  We approximate the backward reachable set of discrete-time autonomous
polynomial systems using the recently developed occupation measure approach. We
formulate the problem as an infinite-dimensional linear programming (LP)
problem on measures and its dual on continuous functions. Then we approximate
the LP by a hierarchy of finite-dimensional semidefinite programming (SDP)
programs on moments of measures and their duals on sums-of-squares polynomials.
Finally we solve the SDP's and obtain a sequence of outer approximations of the
backward reachable set. We demonstrate our approach on three dynamical systems.
As a special case, we also show how to approximate the preimage of a compact
semi-algebraic set under a polynomial map.
