Vector Addition System Reachability Problem: A Short Self-Contained Proof

Jérôme Leroux (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known to be decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney decomposition. This decomposition is used in this paper to prove that the Parikh images of languages recognized by VASs are semi-pseudo-linear; a class that extends the semi-linear sets, a.k.a. the sets definable in 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 semi-linear inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Presburger 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-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas.

Mots clés: Reachability; Petri Nets; Vector Addition Systems; Presburger Arithmetic; Semilinear Sets

Domaines: Autre [cs.OH]

Jérôme Leroux. Vector Addition System Reachability Problem: A Short Self-Contained Proof. Logical Methods in Computer Science, 2010, 6 (3), pp.1-25. ⟨10.2168/LMCS-6(3:22)2010⟩. ⟨hal-00645446⟩