721 research outputs found
The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable
We prove that a semigroup generated by a reversible two-state Mealy automaton
is either finite or free of rank 2. This fact leads to the decidability of
finiteness for groups generated by two-state or two-letter
invertible-reversible Mealy automata and to the decidability of freeness for
semigroups generated by two-state invertible-reversible Mealy automata
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
A characterization of those automata that structurally generate finite groups
Antonenko and Russyev independently have shown that any Mealy automaton with
no cycles with exit--that is, where every cycle in the underlying directed
graph is a sink component--generates a fi- nite (semi)group, regardless of the
choice of the production functions. Antonenko has proved that this constitutes
a characterization in the non-invertible case and asked for the invertible
case, which is proved in this paper
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