Undecidability of MM-QFAs Language Equivalence Problem

TL;DR

This paper proves that determining whether two measure many quantum finite automata recognize the same language is undecidable, highlighting fundamental limits in quantum automata theory.

Contribution

It establishes the undecidability of language equivalence for MM-QFA and EQFA, providing new proofs and deriving related properties from these results.

Findings

Language equivalence is undecidable for MM-QFA and EQFA.

Undecidability of emptiness for MM-QFA and EQFA is proven.

Reductions from emptiness to equivalence problems are demonstrated.

Abstract

Let $L_{>\lambda}(\mathcal{A})$ and $L_{\geq\lambda}(\mathcal{A})$ be the languages recognized by {\em measure many 1-way quantum finite automata (MM-QFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) $\mathcal{A}$ with strict and non-strict cut-point $\lambda$, respectively. We consider the language equivalence problem and show the following 1. both strict and non-strict language equivalence are undecidable; 2. we provide an another proof of the undecidability of non-strict and strict emptiness of MM-QFA and EQFA, and then reducing the language equivalence problem to emptiness problem; 3. lastly, we obtain some other properties which can be derived from the above results.