  The amoebot model [Derakhshandeh et al., 2014] has been proposed as a model
for programmable matter consisting of tiny, robotic elements called amoebots.
We consider the reconfigurable circuit extension [Feldmann et al., JCB 2022] of
the geometric (variant of the) amoebot model that allows the amoebot structure
to interconnect amoebots by so-called circuits. A circuit permits the
instantaneous transmission of signals between the connected amoebots. In this
paper, we examine the structural power of the reconfigurable circuits.
  We start with some fundamental problems like the stripe computation problem
where, given any connected amoebot structure $S$, an amoebot $u$ in $S$, and
some axis $X$, all amoebots belonging to axis $X$ through $u$ have to be
identified. Second, we consider the global maximum problem, which identifies an
amoebot at the highest possible position with respect to some direction in some
given amoebot (sub)structure. A solution to this problem can then be used to
solve the skeleton problem, where a (not necessarily simple) cycle of amoebots
has to be found in the given amoebot structure which contains all boundary
amoebots. A canonical solution to that problem can then be used to come up with
a canonical path, which provides a unique characterization of the shape of the
given amoebot structure. Constructing canonical paths for different directions
will then allow the amoebots to set up a spanning tree and to check symmetry
properties of the given amoebot structure.
  The problems are important for a number of applications like rapid shape
transformation, energy dissemination, and structural monitoring. Interestingly,
the reconfigurable circuit extension allows polylogarithmic-time solutions to
all of these problems.
