  Checking two probabilistic automata for equivalence has been shown to be a
key problem for efficiently establishing various behavioural and anonymity
properties of probabilistic systems. In recent experiments a randomised
equivalence test based on polynomial identity testing outperformed
deterministic algorithms. In this paper we show that polynomial identity
testing yields efficient algorithms for various generalisations of the
equivalence problem. First, we provide a randomized NC procedure that also
outputs a counterexample trace in case of inequivalence. Second, we show how to
check for equivalence two probabilistic automata with (cumulative) rewards. Our
algorithm runs in deterministic polynomial time, if the number of reward
counters is fixed. Finally we show that the equivalence problem for
probabilistic visibly pushdown automata is logspace equivalent to the
Arithmetic Circuit Identity Testing problem, which is to decide whether a
polynomial represented by an arithmetic circuit is identically zero.
