A Tight Lower Bound for the Complementation of Rabin Automata

Complementing ω-automata is a crucial instrument for solving the ω-language containment problem, and therefore it has numerous applications in formal language theory, program analysis and modelchecking. There have been great interests in determining the exact complexity of the complementation problem. However, obtaining nontrivial lower bounds for the complementation problem has been difficult. For complementing Rabin automata, there exists a significant gap between the state-of-the-art lower bound 2 lgN) and the best known upper bound 2 , where k, the number of Rabin pairs, can be as large as 2. In this paper we generalize the full automaton technique and use it to establish a tight lower bound for the complementation problem.We show that for any ǫ > 0, the lower bound for complementing Rabin automata with N states and k Rabin pairs is 2 lgN) if k ≤ 2N(1−ǫ), and is 2Ω(2N lgN) otherwise.