3 research outputs found
Turing degrees of limit sets of cellular automata
Cellular automata are discrete dynamical systems and a model of computation.
The limit set of a cellular automaton consists of the configurations having an
infinite sequence of preimages. It is well known that these always contain a
computable point and that any non-trivial property on them is undecidable. We
go one step further in this article by giving a full characterization of the
sets of Turing degrees of cellular automata: they are the same as the sets of
Turing degrees of effectively closed sets containing a computable point
Complexity of Generic Limit Sets of Cellular Automata
The generic limit set of a topological dynamical system of the smallest
closed subset of the phase space that has a comeager realm of attraction. It
intuitively captures the asymptotic dynamics of almost all initial conditions.
It was defined by Milnor and studied in the context of cellular automata, whose
generic limit sets are subshifts, by Djenaoui and Guillon. In this article we
study the structural and computational restrictions that apply to generic limit
sets of cellular automata. As our main result, we show that the language of a
generic limit set can be at most -hard, and lower in various
special cases. We also prove a structural restriction on generic limit sets
with a global period.Comment: 13 pages, 2 figure