  We present a direct transformation of weak alternating $\omega$-automata into
equivalent backward deterministic $\omega$-automata and show (1) how it can be
used to obtain a transformation of non-deterministic B\"uchi automata into
equivalent backward deterministic automata and (2) that it yields optimal
equivalent backward deterministic automata when applied to linear-time temporal
logic formulas. (1) uses the alternation-free fragment of the linear-time
$\mu$-calculus as an intermediate step; (2) is based on the straightforward
translation of linear-time temporal logic into weak alternating
$\omega$-automata.
