The General Vector Addition System Reachability Problem by Presburger Inductive Invariants

Jérôme Leroux (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The reachability problem for Vector Addition Systems (VAS) is a central problem of net theory. The general problem is known decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney decomposition. This decomposition is used in this paper to prove that Parikh images of languages accepted by VAS are \emph{semi-pseudo-linear}; a class of sets that can be precisely over-approximated by sets definable in the Presburger arithmetic. We provide an application of this result; we prove that if a final configuration is not reachable from an initial one, there exists a Presburger inductive invariant proving this property. Since we can decide with any decision procedure for the Presburger arithmetic if formulas denote inductive invariants, 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-algorithms. A first one that tries to prove the reachability by non-deterministically selecting finite sequences of actions and a second one that tries to prove the non-reachability by non-deterministically selecting Presburger formulas.

Mots clés: Vector Addition Systems; Reachability; Semi-linear; Presburger Arithmetic

Domaines: Autre

Jérôme Leroux. The General Vector Addition System Reachability Problem by Presburger Inductive Invariants. 2009. ⟨hal-00272667v9⟩