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We study the language universality problem for One-Counter Nets, also known as 1-dimensional Vector Addition Systems with States (1-VASS), parameterized either with an initial counter value, or with an upper bound on the allowed counter…

Formal Languages and Automata Theory · Computer Science 2020-07-07 Shaull Almagor , Udi Boker , Piotr Hofman , Patrick Totzke

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

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 and equivalent Petri nets are a well established models of concurrency. The central algorithmic problem for Vector Addition Systems with a long research history is the reachability problem asking whether there exists…

Formal Languages and Automata Theory · Computer Science 2022-10-27 Wojciech Czerwiński , Łukasz Orlikowski

The geometric dimension of a Vector Addition System with States (VASS), emerged in Leroux and Schmitz (2019) and formalized by Fu, Yang, and Zheng (2024), quantifies the dimension of the vector space spanned by cycle effects in the system.…

Formal Languages and Automata Theory · Computer Science 2024-12-20 Yangluo Zheng

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 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

Multi-pushdown systems are a standard model for concurrent recursive programs, but they have an undecidable reachability problem. Therefore, there have been several proposals to underapproximate their sets of runs so that reachability in…

Formal Languages and Automata Theory · Computer Science 2021-08-03 Aneesh K. Shetty , S. Krishna , Georg Zetzsche

We consider history-determinism, a restricted form of non-determinism, for Vector Addition Systems with States (VASS) when used as acceptors to recognise languages of finite words. History-determinism requires that the non-deterministic…

Formal Languages and Automata Theory · Computer Science 2023-07-11 Sougata Bose , David Purser , Patrick Totzke

In this work, we extend undecidability of language equivalence for two-dimensional Vector Addition System with States (VASS) accepting by coverability condition. We show that the problem is undecidable even when one of the two-dimensional…

Formal Languages and Automata Theory · Computer Science 2025-10-27 Wojciech Czerwiński , Łukasz Orlikowski

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

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

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

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

Well-structured transition systems (WSTS) are an abstract family of systems that encompasses a vast landscape of infinite-state systems. By requiring a well-quasi-ordering (wqo) on the set of states, a WSTS enables generic algorithms for…

Formal Languages and Automata Theory · Computer Science 2024-09-17 Ashwani Anand , Sylvain Schmitz , Lia Schütze , Georg Zetzsche

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 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

This paper is about reachability analysis in a restricted subclass of multi-pushdown automata. We assume that the control states of an automaton are partially ordered, and all transitions of an automaton go downwards with respect to the…

Logic in Computer Science · Computer Science 2015-07-01 Wojciech Czerwiński , Piotr Hofman , SŁawomir Lasota

Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…

Formal Languages and Automata Theory · Computer Science 2025-11-27 A. R. Balasubramanian , Matthew Hague , Rupak Majumdar , Ramanathan S. Thinniyam , Georg Zetzsche

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