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$\mu$-Calculus and automata on infinite trees are complementary ways ofdescribing infinite tree languages. The correspondence between $\mu$-Calculusand alternating tree automaton is used to solve the satisfiability and modelchecking problems by compiling the modal $\mu$-Calculus formula into analternating tree automata. Thus advocating an automaton model speciallytailored for working with modal $\mu$-Calculus. The advantage of the automatonmodel is its ability to deal with arbitrary branching in a much simpler way ascompare to the one proposed by Janin and Walukiewicz. Both problems (i.e.,model checking and satisfiability) are solved by reduction to the correspondingproblems of alternating tree automata, namely to the acceptance and thenon-emptiness problems, respectively. These problems, in turn, are solved usingparity games where semantics of alternating tree automata is translated to awinning strategy in an appropriate parity game.

From $\mu$-Calculus to Alternating Tree Automata using Parity Games