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

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.

Four Collaborations with Rūsiņš Freivalds

“…Freivalds with his students visited me at Berkeley, Princeton and Waterloo. He was particularly fascinated by the result that the power of QFAs changed with the success probability and its parallels with his work on FIN and spent a lot of time with his students figuring out when and how exactly the power of QFAs changed, with me also contributing some ideas to this work (Ambainis et al, 1999, Ambainis et al, 2001, Ambainis and Kikusts 2003, Golovkins et al 2011.…”

Automata and Quantum Computing

“…The transformations T σ 's of NaQFA as for LaQFA, the acceptance rule R is as for KWQFA. Bistochastic quantum finite automaton (BiQFA) [49] is a generalization of NaQFA such that the transformations T σ 's can be bistochastic quantum operations and the acceptance rule R is as for KWQFA.…”

“…While 1KWQFAs recognize more languages than MCQFAs and LaQFAs, they cannot recognize some regular languages [62], for example, {{a, b} * a} / ∈ BMM. Researchers [7,30,49] have also shown that if a minimal 1DFA has some certain properties, called forbidden constructions, then its language cannot be recognized by KWQFAs for any or for some error bounds. Using forbidden constructions, it was also shown that BMM is not closed under intersection or union [8].…”

“…Moreover, KWQFAs can recognize more languages if the error bound gets closer to 1 2 [6]. NaQFAs and BiQFA share many of the properties of 1KWQFAs in the bounded error setting [69,49]. In [49], it was shown that any language recognized by a BiQFA with bounded error is in the language class ER which is a proper subset of REG.…”

Quantum Finite Automata: A Modern Introduction

“…4. The details of a random walk on the input w generalizations of this QFA model, e.g., [18,3,12]. In Section 4.5, we saw a twoway QFA model that has classical as well as quantum states, and the classical states govern the computation flow and the determination of whether intermediate measurements or unitary transformations should be performed, depending on both input symbols and previous measurement results.…”