Easy Complementation of History-Deterministic Büchi Automata

A history deterministic (HD) automaton can successfully resolve its nondeterministic choices in a way that only depends on the past. Formally, an HD automaton has a strategy that maps each finite word to the transition to be taken after the word is read, and following this strategy results in accepting all the words in the language of the automaton. HD automata can replace deterministic automata in several applications, most notably reactive synthesis, and they attract a lot of interest in the research community. In 2015, Kuperberg and Skrzypczak proved that an HD Büchi automaton A can be determinized with a quadratic blow-up in the state space. The suggested construction also implicitly induces an HD co-Büchi automaton of linear size that complements A. The construction is based on a series of transformations on an HD strategy for A, which has two drawbacks. First, the construction is conceptually hard to understand. Second, since the state space of the HD strategy is exponential, so is the runtime of a procedure implementing the construction. In this paper, we describe a direct complementation procedure for HD Büchi automata. Our procedure is based on removal of transitions from A itself, which can be done in polynomial time. It results in an HD Büchi automaton equivalent to A on top of which we define a complementing HD co-Büchi automaton of linear size. A deterministic equivalent automaton of size quadratic in A can then be defined in polynomial time too. We also prove a quadratic lower bound on the size of an HD strategy for HD Büchi automata.