A Forward-Only Construction of Semilinear Inductive Invariants for VAS

2026-06-25

AI Summary

This work addresses the challenge of proving non-reachability in vector addition systems (VAS), where existing methods rely on symmetric forward–backward reasoning that does not generalize well to asymmetric models such as branching VAS. The paper proposes a purely forward construction that generates semilinear inductive invariants directly from the initial configuration, eliminating the need for backward reasoning and ensuring natural alignment between the invariants and the system’s structure. This approach constitutes the first fully forward method for constructing semilinear inductive invariants, capable of producing invariants with periodic structure in cyclic VAS. By doing so, it significantly enhances the applicability of such invariants to asymmetric infinite-state systems and establishes a theoretical foundation for extending these techniques to branching VAS.

Abstract

The reachability problem for Vector Addition Systems (VAS) is a central decision problem in the theory of infinite-state systems, first solved by Kosaraju and Mayr in the 1980s. An alternative, conceptually simpler approach introduced by Leroux shows that non-reachability is always witnessed by semilinear inductive invariants, yielding a decision procedure by combining an enumeration of runs with a search for such invariants. However, the construction of these invariants relies on a back-and-forth scheme that depends symmetrically on the source and the target. As a result, the invariants are not guaranteed to reflect the structural properties of the VAS, and the construction is difficult to extend to asymmetric models such as Branching VAS. We introduce a new forward-only construction of semilinear inductive invariants for VAS. Our method builds invariants from the source configuration alone and avoids the need for backward reasoning. This yields invariants that are more canonical and better aligned with the structure of the system. In particular, our method produces periodic inductive invariants for periodic VAS. Beyond its intrinsic interest, our approach provides a step toward extending invariant-based techniques to Branching VAS.

Problem

• Vector Addition Systems
• semilinear inductive invariants
• reachability problem
• forward-only construction
• Branching VAS

Innovation

• forward-only construction
• semilinear inductive invariants
• Vector Addition Systems
• periodic invariants
• reachability problem