Reachability is in DynFO

We consider the dynamic complexity of some central graph problems such as Reachability and Matching and linear algebraic problems such as Rank and Inverse. As elementary change operations we allow insertion and deletion of edges of a graph and the modification of a single entry in a matrix, and we are interested in the complexity of maintaining a property or query. Our main results are as follows: 1. Rank of a matrix is in DynFO(+,×); 2. Reachability is in DynFO; 3. Maximum Matching (decision) is in non-uniform DynFO. Here, DynFO allows updates of the auxiliary data structure defined in first-order logic, DynFO(+,×) additionally has arithmetics at initialization time and non-uniform DynFO allows arbitrary auxiliary data at initialization time. Alternatively, DynFO(+,×) and non-uniform DynFO allow updates by uniform and non-uniform families of poly-size, boundeddepth circuits, respectively. The second result confirms a two decade old conjecture of Patnaik and Immerman [27]. The proofs rely mainly on elementary Linear Algebra. The second result can also be concluded from [13].