(Extended Version) Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

TL;DR

This paper characterizes the class of languages recognized by measure-only quantum automata using algebraic structures, linking quantum automata theory with formal language and logic classifications.

Contribution

It provides an algebraic characterization of measure-only quantum automata languages and establishes their equivalence with certain logical and language classes.

Findings

Languages in the class have syntactic monoids that are J-trivial and literally idempotent.

The class coincides with a variety of literally idempotent piecewise testable languages.

A polynomial time algorithm is provided for deciding language membership in this class.

Abstract

We study a model of one-way quantum automaton where only measurement operations are allowed ($\mon$). We give an algebraic characterization of $\lmo(\Sigma)$, showing that the syntactic monoids of the languages in $\lmo(\Sigma)$ are exactly the $J$-trivial literally idempotent syntactic monoids, where $J$ is the Green's relation determined by two-sided ideals. We also prove that $\lmo(\Sigma)$ coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to $\lmo(\Sigma)$ and to discuss definability issues in terms of the existential first-order logic $\Sigma_1[<]$ and the linear temporal logic without the next operator LTLWN.