Automated Verification of Nested DFS

In this paper we demonstrate the automated verification of the Nested Depth-First Search (NDFS) algorithm for detecting accepting cycles. The starting point is a recursive formulation of the NDFS algorithm. We use Dafny to annotate the algorithm with invariants and a global specification. The global specification requires that NDFS indeed solves the accepting cycle problem. The invariants are proved automatically by the SMT solver Z3 underlying Dafny. The global specifications, however, need some inductive reasoning on paths in a graph. To prove these properties, some auxiliary lemmas had to be provided. The full specification is contained in this paper. It fits on 4 pages, is verified by Dafny in about 2 minutes, and was developed in a couple of weeks.