Büchi Complementation Made Tight

The precise complexity of complementing Büchi automata is an intriguing and long standing problem. While optimal complementation techniques for finite automata are simple – it suffices to determinize them using a simple subset construction and to dualize the acceptance condition of the resulting automaton – Büchi complementation is more involved. Indeed, the construction of an EXPTIME complementation procedure took a quarter of a century from the introduction of Büchi automata in the early 60s, and stepwise narrowing the gap between the upper and lower bound to a simple exponent (of (6e) for Büchi automata with n states) took four decades. While the distance between the known upper (O ` (0.96n) ́ ) and lower (Ω ` (0.76n) ́ ) bound on the required number of states has meanwhile been significantly reduced, an exponential factor remains between them. Also, the upper bound on the size of the complement automaton is not linear in the bound of its state space. These gaps are unsatisfactory from a theoretical point of view, but also because Büchi complementation is a useful tool in formal verification, in particular for the language containment problem. This paper proposes a Büchi complementation algorithm whose complexity meets, modulo a quadratic (O(n)) factor, the known lower bound for Büchi complementation. It thus improves over previous constructions by an exponential factor and concludes the quest for optimal Büchi complementation algorithms.