Progress measures, immediate determinacy, and a subset construction for tree automata

Abstract

<p>Using the concept of progress measure, we give a new proof of Rabin's fundamental result that the languages defined by tree automata are closed under complementation.</p><p>To do this we show that for certain infinite games based on tree automata, an <em>immediate determinacy</em> property holds for the player who is trying to win according to a Rabin acceptance condition. Immediate determinacy is stronger than the <em> forgetful determinacy</em> of Gurevich and Harrington, which depends on more information about the past, but applies to another class of games.</p><p>Next, we show a graph theoretic duality theorem for winning conditions. Finally, we present an extended version of Safra's determinization construction. Together, these ingredients and the determinacy of Borel games yield a straightforward recipe for complementing tree automata.</p><p>Our construction is almost optimal, i.e. the state space blow-up is essentially exponential --- thus roughly the same as for automata on finite or infinite words.</p><p>To our knowledge, no prior constructions have been better than double exponential.</p>