Language Equivalence is Undecidable in VASS with Restricted Nondeterminism

TL;DR

This paper proves that determining language equivalence in two-dimensional Vector Addition Systems with States (VASS) remains undecidable even under certain restrictions, highlighting fundamental limits in analyzing these computational models.

Contribution

It extends undecidability results to VASS with restricted nondeterminism and establishes equivalence conditions for history-deterministic VASSs.

Findings

Language equivalence is undecidable for certain restricted VASSs.

Equivalence of languages of two history-deterministic VASSs iff they simulate each other.

Undecidability extends to various simulation-based equivalence relations.

Abstract

In this work, we extend undecidability of language equivalence for two-dimensional Vector Addition System with States (VASS) accepting by coverability condition. We show that the problem is undecidable even when one of the two-dimensional VASSs is deterministic and the other is history-deterministic. Moreover, we observe, that the languages of two history-deterministic VASSs are equal if and only if each can simulate the other. This observation allows us to extend the undecidability to any equivalence relation between two-sided simulation and language equivalence.