Linear Algebra Based Bounds for One-Dimensional Cellular Automata

  • Jarkko Kari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6808)


One possible complexity measure for a cellular automaton is the size of its neighborhood. If a cellular automaton is reversible with a small neighborhood, the inverse automaton may need a much larger neighborhood. Our interest is to find good upper bounds for the size of this inverse neighborhood. It turns out that a linear algebra approach provides better bounds than any known combinatorial methods. We also consider cellular automata that are not surjective. In this case there must exist so-called orphans, finite patterns without a pre-image. The length of the shortest orphan measures the degree of non-surjectiveness of the map. Again, a linear algebra approach provides better bounds on this length than known combinatorial methods. We also use linear algebra to bound the minimum lengths of any diamond and any word with a non-balanced number of pre-images. These both exist when the cellular automaton in question is not surjective. All our results deal with one-dimensional cellular automata. Undecidability results imply that in higher dimensional cases no computable upper bound exists for any of the considered quantities.


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jarkko Kari
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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