We introduce a new class of automata on infinite trees called alternating nonzero automata, which extends the class of non-deterministic nonzero automata. We reduce the emptiness problem for alternating nonzero automata to the same problem for non-deterministic ones, which implies decidability. We obtain as a corollary algorithms for the satisfiability of a probabilistic temporal logic extending both CTL* and the qualitative fragment of pCTL*. The theory of automata on infinite trees is rooted in Rabin's seminal theorem which establishes an effective correspondence between the monadic second order logic (MSO) theory of the infinite binary tree and the non-deterministic automata on this tree [20]. In this correspondence, the satisfiability of the logic is dual to the emptiness of the algorithm and both these algorithmic problems are mutually reducible to one another. This elegant setting has been partially extended to probabilistic logics [15, 7, 16, 17, 3] and automata with probabilistic winning conditions [20, 19, 1, 8, 3] . In this paper we make another step in this direction: we show a correspondence between the logic CTL * [∃, ∀, P >0 , P =1 ] and nonzero alternating automata with limited choice. Moreover we show that the emptiness problem of the automata is decidable and obtain as a corollary the decidability of the satisfiability of the logic. Alternating nonzero automata are an alternating version of non-deterministic nonzero automata introduced in [4] , which themselves are equivalent to non-deterministic zero automata introduced in [3] . An alternating nonzero automaton takes as input a binary tree. Some states of the automaton are controlled by Eve, while other states are controlled by Adam, and the player controlling the current state chooses the next transition. Some transitions are local transitions, in which case the automaton stays on the same node of the input tree while other are split transitions in which case the automaton proceeds to the left son or to the right son of the current node with equal probability 1 2 . This interaction between Eve and Adam is seen as a game where Eve and Adam play according to some strategies. Once the strategies are fixed, one obtains a Markov chain whose trajectories are all possible plays consistent with the strategies. The winner is determined with respect to winning conditions introduced in [3, 4] , using a total order on the set of states (used to compute the limsup of a play which is the largest state seen infinitely often during the play) and three subsets of states, respectively called the sure, almost-sure and positive states. Eve wins if and only if the three acceptance conditions hold: sure winning: every play has limsup in sure states; and almost-sure winning: almost-every play has limsup in almost-sure states; and positive winning: whenever the play enters a positive state there is positive probability that the play never exits positive states.
A second decidability result concerned the class of alternating zero automata REF , restricting the abilities of the second player.
Alternating Nonzero Automata
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