  Tableaux-based decision procedures for satisfiability of modal and
description logics behave quite well in practice, but it is sometimes hard to
obtain exact worst-case complexity results using these approaches, especially
for EXPTIME-complete logics. In contrast, automata-based approaches often yield
algorithms for which optimal worst-case complexity can easily be proved.
However, the algorithms obtained this way are usually not only worst-case, but
also best-case exponential: they first construct an automaton that is always
exponential in the size of the input, and then apply the (polynomial) emptiness
test to this large automaton. To overcome this problem, one must try to
construct the automaton "on-the-fly" while performing the emptiness test.
  In this paper we will show that Voronkov's inverse method for the modal logic
K can be seen as an on-the-fly realization of the emptiness test done by the
automata approach for K. The benefits of this result are two-fold. First, it
shows that Voronkov's implementation of the inverse method, which behaves quite
well in practice, is an optimized on-the-fly implementation of the
automata-based satisfiability procedure for K. Second, it can be used to give a
simpler proof of the fact that Voronkov's optimizations do not destroy
completeness of the procedure. We will also show that the inverse method can
easily be extended to handle global axioms, and that the correspondence to the
automata approach still holds in this setting. In particular, the inverse
method yields an EXPTIME-algorithm for satisfiability in K w.r.t. global
axioms.
