  Motivated by advances is nanoscale applications and simplistic robot agents,
we look at problems based on using a global signal to move all agents when
given a limited number of directional signals and immovable geometry. We study
a model where unit square particles move within a 2D grid based on uniform
external forces. Movement is based on a sequence of uniform commands which
cause all particles to move 1 step in a specific direction. The 2D grid board
additionally contains "blocked" spaces which prevent particles from entry.
Within this model, we investigate the complexity of deciding 1) whether a
target location on the board can be occupied (by any) particle (\emph{occupancy
problem}), 2) whether a specific particle can be relocated to another specific
position in the board (\emph{relocation problem}), and 3) whether a board
configuration can be transformed into another configuration
(\emph{reconfiguration problem}). We prove that while occupancy is solvable in
polynomial time, the relocation and reconfiguration problems are both
NP-Complete even when restricted to only 2 or 3 movement directions. We further
define a hierarchy of board geometries and show that this hardness holds for
even very restricted classes of board geometry.
