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We study the complexity of the reachability problem for Vector Addition Systems with States (VASSes) in fixed dimensions. We provide four lower bounds improving the currently known state-of-the-art: 1) \np-hardness for unary flat $4$-VASSes…

Formal Languages and Automata Theory · Computer Science 2022-03-09 Wojciech Czerwiński , Łukasz Orlikowski

The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known to be decidable by algorithms exclusively based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney…

Logic in Computer Science · Computer Science 2015-07-01 leroux jerome

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

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

A pushdown vector addition system with states (PVASS) extends the model of vector addition systems with a pushdown stack. The algorithmic analysis of PVASS has applications such as static analysis of recursive programs manipulating integer…

Logic in Computer Science · Computer Science 2024-02-21 Guillermo A. Perez , Shrisha Rao

We consider the model of one-dimensional Pushdown Vector Addition Systems (1-PVAS), a fundamental computational model simulating both recursive and concurrent behaviours. Our main result is decidability of the reachability problem for…

Formal Languages and Automata Theory · Computer Science 2024-11-05 Clotilde Bizière , Wojciech Czerwiński

More than 30 years after their inception, the decidability proofs for reachability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr,…

Logic in Computer Science · Computer Science 2015-08-11 Jérôme Leroux , Sylvain Schmitz

Despite recent progress which settled the complexity of the reachability problem for Vector Addition Systems with States (VASSes) as being Ackermann-complete we still lack much understanding for that problem. A striking example is the…

Formal Languages and Automata Theory · Computer Science 2022-09-14 Wojciech Czerwiński , Adam Jędrych

A complete characterization of the complexity of the reachability problem for vector addition system has been open for a long time. The problem is shown to be Tower complete.

Logic in Computer Science · Computer Science 2020-03-24 Yuxi Fu , Qizhe Yang

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

The reachability problem in vector addition systems is a central question, not only for the static verification of these systems, but also for many inter-reducible decision problems occurring in various fields. The currently best known…

Logic in Computer Science · Computer Science 2019-08-20 Jérôme Leroux , Sylvain Schmitz

We consider a variant of VASS extended with integer counters, denoted VASS+Z. These are automata equipped with N and Z counters; the N-counters are required to remain nonnegative and the Z-counters do not have this restriction. We study the…

Formal Languages and Automata Theory · Computer Science 2026-03-06 Clotilde Bizière , Wojciech Czerwiński , Roland Guttenberg , Jérôme Leroux , Vincent Michielini , Łukasz Orlikowski , Antoni Puch , Henry Sinclair-Banks

Reachability problems in infinite-state systems are often subject to extremely high complexity. This motivates the investigation of efficient overapproximations, where we add transitions to obtain a system in which reachability can be…

Formal Languages and Automata Theory · Computer Science 2022-06-28 Moses Ganardi , Rupak Majumdar , Georg Zetzsche

We investigate the parameterised complexity of the classic coverability problem for vector addition systems (VAS): given a finite set of vectors $V \subseteq\mathbb{Z}^d$, an initial configuration $s\in\mathbb{N}^d$, and a target…

Computational Complexity · Computer Science 2025-12-16 Michał Pilipczuk , Sylvain Schmitz , Henry Sinclair-Banks

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 study a class of reachability problems in weighted graphs with constraints on the accumulated weight of paths. The problems we study can equivalently be formulated in the model of vector addition systems with states (VASS). We consider a…

Logic in Computer Science · Computer Science 2020-09-08 Shaull Almagor , Nathann Cohen , Guillermo A. Pérez , Mahsa Shirmohammadi , James Worrell

We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in…

Logic in Computer Science · Computer Science 2023-06-22 Michael Blondin , Christoph Haase , Filip Mazowiecki , Mikhail Raskin

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

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

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