  We state an algorithm that, given an automata network and a block-sequential
update schedule, produces an automata network of the same size or smaller with
the same limit dynamics under the parallel update schedule. Then, we focus on
the family of automata cycles which share a unique path of automata, called
tangential cycles, and show that a restriction of our algorithm allows to
reduce any instance of these networks under a block-sequential update schedule
into a smaller parallel network of the family and to characterize the number of
reductions operated while conserving their limit dynamics. We also show that
any tangential cycles reduced by our main algorithm are transformed into a
network whose size is that of the largest cycle of the initial network. We end
by showing that the restricted algorithm allows the direct characterization of
block-sequential double cycles as parallel ones.
