The Strength of Safra’s Construction

Automata on Infinite Words. Automata running on infinite words (such as non-deterministic Büchi automata) provide an established framework for the specification and verification of nonterminating programs (in particular via the model checking technique). However, some of their basic properties are known to require non-trivial reasoning principles. This is most notably the case of closure under complement of non-deterministic Büchi automata: Given an automaton A, to construct an automaton à which accepts exactly the infinite words rejected by A. There are different known constructions of à from A, but all of them require non-trivial non-constructive mathematical principles to be proven correct. Complementation can be achieved via determinization: Given a non-deterministic Büchi automaton, to construct an equivalent determinisitic automaton (with a different type of acceptance condition). As it is usually trivial to complement a deterministic automaton, determinization constructions for automata on infinite words are at least as hard as complementation of Büchi automata. We are primarily interested in Safra’s construction for determinization (see e.g. [PP04] for details), which gives (asymptotically) optimal automata for complementation. Safra’s construction relies on Weak König Lemma, which states that every infinite tree on a finite alphabet has an infinite path.