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Natural Computing

, Volume 16, Issue 3, pp 367–381 | Cite as

Computing the periods of preimages in surjective cellular automata

  • Luca Mariot
  • Alberto Leporati
  • Alberto Dennunzio
  • Enrico Formenti
Article

Abstract

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.

Keywords

Cellular automata Surjectivity De Bruijn graph Bipermutivity Linear recurring sequences Linear feedback shift registers 

Mathematics Subject Classification

37B15 68Q80 94A55 

Notes

Acknowledgements

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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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi Milano-BicoccaMilanoItaly
  2. 2.Laboratoire I3SUniversité Nice-Sophia AntipolisSophia AntipolisFrance

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