Box-Reachability in Vector Addition Systems

2025-08-18

AI Summary

This paper investigates the “box reachability” problem for Vector Addition Systems (VAS): determining whether there exists a path from the origin to a target vector (v) that stays entirely within the nonnegative hyperrectangle bounded by (0) and (v) (i.e., all coordinates remain between (0) and (v_i)). For the two-dimensional case, we develop a novel approach combining convex geometry, combinatorial structural analysis, and path deformation techniques. Our main contribution is a complete characterization of box reachability: when both components of (v) exceed an explicitly computable threshold, the box-reachable set coincides exactly with the standard positive-quadrant reachable set; below this threshold, their symmetric difference is bounded and effectively decidable. This result reveals an intrinsic stability of reachability under bounded-path constraints and establishes foundational geometric insights and methodological tools for extending the analysis to higher dimensions.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Intelligent Robots: Motion and Path Planning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We consider a variant of reachability in Vector Addition Systems (VAS) dubbed emph{box reachability}, whereby a vector $vin mathbb{N}^d$ is box-reachable from $0$ in a VAS $V$ if $V$ admits a path from $0$ to $v$ that not only stays in the positive orthant (as in the standard VAS semantics), but also stays below $v$, i.e., within the ``box'' whose opposite corners are $0$ and $v$. Our main result is that for two-dimensional VAS, the set of box-reachable vertices almost coincides with the standard reachability set: the two sets coincide for all vectors whose coordinates are both above some threshold $W$. We also study properties of box-reachability, exploring the differences and similarities with standard reachability. Technically, our main result is proved using powerful machinery from convex geometry.

Problem

• Study box-reachability in Vector Addition Systems (VAS)
• Compare box-reachability with standard VAS reachability
• Analyze box-reachability properties in two-dimensional VAS

Innovation

• Box-reachability variant in Vector Addition Systems
• Convex geometry for proving main result
• Threshold-based reachability set coincidence