Explicit Muller Games are PTIME

Regular games provide a very useful model for the synthesis of controllers in reactive systems. The complexity of these games depends on the representation of the winning condition: if it is represented through a win-set, a coloured condition, a Zielonka-DAG or Emerson-Lei formulae, the winner problem is pspace-complete; if the winning condition is represented as a Zielonka tree, the winner problem belongs to np and conp. In this paper, we show that explicit Muller games can be solved in polynomial time, and provide an effective algorithm to compute the winning regions