# Deciding Fast Termination for Probabilistic VASS with Nondeterminism

## TL;DR

This paper investigates the problem of determining whether probabilistic vector addition systems with states (pVASS) with nondeterminism have linear expected termination time, providing polynomial-time decidability results and a quadratic lower bound dichotomy.

## Contribution

The paper introduces techniques for checking fast termination in pVASS with nondeterminism and establishes a polynomial-time decision procedure for linear expected termination time.

## Key findings

- Decidability of linear expected termination time in certain pVASS classes.
- A polynomial-time algorithm for checking fast termination.
- A quadratic lower bound for non-linear expected termination times.

## Abstract

A probabilistic vector addition system with states (pVASS) is a finite state Markov process augmented with non-negative integer counters that can be incremented or decremented during each state transition, blocking any behaviour that would cause a counter to decrease below zero. The pVASS can be used as abstractions of probabilistic programs with many decidable properties. The use of pVASS as abstractions requires the presence of nondeterminism in the model. In this paper, we develop techniques for checking fast termination of pVASS with nondeterminism. That is, for every initial configuration of size n, we consider the worst expected number of transitions needed to reach a configuration with some counter negative (the expected termination time). We show that the problem whether the asymptotic expected termination time is linear is decidable in polynomial time for a certain natural class of pVASS with nondeterminism. Furthermore, we show the following dichotomy: if the asymptotic expected termination time is not linear, then it is at least quadratic, i.e., in $\Omega(n^2)$.