Abstract
The limit sets of cellular automata, defined by Wolfram, play an important role in applications of cellular automata. The emptiness problem for CA limit sets is whether the limit set of a given CA is empty. Our main concern is the decidability of this problem. Since, as we show in this paper, the limit set of any CA is non-empty, the emptiness problem for CA limit sets becomes whether the limit set of a given CA contains a non-quiescent configuration. We show that this problem is undecidable for n-dimensional CA where n ≥ 2. This problem is still open for 1-dimensional CA, i.e. linear CA. As a partial result, we show that when only the finite confifurations are concerned, this problem is undecidable even for linear CA.
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