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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

Vector addition system with states is an ubiquitous model of computation with extensive applications in computer science. The reachability problem for vector addition systems is central since many other problems reduce to that question. The…

Logic in Computer Science · Computer Science 2020-07-20 Jérôme Leroux , Grégoire Sutre

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

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 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

A pushdown vector addition system with states (PVASS) extends the model of vector addition systems with a pushdown store. A PVASS is said to be \emph{bidirected} if every transition (pushing/popping a symbol or modifying a counter) has an…

Formal Languages and Automata Theory · Computer Science 2022-04-26 Moses Ganardi , Rupak Majumdar , Andreas Pavlogiannis , Lia Schütze , Georg Zetzsche

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

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

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

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

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

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

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

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 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 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 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

A vector addition system with states (VASS) consists of a finite set of states and counters. A configuration is a state and a value for each counter; a transition changes the state and each counter is incremented, decremented, or left…

Formal Languages and Automata Theory · Computer Science 2019-05-15 Krishnendu Chatterjee , Thomas A. Henzinger , Jan Otop

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

By adapting the iterative yardstick construction of Stockmeyer, we show that the reachability problem for vector addition systems with a stack does not have elementary complexity. As a corollary, the same lower bound holds for the…

Formal Languages and Automata Theory · Computer Science 2013-10-08 Ranko Lazic