  We consider the algorithmic task of computing a maximal autarky for a
clause-set F, i.e., a partial assignment which satisfies every clause of F it
touches, and where this property is destroyed by adding any non-empty set of
further assignments. We employ SAT solvers as oracles, using various
capabilities. Using the standard SAT oracle, log_2(n(F)) oracle calls suffice,
where n(F) is the number of variables, but the drawback is that (translated)
cardinality constraints are employed, which makes this approach less efficient
in practice. Using an extended SAT oracle, motivated by the capabilities of
modern SAT solvers, we show how to compute maximal autarkies with 2 n(F)^{1/2}
simpler oracle calls. This novel algorithm combines the previous two main
approaches, based on the autarky-resolution duality and on SAT translations.
