  We build a general theory for characterizing the computational complexity of
motion planning of robot(s) through a graph of "gadgets", where each gadget has
its own state defining a set of allowed traversals which in turn modify the
gadget's state. We study two families of such gadgets, one which naturally
leads to motion planning problems with polynomially bounded solutions, and
another which leads to polynomially unbounded (potentially exponential)
solutions. We also study a range of competitive game-theoretic scenarios, from
one player controlling one robot to teams of players each controlling their own
robot and racing to achieve their team's goal. Under small restrictions on
these gadgets, we fully characterize the complexity of bounded 1-player motion
planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs.
PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete),
and we partially characterize the complexity of unbounded 2-player motion
planning (P vs. EXPTIME-complete), bounded 2-team motion planning (P vs.
NEXPTIME-complete), and unbounded 2-team motion planning (P vs. undecidable).
These results can be seen as an alternative to Constraint Logic (which has
already proved useful as a basis for hardness reductions), providing a wide
variety of agent-based gadgets, any one of which suffices to prove a problem
hard.
