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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

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

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

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

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

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

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

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

A vector addition system with states (VASS) consists of a finite set of states and counters. A transition changes the current state to the next state, and every counter is either incremented, or decremented, or left unchanged. A state and…

Formal Languages and Automata Theory · Computer Science 2020-07-20 Krishnendu Chatterjee , Thomas A. Henzinger , Jan Otop

Multi-stack pushdown systems are a well-studied model of concurrent computation using threads with first-order procedure calls. While, in general, reachability is undecidable, there are numerous restrictions on stack behaviour that lead to…

Formal Languages and Automata Theory · Computer Science 2013-10-11 Matthew Hague

Vector addition systems (VAS), also known as Petri nets, are a popular model of concurrent systems. Many problems from many areas reduce to the reachability problem for VAS, which consists of deciding whether a target configuration of a VAS…

Formal Languages and Automata Theory · Computer Science 2024-05-01 Roland Guttenberg

We consider two-player turn-based games with zero-reachability and zero-safety objectives generated by extended vector addition systems with states. Although the problem of deciding the winner in such games is undecidable in general, we…

Computer Science and Game Theory · Computer Science 2010-02-15 Tomas Brazdil , Petr Jancar , Antonin Kucera

In many kinds of infinite-state systems, the coverability problem has significantly lower complexity than the reachability problem. In order to delineate the border of computational hardness between coverability and reachability, we propose…

Formal Languages and Automata Theory · Computer Science 2025-05-21 Yousef Shakiba , Henry Sinclair-Banks , Georg Zetzsche

An $\mathsf{F}_{d}$ upper bound for the reachability problem in vector addition systems with states (VASS) in fixed dimension is given, where $\mathsf{F}_d$ is the $d$-th level of the Grzegorczyk hierarchy of complexity classes. The new…

Logic in Computer Science · Computer Science 2024-04-24 Yuxi Fu , Qizhe Yang , Yangluo Zheng

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

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

Pushdown systems (PDSs) are a natural model for sequential programs, but they can fail to accurately represent the way an assembly stack actually operates. Indeed, one may want to access the part of the memory that is below the current…

Formal Languages and Automata Theory · Computer Science 2018-11-08 Adrien Pommellet , Marcio Diaz , Tayssir Touili

A vector addition system (VAS) with an initial and a final marking and transition labels induces a language. In part because the reachability problem in VAS remains far from being well-understood, it is difficult to devise decision…

Formal Languages and Automata Theory · Computer Science 2018-02-20 Wojciech Czerwiński , Piotr Hofman , Georg Zetzsche

We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability…

Formal Languages and Automata Theory · Computer Science 2020-07-22 Wojciech Czerwiński , Diego Figueira , Piotr Hofman

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