  Patnaik and Immerman introduced the dynamic complexity class DynFO of
database queries that can be maintained by first-order dynamic programs with
the help of auxiliary relations under insertions and deletions of edges
(Patnaik and Immerman 1997). This article confirms their conjecture that the
Reachability query is in DynFO.
  As a byproduct it is shown that the rank of a matrix with small values can be
maintained in DynFO(+,x). It is further shown that the (size of the) maximum
matching of a graph can be maintained in non-uniform DynFO, another extension
of DynFO, with non-uniform initialisation of the auxiliary relations.
