Cut-elimination and the decidability of reachability in alternating pushdown systems

We give a new proof of the decidability of reachability in alternating pushdown systems, showing that it is a simple consequence of a cut-elimination theorem for some natural-deduction style inference systems. Then, we show how this result can be used to extend an alternating pushdown system into a complete system where for every configuration $A$, either $A$ or $\neg A$ is provable.