  Consider the problem of planning collision-free motion of $n$ objects in the
plane movable through contact with a robot that can autonomously translate in
the plane and that can move a maximum of $m \leq n$ objects simultaneously.
This represents the abstract formulation of a manipulation planning problem
that is proven to be decidable in this paper. The tools used for proving
decidability of this simplified manipulation planning problem are, in fact,
general enough to handle the decidability problem for the wider class of
systems characterized by a stratified configuration space. These include, for
example, problems of legged and multi-contact locomotion, bi-manual
manipulation. In addition, the described approach does not restrict the
dynamics of the manipulation system to be considered.
