3 Determinization of Büchi -

To determinize Büchi automata it is necessary to switch to another class of ω-automata, e.g. Muller or Rabin automata. The reason is that there exist languages which are accepted by some nondeterministic Büchi-automaton, but not by any deterministic Büchi-automaton (c.f. section 3.1). The history of constructions for determinizing Büchi automata is long: it starts in 1963 with a faulty construction [133]. In 1966 McNaughton showed, that a Büchi automaton can be transformed effectively into an equivalent deterministic Muller automaton [125]. Safra’s construction [158] of 1988 leads to deterministic Rabin or Muller automata (c.f. section 3.2): given a nondeterministic Büchi automaton with n states, the equivalent deterministic automaton has 2 logn) states. For Rabin automata, Safra’s construction is optimal (c.f. section 3.3). The question whether it can be improved for Muller automata is open. Safra’s construction is often felt to be difficult. Thus, in 1995 Muller and Schupp [137] presented a ‘more intuitive’ alternative, which is also optimal for Rabin automata. Although Safra’s construction is optimal for Rabin automata, the resulting automata often contain equivalent states, which can be eliminated. An example for this effect is presented in Exercise3.6. As it is completely open how to minimize ω-automata, it would be quite interesting to develop procedures for ‘fine tuning’ Safra’s construction. Some ideas in this direction can be found e.g. in [153]. Considering the languages recognizable by different classes of automata we obtain the following picture: