On the difference set of two transductions

The difference set ${\Delta }_{s,t}$ of two (nondeterministic, in general) transducers $s,t$ is the set of all input words for which the output sets of the two transducers are not equal. When the two transducers realize homomorphisms, their difference set is the complement of the well known equality set of the two homomorphisms. However, we show that transducer difference sets result in Chomsky-like classes of languages that are different than the classes resulting from equality sets. We also consider the following word problem: given transducers $s,t$ and input w, tell whether the output sets $s\left(w\right)$ and $t\left(w\right)$ are different. In general the problem is PSPACE-complete, but it becomes NP-complete when at least one of the given transducers has finite outputs. We also provide a PRAX (polynomial randomized approximation) algorithm for the word problem as well as for the NFA (in)equivalence problem. Our presentation of PRAX algorithms improves the original presentation.