353 research outputs found
On injective endomorphisms of symbolic schemes
Building on the seminal work of Gromov on endomorphisms of symbolic algebraic
varieties [10], we introduce a notion of cellular automata over schemes which
generalize affine algebraic cellular automata in [7]. We extend known results
to this more general setting. We also establish several new ones regarding the
closed image property, surjunctivity, reversibility, and invertibility for
cellular automata over algebraic varieties with coefficients in an
algebraically closed field. As a byproduct, we obtain a negative answer to a
question raised in [7] on the existence of a bijective complex affine algebraic
cellular automaton whose inverse
is not algebraic
An automaton-theoretic approach to the representation theory of quantum algebras
We develop a new approach to the representation theory of quantum algebras
supporting a torus action via methods from the theory of finite-state automata
and algebraic combinatorics. We show that for a fixed number , the
torus-invariant primitive ideals in quantum matrices can be seen as
a regular language in a natural way. Using this description and a semigroup
approach to the set of Cauchon diagrams, a combinatorial object that
paramaterizes the primes that are torus-invariant, we show that for fixed,
the number of torus-invariant primitive ideals in quantum matrices
satisfies a linear recurrence in over the rational numbers. In the case we give a concrete description of the torus-invariant primitive ideals
and use this description to give an explicit formula for the number P(3,n).Comment: 31 page
On the structure of Clifford quantum cellular automata
We study reversible quantum cellular automata with the restriction that these
are also Clifford operations. This means that tensor products of Pauli
operators (or discrete Weyl operators) are mapped to tensor products of Pauli
operators. Therefore Clifford quantum cellular automata are induced by
symplectic cellular automata in phase space. We characterize these symplectic
cellular automata and find that all possible local rules must be, up to some
global shift, reflection invariant with respect to the origin. In the one
dimensional case we also find that every uniquely determined and
translationally invariant stabilizer state can be prepared from a product state
by a single Clifford cellular automaton timestep, thereby characterizing these
class of stabilizer states, and we show that all 1D Clifford quantum cellular
automata are generated by a few elementary operations. We also show that the
correspondence between translationally invariant stabilizer states and
translationally invariant Clifford operations holds for periodic boundary
conditions.Comment: 28 pages, 2 figures, LaTe
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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