On the Relative Succinctness of Nondeterministic Büchi and co-Büchi Word Automata

The practical importance of automata on infinite objects has motivated a re-examination of the complexity of automata-theoretic constructions. One such construction is the translation, when possible, of nondeterministic Büchi word automata (NBW) to nondeterministic co-Büchi word automata (NCW). Among other applications, it is used in the translation (when possible) of LTL to the alternation-free μ-calculus. The best known upper bound for the translation of NBW to NCW is exponential (given an NBW with n states, the best translation yields an equivalent NCW with 2 log n) states). On the other hand, the best known lower bound is trivial (no NBW with n states whose equivalent NCW requires even n+1 states is known). In fact, only recently was it shown that there is an NBW whose equivalent NCW requires a different structure. In this paper we improve the lower bound by showing that for every integer k ≥ 1 there is a language Lk over a two-letter alphabet, such that Lk can be recognized by an NBW with 2k+1 states, whereas the minimal NCW that recognizes Lk has 3k states. Even though this gap is not asymptotically very significant, it nonetheless demonstrates for the first time that NBWs are more succinct than NCWs. In addition, our proof points to a conceptual advantage of the Büchi condition: an NBW can abstract precise counting by counting to infinity with two states. To complete the picture, we consider also the reverse NCW to NBW translation, and show that the known upper bound, which duplicates the state space, is tight.