10,991 research outputs found
Conjugacy of one-dimensional one-sided cellular automata is undecidable
Two cellular automata are strongly conjugate if there exists a
shift-commuting conjugacy between them. We prove that the following two sets of
pairs of one-dimensional one-sided cellular automata over a full shift
are recursively inseparable: (i) pairs where has strictly larger
topological entropy than , and (ii) pairs that are strongly conjugate and
have zero topological entropy.
Because there is no factor map from a lower entropy system to a higher
entropy one, and there is no embedding of a higher entropy system into a lower
entropy system, we also get as corollaries that the following decision problems
are undecidable: Given two one-dimensional one-sided cellular automata and
over a full shift: Are and conjugate? Is a factor of ? Is
a subsystem of ? All of these are undecidable in both strong and weak
variants (whether the homomorphism is required to commute with the shift or
not, respectively). It also immediately follows that these results hold for
one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201
Characterization for entropy of shifts of finite type on Cayley trees
The notion of tree-shifts constitutes an intermediate class in between
one-sided shift spaces and multidimensional ones. This paper proposes an
algorithm for computing of the entropy of a tree-shift of finite type.
Meanwhile, the entropy of a tree-shift of finite type is for some , where is a Perron number. This
extends Lind's work on one-dimensional shifts of finite type. As an
application, the entropy minimality problem is investigated, and we obtain the
necessary and sufficient condition for a tree-shift of finite type being
entropy minimal with some additional conditions
- …