Backward deterministic and weak alternating $\omega$-automata

We present a direct transformation of weak alternating ω-automata into equivalent backward deterministic ω-automata and show (1) how it can be used to obtain a transformation of non-deterministic Büchi 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 lineartime μ-calculus as an intermediate step; (2) is based on the straightforward translation of linear-time temporal logic into weak alternating ω-automata.