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.

Source : oai:arXiv.org:1009.1076

DOI : 10.2168/LMCS-6(3:22)2010

Volume: Volume 6, Issue 3

Published on: September 9, 2010

Submitted on: January 4, 2010

Keywords: Computer Science - Logic in Computer Science,F.1

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