# Simulation Relations for Alternating Büchi Automata

We adapt direct, delayed, and fair simulation to alternating Büchi automata. Unlike with nondeterministic Büchi automata, naive quotients do not preserve the recognized language. As a remedy, we present specifically designed definitions of quotients, namely minimax and semi-elective quotients: minimax quotients, which are simple and have a minimum number of transitions, preserve the recognized language when used with direct but not with delayed simulation, while semi-elective quotients, which are more complicated and have more transitions, preserve the recognized language when used with direct simulation as well as delayed simulation. Just as in the case of nondeterministic Büchi automata, fair simulation cannot be used for quotienting. We show that all three types of simulations imply language containment in the sense that if one automaton simulates another automaton with respect to any of the three simulations, then the language recognized by the latter is contained in the language recognized by the former. Our approach is game-theoretic; the proofs rely then the language recognized by the latter is contained in the language recognized by the former. Our approach is game-theoretic; the proofs rely on a specifically tailored join operation for strategies in simulation games which is interesting in its own right. Computing all three types of simulation relations and the described quotients is not more difficult than computing the corresponding relations and the naive quotient for nondeterministic Büchi automata. For weak alternating Büchi automata, which are known to recognize all regular $\omega$-languages, we present a particularly efficient algorithm running in time quadratic in the input size.

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