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Reachability in pushdown vector addition systems with states (PVASS) is among the longest standing open problems in Theoretical Computer Science. We show that the problem is decidable in full generality. Our decision procedure is similar in…

Logic in Computer Science · Computer Science 2026-02-19 Roland Guttenberg , Eren Keskin , Roland Meyer

Vector Addition Systems with States (VASS), equivalent to Petri nets, are a well-established model of concurrency. The central algorithmic challenge in VASS is the reachability problem: is there a run from a given starting state and counter…

Formal Languages and Automata Theory · Computer Science 2024-12-24 Dmitry Chistikov , Wojciech Czerwiński , Filip Mazowiecki , Łukasz Orlikowski , Henry Sinclair-Banks , Karol Węgrzycki

We investigate the reachability problem in symmetric vector addition systems with states (VASS), where transitions are invariant under a group of permutations of coordinates. One extremal case, the trivial groups, yields general VASS. In…

Formal Languages and Automata Theory · Computer Science 2026-01-12 Łukasz Kamiński , Sławomir Lasota

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof. A ZVASS comprises a finite-state controller with a finite number of…

Formal Languages and Automata Theory · Computer Science 2014-10-01 Christoph Haase , Simon Halfon

We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASS) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed…

Formal Languages and Automata Theory · Computer Science 2025-03-26 Wojciech Czerwiński , Piotr Hofman

The reachability problem for vector addition systems is a central problem of net theory. This problem is known to be decidable but the complexity is still unknown. Whereas the problem is EXPSPACE-hard, no elementary upper bounds complexity…

Logic in Computer Science · Computer Science 2016-08-11 Jérôme Leroux

The geometric dimension $g$ of a Vector Addition System with States (VASS) is the dimension of the vector space generated by cycles in the VASS; this parameter refines the standard dimension $d$, the number of counters. Recently, it was…

Formal Languages and Automata Theory · Computer Science 2026-02-18 Wojciech Czerwiński , Roland Guttenberg , Łukasz Orlikowski , Henry Sinclair-Banks , Yangluo Zheng

Vector addition systems with states (VASS) are a classic model in concurrency theory. Grammar vector addition systems (GVAS), equivalently, pushdown VASS, extend VASS by using a context-free grammar to control addition. In this paper, our…

Logic in Computer Science · Computer Science 2026-02-06 Chengfeng Xue , Yuxi Fu

Branching VASS (BVASS) generalise vector addition systems with states by allowing for special branching transitions that can non-deterministically distribute a counter value between two control states. A run of a BVASS consequently becomes…

Formal Languages and Automata Theory · Computer Science 2016-05-09 Stefan Göller , Christoph Haase , Ranko Lazić , Patrick Totzke

We consider a variant of reachability in Vector Addition Systems (VAS) dubbed \emph{box reachability}, whereby a vector $v\in \mathbb{N}^d$ is box-reachable from $0$ in a VAS $V$ if $V$ admits a path from $0$ to $v$ that not only stays in…

Formal Languages and Automata Theory · Computer Science 2025-08-19 Shaull Almagor , Itay Hasson , Michał Pilipczuk , Michael Zaslavski

Vector addition systems with states (VASS) are a popular model for concurrent systems. However, many decision problems have prohibitively high complexity. Therefore, it is sometimes useful to consider overapproximating semantics in which…

Logic in Computer Science · Computer Science 2023-09-14 Pascal Baumann , Khushraj Madnani , Filip Mazowiecki , Georg Zetzsche

Petri nets, equivalently presentable as vector addition systems with states, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there…

Formal Languages and Automata Theory · Computer Science 2022-01-14 Sławomir Lasota

Vector Addition Systems (VAS), aka Petri nets, are a popular model of concurrency. The reachability set of a VAS is the set of configurations reachable from the initial configuration. Leroux has studied the geometric properties of VAS…

Formal Languages and Automata Theory · Computer Science 2023-07-25 Roland Guttenberg , Mikhail Raskin , Javier Esparza

In this paper we consider the reachability problem for bounded branching VASS. Bounded VASS are a variant of the classic VASS model where all values in all configurations are upper bounded by a fixed natural number, encoded in binary in the…

Formal Languages and Automata Theory · Computer Science 2019-08-20 Filip Mazowiecki , Michał Pilipczuk

We study the geometry of reachability sets of continuous vector addition systems with states (VASS). In particular we establish that they are almost Minkowski sums of convex cones and zonotopes generated by the vectors labelling the…

Logic in Computer Science · Computer Science 2022-11-15 Shaull Almagor , Arka Ghosh , Tim Leys , Guillermo A. Perez

Seminal results establish that the coverability problem for Vector Addition Systems with States (VASS) is in EXPSPACE (Rackoff, '78) and is EXPSPACE-hard already under unary encodings (Lipton, '76). More precisely, Rosier and Yen later…

Formal Languages and Automata Theory · Computer Science 2023-05-03 Marvin Künnemann , Filip Mazowiecki , Lia Schütze , Henry Sinclair-Banks , Karol Węgrzycki

This note is a product of digestion of the famous proof of decidability of the reachability problem for vector addition systems with states (VASS), as first established by Mayr in 1981 and then simplified by Kosaraju in 1982. The note is…

Logic in Computer Science · Computer Science 2020-05-28 Sławomir Lasota

Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations,…

Logic in Computer Science · Computer Science 2023-06-22 Michael Blondin , Mikhail Raskin

Vector addition system with states (VASS) is a popular model for the verification of concurrent systems. VASS consists of finitely many control states and a set of counters which can be incremented and decremented, but not tested for zero.…

Logic in Computer Science · Computer Science 2024-05-21 A. R. Balasubramanian

The recent years have seen remarkable progress in establishing the complexity of the reachability problem for vector addition systems with states (VASS), equivalently known as Petri nets. Existing work primarily considers the case in which…

Formal Languages and Automata Theory · Computer Science 2023-10-26 Andrei Draghici , Christoph Haase , Andrew Ryzhikov