We show how alternating automata provide decision procedures for the
equality of inductively defined Boolean functions and present
applications to reasoning about parameterized families of circuits. We
use alternating word automata to formalize families of linearly
structured circuits and alternating tree automata to formalize
families of tree structured circuits. We provide complexity bounds for
deciding the equality of function (or circuit) families and show how
our decision procedures can be implemented using BDDs. In comparison
to previous work, our approach is simpler, has better complexity
bounds, and, in the case of tree-structured families, is more general.