  Probabilistic automata were introduced by Rabin in 1963 as language
acceptors. Two automata are equivalent if and only if they accept each word
with the same probability. On the other side, in the process algebra community,
probabilistic automata were re-proposed by Segala in 1995 which are more
general than Rabin's automata. Bisimulations have been proposed for Segala's
automata to characterize the equivalence between them. So far the two notions
of equivalences and their characteristics have been studied mostly
independently. In this paper, we consider Segala's automata, and propose a
novel notion of distribution-based bisimulation by joining the existing
equivalence and bisimilarities. We demonstrate the utility of our definition by
studying distribution-based bisimulation metrics, which gives rise to a robust
notion of equivalence for Rabin's automata. We compare our notions of
bisimulation to some existing distribution-based bisimulations and discuss
their compositionality and relations to trace equivalence. Finally, we show the
decidability and complexity of all relations.
