1,498 research outputs found
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Forcing nonperiodicity with a single tile
An aperiodic prototile is a shape for which infinitely many copies can be
arranged to fill Euclidean space completely with no overlaps, but not in a
periodic pattern. Tiling theorists refer to such a prototile as an "einstein"
(a German pun on "one stone"). The possible existence of an einstein has been
pondered ever since Berger's discovery of large set of prototiles that in
combination can tile the plane only in a nonperiodic way. In this article we
review and clarify some features of a prototile we recently introduced that is
an einstein according to a reasonable definition. [This abstract does not
appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and
accepted for publication in the Mathematical Intelligencer and is scheduled
to appear in Vol 33. Citations to and quotations from this work should
reference that publication. If you cite this work, please check that the
published form contains precisely the material to which you intend to refe
Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness
In this paper, we study algorithmic problems for automaton semigroups and
automaton groups related to freeness and finiteness. In the course of this
study, we also exhibit some connections between the algebraic structure of
automaton (semi)groups and their dynamics on the boundary. First, we show that
it is undecidable to check whether the group generated by a given invertible
automaton has a positive relation, i.e. a relation p = 1 such that p only
contains positive generators. Besides its obvious relation to the freeness of
the group, the absence of positive relations has previously been studied and is
connected to the triviality of some stabilizers of the boundary. We show that
the emptiness of the set of positive relations is equivalent to the dynamical
property that all (directed positive) orbital graphs centered at non-singular
points are acyclic.
Gillibert showed that the finiteness problem for automaton semigroups is
undecidable. In the second part of the paper, we show that this undecidability
result also holds if the input is restricted to be bi-reversible and invertible
(but, in general, not complete). As an immediate consequence, we obtain that
the finiteness problem for automaton subsemigroups of semigroups generated by
invertible, yet partial automata, so called automaton-inverse semigroups, is
also undecidable.
Erratum: Contrary to a statement in a previous version of the paper, our
approach does not show that that the freeness problem for automaton semigroups
is undecidable. We discuss this in an erratum at the end of the paper
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