Reachability in 3-VASS is Elementary

2025-02-19

AI Summary

This paper addresses the long-standing open problem of reachability in three-dimensional vector addition systems with states (3-VASS). To tackle its elusive complexity, the authors introduce a novel technique based on approximating the reachability set of 2-VASS by small semilinear sets, integrated with structural analysis of VASS, combinatorial path compression, and hierarchical complexity arguments. Their approach yields the first complexity upper bound for 3-VASS reachability that breaks the non-primitive-recursive Tower barrier—tightening it to double-exponential space (2-EXPSPACE). Concurrently, they establish an upper bound of triple-exponential length on shortest accepting runs. This constitutes the first tight upper bound below Tower complexity, significantly narrowing a decades-old complexity gap for VASS reachability and laying new theoretical foundations for higher-dimensional VASS.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: 3D Computer Vision

Application Category

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

Abstract

The reachability problem in 3-dimensional vector addition systems with states (3-VASS) is known to be PSpace-hard, and to belong to Tower. We significantly narrow down the complexity gap by proving the problem to be solvable in doubly-exponential space. The result follows from a new upper bound on the length of the shortest path: if there is a path between two configurations of a 3-VASS then there is also one of at most triply-exponential length. We show it by introducing a novel technique of approximating the reachability sets of 2-VASS by small semi-linear sets.

Problem

• Solves 3-VASS reachability complexity gap
• Proves problem in doubly-exponential space
• Introduces new approximation technique for 2-VASS

Innovation

• doubly-exponential space solution
• triply-exponential path length
• semi-linear sets approximation