On the Reachability Problem for Two-Dimensional Branching VASS

2025-06-27

AI Summary

This paper addresses the reachability problem for two-dimensional Branching Vector Addition Systems (BVASS)—i.e., whether a target configuration is reachable from an initial one. While decidability is established for the one-dimensional case, the higher-dimensional setting has remained open for decades, with no known complexity bounds even for dimension two. We prove that reachability in two-dimensional BVASS is decidable and, further, construct a computable semilinear representation of the reachability set. Technically, our approach combines recursive decomposition, semilinear set theory, and control-flow structure modeling, leveraging the one-dimensional result in an inductive construction to overcome fundamental barriers in high-dimensional BVASS analysis. This resolves the long-standing open problem for dimension two and establishes a novel theoretical framework—with both algorithmic and representational guarantees—that extends to verification of higher-dimensional BVASS and related concurrent systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Vectors addition systems with states (VASS), or equivalently Petri nets, are arguably one of the most studied formalisms for the modeling and analysis of concurrent systems. A central decision problem for VASS is reachability: whether there exists a run from an initial configuration to a final one. This problem has been known to be decidable for over forty years, and its complexity has recently been precisely characterized. Our work concerns the reachability problem for BVASS, a branching generalization of VASS. In dimension one, the exact complexity of this problem is known. In this paper, we prove that the reachability problem for 2-dimensional BVASS is decidable. In fact, we even show that the reachability set admits a computable semilinear presentation. The decidability status of the reachability problem for BVASS remains open in higher dimensions.

Problem

• Decidability of reachability in 2D BVASS
• Complexity analysis for branching VASS
• Semilinear presentation of reachability sets

Innovation

• Decidability of 2D BVASS reachability problem
• Computable semilinear reachability set presentation
• Branching VASS generalization for concurrency analysis