# Coverability is Undecidable in One-dimensional Pushdown Vector Addition Systems with Resets

## TL;DR

This paper proves that the coverability problem becomes undecidable when a pushdown stack and reset instructions are combined in one-dimensional vector addition systems, highlighting the increased complexity of this model.

## Contribution

It establishes the undecidability of coverability in one-dimensional pushdown vector addition systems with resets, a previously open problem.

## Key findings

- Coverability is decidable without resets.
- Coverability is undecidable with resets alone.
- Undecidability is proven for the combined model.

## Abstract

We consider the model of pushdown vector addition systems with resets. These consist of vector addition systems that have access to a pushdown stack and have instructions to reset counters. For this model, we study the coverability problem. In the absence of resets, this problem is known to be decidable for one-dimensional pushdown vector addition systems, but decidability is open for general pushdown vector addition systems. Moreover, coverability is known to be decidable for reset vector addition systems without a pushdown stack. We show in this note that the problem is undecidable for one-dimensional pushdown vector addition systems with resets.