  $\mu$-Calculus and automata on infinite trees are complementary ways of
describing infinite tree languages. The correspondence between $\mu$-Calculus
and alternating tree automaton is used to solve the satisfiability and model
checking problems by compiling the modal $\mu$-Calculus formula into an
alternating tree automata. Thus advocating an automaton model specially
tailored for working with modal $\mu$-Calculus. The advantage of the automaton
model is its ability to deal with arbitrary branching in a much simpler way as
compare to the one proposed by Janin and Walukiewicz. Both problems (i.e.,
model checking and satisfiability) are solved by reduction to the corresponding
problems of alternating tree automata, namely to the acceptance and the
non-emptiness problems, respectively. These problems, in turn, are solved using
parity games where semantics of alternating tree automata is translated to a
winning strategy in an appropriate parity game.
