# VASS reachability in three steps

## TL;DR

This paper provides an intuitive, three-step explanation of the decidability proof for VASS reachability, highlighting a condition that reduces problem complexity iteratively.

## Contribution

It offers a simplified, conceptual overview of the key ideas in the classic VASS reachability proof, emphasizing the reduction technique through a specific condition.

## Key findings

- Decidable condition Theta implies reachability.
- Size reduction of VASS via negation of Theta.
- Method applies to various VASS generalizations.

## Abstract

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 neither intended to be rigorously formal nor complete; it is rather intended to be an intuitive but precise enough description of main concepts exploited in the proof. Very roughly, the overall idea is to provide a decidable condition Theta on a VASS such that Theta implies reachability and its negation implies that the size of VASS can be reduced. With these two properties, the size of input can be incrementally reduced until the problem becomes trivial. We proceed in three steps: we first formulate the condition Theta for plain VASS, then adapt it to more general VASS with unconstrained coordinates, and finally to generalized VASS of Kosaraju.