  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.
