Abstract
Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation of every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the idempotent behavior of these matrices. The significance of this property lies in the implementation of a matrix technique for detecting whether a one-dimensional cellular automaton is reversible or not. In particular, we present a procedure using the eigenvectors of these matrices to find the inverse rule of a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided.
Highlights
Cellular automata were invented by John von Neumann to prove the existence of self-reproducing systems [17]
In this paper we only study reversible one-dimensional cellular automata with neighborhood size 2 since all the other cases can be transformed to this one
A relevant tool in this sense is the presentation of every automaton by another of neighborhood size 2, this simulation yields that the connectivity matrices have a very suitable shape to analyze them
Summary
Cellular automata were invented by John von Neumann to prove the existence of self-reproducing systems [17]. The idea is to use a relevant result presented in the paper of Tim Boykett [2] which shows that any one-dimensional automaton can be transformed into another of neighborhood size 2 In this way it is just necessary to study this case to understand the rest. For the sequences in Kn−1, their ancestors belong to K2n−2 and the evolution rule defines a mapping φ : K2n−2 → Kn−1.Take a new set S of states with cardinality equal to kn−1. This mapping presents the same behavior that the evolution of the sequences in K2n−2, but τ is an evolution rule of neighborhood size 2 In this way the original automaton is simulated by another, the new automaton has a greater number of states than the original one. The value of each entry is the evolution of the neighborhood formed by the coordinates of the entry
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