Limit Sets Of Cellular Automata Research Articles

Arithmetical complexity of the language of generic limit sets of cellular automata

The generic limit set of a dynamical system is the smallest set that attracts most of the space in a topological sense: it is the smallest closed set with a comeager basin of attraction. Introduced by Milnor, it has been studied in the context of one-dimensional cellular automata by Djenaoui and Guillon, Delacourt, and Törmä. In this article we present complexity bounds on realizations of generic limit sets of cellular automata with prescribed properties. We show that generic limit sets have a Π20 language if they are inclusion-minimal, a Σ10 language if the cellular automaton has equicontinuous points, and that these bounds are tight. We also prove that many chain mixing Π20 subshifts and all chain mixing Δ20 subshifts are realizable as generic limit sets. As a corollary, we characterize the minimal subshifts that occur as generic limit sets.

Limit Sets of Cellular Automata Associated to Probability Measures

We introduce the concept of limit set associated to a cellular automaton (CA) and a shift invariant probability measure. This is a subshift whose forbidden blocks are exactly those, whose probabilities tend to zero as time tends to infinity. We compare this probabilistic concept of limit set with the concepts of attractors, both in topological and measure-theoretic sense. We also compare this notion with that of topological limit set in different dynamical situations.

On the sofic limit sets of cellular automata

AbstractIt is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

Rice's theorem for the limit sets of cellular automata

Rice's theorem is a well-known result in the theory of recursive functions. A corresponding theorem for cellular automata limit sets is proved: All nontrivial properties of limit sets of cellular automata (CAs) are shown undecidable. The theorem remains valid even if only one-dimensional CAs are considered.

On the Limit Sets of Cellular Automata

The limit sets of cellular automata, defined by Wolfram, play an important role in applications of cellular automata to complex systems. A number of results on limit sets are proved, considering both finite and infinite configurations of cellular automata. The main concern of this paper is with testing membership and (essential) emptiness of limit sets for linear and two-dimensional cellular automata.

The emptiness problem for CA limit sets

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.