The rabin index and chain automata, with applications to automata and games

Abstract

In this paper we relate the Rabin Index of an ω-language to the complexity of translation amongst automata, strategies for two-person regular games, and the complexity of controller-synthesis and verification of finite state systems, via a new construction to transform Rabin automata to Chain automata. The Rabin Index is the minimum number of pairs required to realize the language as a deterministic Rabin automaton (DRA), and is a measure of the inherent complexity of the ω-language. Chain automata are a special kind of Rabin automata where the sets comprising the acceptance condition form a chain. Our main construction translates a DRA with n states and h pairs to a deterministic chain automaton (DCA) with n.h k states, where k is the Rabin Index of the language. Using this construction, we can transform a DRA into a minimum-pair DRA or a minimum-pair deterministic Streett automaton (DSA), each with n.h k states. Using a simple correspondence between tree automata (TA) and games, we extend the constructions to translate between nondeterministic Rabin and Streett TA while simultaneously reducing the number of pairs; for the class of “trim” deterministic Rabin TA our construction gives a minimum-index deterministic Chain TA, or a minimum-pair DRTA or DSTA, each with n.h k states, where k is RI of the tree-language.Using these results, we obtain upper bounds on the memory required to implement strategies in infinite games. In particular, the amount of memory required in a game presented as a DRA, or DSA, is bounded by nh k , where k is the RI of the game language.