Computing the periods of preimages in surjective cellular automata
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A basic property of one-dimensional surjective cellular automata (CA) is that any preimage of a spatially periodic configuration (SPC) is spatially periodic as well. This paper investigates the relationship between the periods of SPC and the periods of their preimages for various classes of CA. When the CA is only surjective and y is a SPC of least period p, the least periods of all preimages of y are multiples of p. By leveraging on the De Bruijn graph representation of CA, we devise a general algorithm to compute the least periods appearing in the preimages of a SPC, along with their corresponding multiplicities (i.e. how many preimages have a particular least period). Next, we consider the case of linear and bipermutive cellular automata (LBCA) defined over a finite field as state alphabet. In particular, we show an equivalence between preimages of LBCA and concatenated linear recurring sequences (LRS) that allows us to give a complete characterization of their periods. Finally, we generalize these results to LBCA defined over a finite ring as alphabet.
KeywordsCellular automata Surjectivity De Bruijn graph Bipermutivity Linear recurring sequences Linear feedback shift registers
Mathematics Subject Classification37B15 68Q80 94A55
The authors wish to thank Ilkka Törma for suggesting that Lemma 4 holds in the general surjective case, and Marco Previtali for insightful comments about the computational complexity of the u-closure graph building procedure. Further, the authors are grateful to the anonymous reviewers for their helpful comments on how to improve the paper.
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