The General Vector Addition System Reachability Problem by Presburger Inductive Invariants

The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-SacerdoteTenney decomposition. This decomposition is used in this paper to prove that the Parikh images of languages accepted by VASs are semi-pseudo-linear; a class that extends the semi-linear sets, a.k.a. the sets definable in the Presburger arithmetic. We provide an application of this result; we prove that a final configuration is not reachable from an initial one if and only if there exists a Presburger formula denoting a forward inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Preburger formula denotes an inductive invariant, we deduce that there exist checkable certificates of non-reachability. In particular, there exists a simple algorithm for deciding the general VAS reachability problem based on two semi-alg...

• Type: Conference
• Year: 2009
• Where: LICS
• Authors: Jérôme Leroux
• Keywords: Computer Science, Inductive Invariant, LICS 2009, Presburger Formulas, Reachability Problem