  We investigate the application of Courcelle's Theorem and the logspace
version of Elberfeld etal. in the context of the implication problem for
propositional sets of formulae, the extension existence problem for default
logic, as well as the expansion existence problem for autoepistemic logic and
obtain fixed-parameter time and space efficient algorithms for these problems.
On the other hand, we exhibit, for each of the above problems, families of
instances of a very simple structure that, for a wide range of different
parameterizations, do not have efficient fixed-parameter algorithms (even in
the sense of the large class XPnu), unless P=NP.
