The General Vector Addition System Reachability Problem by Presburger Inductive Invariants

TL;DR

This paper proves that the reachability problem for Vector Addition Systems can be decided using semi-linear inductive invariants, providing a new approach with checkable certificates for non-reachability based on Presburger arithmetic.

Contribution

It introduces a novel method for deciding VAS reachability using semi-linear inductive invariants and Presburger formulas, extending previous decomposition techniques.

Findings

The Parikh images of VAS languages are semi-pseudo-linear.

Existence of checkable certificates for non-reachability.

A semi-algorithmic decision procedure for VAS reachability.

Abstract

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.