Decidability of Sensitivity and Equicontinuity for Linear Higher-Order Cellular Automata

  • Alberto Dennunzio
  • Enrico Formenti
  • Luca ManzoniEmail author
  • Luciano Margara
  • Antonio E. Porreca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)


We study the dynamical behavior of linear higher-order cellular automata (HOCA) over \(\mathbb {Z}_m\). In standard cellular automata the global state of the system at time t only depends on the state at time \(t-1\), while in HOCA it is a function of the states at time \(t-1\), ..., \(t-n\), where \(n\ge 1\) is the memory size. In particular, we provide easy-to-check necessary and sufficient conditions for a linear HOCA over \(\mathbb {Z}_m\) of memory size n to be sensitive to the initial conditions or equicontinuous. Our characterizations of sensitivity and equicontinuity extend the ones shown in [23] for linear cellular automata (LCA) over \(\mathbb {Z}_m^n\) in the case \(n=1\). We also prove that linear HOCA over \(\mathbb {Z}_m\) of memory size n are indistinguishable from a subclass of LCA over \(\mathbb {Z}_m^n\). This enables to decide injectivity and surjectivity for linear HOCA over \(\mathbb {Z}_m\) of memory size n by means of the decidable characterizations of injectivity and surjectivity provided in [2] and [20] for LCA over \(\mathbb {Z}^n_m\).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alberto Dennunzio
    • 1
  • Enrico Formenti
    • 2
  • Luca Manzoni
    • 1
    Email author
  • Luciano Margara
    • 3
  • Antonio E. Porreca
    • 1
    • 4
  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-BicoccaMilanItaly
  2. 2.Universite Côte d’Azur, CNRS, I3SNice CedexFrance
  3. 3.Department of Computer Science and EngineeringUniversity of BolognaCesenaItaly
  4. 4.Aix Marseille Université, Université de Toulon, CNRS, LISMarseilleFrance

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