Quantum finite automata and linear context-free languages: a decidable problem

TL;DR

This paper addresses the decidability of the intersection emptiness problem between a language L from classical families and the threshold language recognized by a measure-once quantum automaton Q. By proving that the topological closure Cl(φ(L)) is semialgebraic (or effectively definable under rationality) for L in the classes of bounded semilinear and linear context-free languages, the authors leverage Tarski-Seidenberg elimination to decide whether L is contained in the non-thresholded region, equivalently solving L ∩ |Q_>|=∅. The main contributions are the decidability results for bounded semilinear and linear context-free L with rational Q, the establishment of semialgebraic/effectively definable closures for these L, and a construction showing limits of the approach via non-semialgebraic closures when taking CFL complements. This extends Blondel et al.'s earlier result from the free monoid to richer language families, connecting quantum automata decision problems with semialgebraic geometry and real-closed field theory. The work advances understanding of when quantum automata decision problems remain tractable and highlights the role of effective definability in enabling algorithms.

Abstract

We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is recursively decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages.

Theorems & Definitions (25)

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