Universality Problem for Unambiguous VASS

TL;DR

This paper investigates the universality problem for unambiguous Vector Addition Systems with States (VASS), revealing complexity classifications that differ significantly from general VASS, with results varying based on the system's dimension.

Contribution

It establishes the complexity of the universality problem for unambiguous VASS, showing ExpSpace-completeness in general and PSpace/coNP-hardness in fixed dimensions.

Findings

Universality for unambiguous VASS is ExpSpace-complete.

In fixed dimension, the problem is PSpace-complete for dimension ≥ 2.

For 1-dimensional VASS, the problem is coNP-hard.

Abstract

We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d is fixed, the universality problem is PSpace-complete if d is at least 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets).