Measure Many Quantum Finite Automata on Infinite Words

2026-05-08

AI Summary

This work investigates the extension of quantum finite automata to infinite words by introducing the model of measure-many quantum Büchi automata (MMQBA). The proposed model integrates unitary evolution with repeated projective measurements and employs Büchi acceptance conditions together with cumulative acceptance probabilities to define language semantics. As the first formalization of measure-many quantum automata over infinite words, this study characterizes the asymptotic behavior of their recognized languages, establishing that MMQBAs are closed under union but not under intersection or complementation. Furthermore, it shows that the emptiness problem is semi-decidable, while universality, inclusion, equivalence, and membership problems are all undecidable.

Abstract

We define a quantum computational model over infinite words, called Measure-Many Quantum Büchi Automata (MMQBA), which extends Measure-many Quantum Finite automata (MMQFA) to the infinite word setting with Büchi acceptance condition. In MMQBA, the quantum state evolves through unitary transformations followed by repeated projective measurements. An infinite word is accearaq2ppted with respect to a cutpoint p is in (0, 1] if (i) the run visits accepting states infinitely often, (ii) the limiting cumulative acceptance probability is at least p, and (iii) the limiting cumulative rejection lprobability is strictly less than p. We formalize the semantics of MMQBA, establish a language-theoretic characterization showing that MMQBA languages are precisely of the form lim(L(M, p)) for MMQFA M , and develop a decomposition of the non-halting subspace. We prove that MMQBA is closed under union but not under intersection or complementation. On the algorithmic side, we show that the emptiness problem is semi-decidable, while universality, inclusion, equivalence, and membership remain undecidable.

Problem

• Quantum Finite Automata
• Infinite Words
• Büchi Acceptance
• Language Recognition
• Decidability

Innovation

• Quantum Büchi Automata
• Infinite Words
• Projective Measurement
• Language Characterization
• Undecidability