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Torma that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.
We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Torma that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.
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