  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 most 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. Our bisimulation bridges the two closely related concepts in
the community, and provides a uniform way of studying their characteristics. 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.
