Quantum Finite Automata and Probabilistic Reversible Automata: R-trivial Idempotent Languages

TL;DR

This paper characterizes which R-trivial idempotent languages can be recognized by various quantum and probabilistic automata models, introducing a new bistochastic QFA model and analyzing their capabilities.

Contribution

It introduces the bistochastic QFA model and provides a complete algebraic characterization of R1 languages recognized by these automata.

Findings

Bistochastic QFA generalizes existing models.

Complete algebraic characterization of recognized R1 languages.

Forbidden constructions do not fully explain recognition limitations.

Abstract

We study the recognition of R-trivial idempotent (R1) languages by various models of "decide-and-halt" quantum finite automata (QFA) and probabilistic reversible automata (DH-PRA). We introduce bistochastic QFA (MM-BQFA), a model which generalizes both Nayak's enhanced QFA and DH-PRA. We apply tools from algebraic automata theory and systems of linear inequalities to give a complete characterization of R1 languages recognized by all these models. We also find that "forbidden constructions" known so far do not include all of the languages that cannot be recognized by measure-many QFA.