Rabin Games Revisited

Abstract

Rabin games have their roots in Michael Rabin's seminal work on automata over infinite trees and form a classical model for infinite-round games with ω-regular winning conditions. They play a central role in automata theory and temporal synthesis: translating an LTL specification into an equivalent deterministic Rabin automaton reduces synthesis to solving a Rabin game. In practice, however, Rabin games are typically solved indirectly, via parity games. One possible reason is that many direct algorithms have a factorial dependence on the Rabin index k, the number of Rabin pairs. In this work, we revisit Rabin's 1972 recursive algorithm. The algorithm is implicit in Rabin's proof and intertwined with finite-state determinacy. First, we "excavate" it as an explicit arena-based solver using standard reachability- and Büchi-game solving as basic building blocks, thus avoiding the exponential dependence on the alphabet in Rabin's original presentation. Second, we show that memoization removes the factorial term from the recursion-tree analysis. Finally, we introduce a variation that recurses on the Rabin index. These results provide a modern analysis of Rabin's recursive method and a basis for studying direct Rabin-game solvers for temporal synthesis.