  We introduce some classical complexity-theoretic techniques to Parameterized
Complexity. First, we study relativization for the machine models that were
used by Chen, Flum, and Grohe (2005) to characterize a number of parameterized
complexity classes. Here we obtain a new and non-trivial characterization of
the A-Hierarchy in terms of oracle machines, and parameterize a famous result
of Baker, Gill, and Solovay (1975), by proving that, relative to specific
oracles, FPT and A[1] can either coincide or differ (a similar statement holds
for FPT and W[P]). Second, we initiate the study of interactive proof systems
in the parameterized setting, and show that every problem in the class AW[SAT]
has a proof system with "short" interactions, in the sense that the number of
rounds is upper-bounded in terms of the parameter value alone.
