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We give a uniform treatment of the logical properties of alternating weak automata on infinite strings, extending and refining work of Muller, Saoudi, and Schupp (1984) and Kupferman and Vardi (1997). Two ideas are essential in the present set-up: There is no acyclicity requirement on the transition structure of weak alternating automata, and acceptance is defined only in terms of reachability of states; moreover, the run trees of the standard framework are replaced by run dags of bounded width. As applications, one obtains a new normal form for monadic second order logic, a simple complementation proof for weak alternating automata, and elegant connections to temporal logic.