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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.

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Cellular automata are discrete dynamical systems based on simple local interactions among its components, but sometimes they are able to yield quite a complex global behavior. A special kind of cellular automaton is the one where the global behavior is invertible, this type of cellular automaton is called reversible. In this paper we expose the graph representation provided by de Bruijn diagrams of reversible one-dimensional cellular automata and we define the distinct types of paths between self-loops in such diagrams. With this, we establish the way in which a reversible one-dimensional cellular automaton generates sequences composed by subsequences produced by the undefined repetition of a single state. Using this graph presentation, we define Welch diagrams which will be useful for proving that all the extensions of the ancestors in reversible one-dimensional cellular automata are equivalent to the full shift. In this way an important result of this paper is that we understand and classify the behavior of a reversible automaton analyzing the extensions of the ancestors of a given sequence by means of symbolic dynamics tools. A final example illustrates the results exposed in the paper.