initial state, thus E(j,i,i) and i favors j ( the initial state has to be accepting state). For some other active Aβ* machine M' activated at k, where j< k <i, k favors j (the master has record of this) and i is the earliest time at which E(j,k,i) ( M' goes dormant at time i ). Note that the green light for M does not flash every time when a machine goes dormant due to M. Lemma 3 If there exist a time equivalent to infinitely many times that favor it and i is the earliest such time, then a Aβ* machine M is activated at i, remained active forever (no corresponding red light flash thereafter), and flashes the green light infinitely many times. Proof: Let's suppose a Aβ* machine was activated at time j such that j < i and this machine goes to initial state at time i. Therefore, if any time is equivalent and favors i then the time must be in the same relation with j. This contradicts the hypothesis that i is the earliest time such that infinitely many times equivalent to i and favoring i. So at time i, no active machine can be in the initial state. Hence, the master has to activate a new Aβ* machine at time i, which is our M. This machine will never go dormant because if some other machine, which was activated at time l, makes it dormant at time k then E(l,i,k). Again, the same contradiction is implied. By construction and due to infinitely many times are equivalent to i and favor i, the green light will flash infinitely often. Lemma 4 Conversely, if there is a Aβ* machine M whose green light flashed infinitely often and red light only finitely often then there are infinitely many times equivalent to and favoring the last time at which M became active. Proof: True by construction. Lemma 5 w ∈ αβω if and only if, for some Aβ* machine, the green light flashes infinitely often and the red light flashes only finitely often. Proof: Due to Lemmas 2-4. The above description of a full machine can be viewed as a large deterministic automaton. Now, it is left to define the Muller acceptance condition. In this large automaton, we define μn to be the set of states in which the green light flashes and the red light does