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On von Neumann regularity of cellular automata
Tekijät: Salo Ville
Kustantaja: SPRINGER
Julkaisuvuosi: 2023
Journal: Natural Computing
Tietokannassa oleva lehden nimi: NATURAL COMPUTING
Lehden akronyymi: NAT COMPUT
Sivujen määrä: 12
ISSN: 1567-7818
eISSN: 1572-9796
DOI: https://doi.org/10.1007/s11047-022-09935-w
Verkko-osoite: https://doi.org/10.1007/s11047-022-09935-w
Rinnakkaistallenteen osoite: https://research.utu.fi/converis/portal/detail/Publication/178723717
Tiivistelmä
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.
Ladattava julkaisu This is an electronic reprint of the original article. |  |
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